2555:
such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the
Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.
5963:
4923:
85:
5975:
49:
5999:
5987:
3047:(MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.
2238:, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.
861:
A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original
2257:
While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative
2554:
One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for
2804:
on a constraint manifold; the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by
2119:
The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated
2103:
Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an
2135:
Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a
2139:
Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm.
2214:
states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.
809:
2115:
A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.
2428:
More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on
2069:
problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the
1561:
4168:
Piryonesi, S. Madeh; Nasseri, Mehran; Ramezani, Abdollah (9 July 2018). "Piryonesi, S. M., Nasseri, M., & Ramezani, A. (2018). Resource leveling in construction projects with activity splitting and resource constraints: a simulated annealing optimization".
534:
1289:
1965:
is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty
2002:
make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization
3002:
are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,
1436:
3107:
has been applied to calculate the maximal possible yields of fermentation products, and to infer gene regulatory networks from multiple microarray datasets as well as transcriptional regulatory networks from high-throughput data.
858:, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.
3672:
2547:, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the
1181:
1948:
studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving
2969:
structures, handset antennas, electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical
3967:
2605:
for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)
2645:
objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.
953:
2186:
at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.
1935:
allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex
2053:
Is a branch of infinite-dimensional optimization concerned with finding the best way to achieve some goal, such as finding a surface whose boundary is a specific curve, but with the least possible area.
2672:
Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.
2611:(alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.
736:
2394:
definite at a critical point, then the point is a local minimum; if the
Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of
2722:. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:
2379:, which meet in loss function minimization of the neural network. The positive-negative momentum estimation lets to avoid the local minimum and converges at the objective function global minimum.
3587:
1447:
6449:
1051:
3669:
4820:
3819:
De, Bishnu Prasad; Kar, R.; Mandal, D.; Ghoshal, S.P. (2014-09-27). "Optimal selection of components value for analog active filter design using simplex particle swarm optimization".
434:
874:
that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.
2270:'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
1192:
4098:
Piryonesi, Sayed Madeh; Tavakolan, Mehdi (9 January 2017). "A mathematical programming model for solving cost-safety optimization (CSO) problems in the maintenance of structures".
3103:
Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.
2104:
infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the
2077:
614:
4433:
Vo, Thuy D.; Paul Lee, W.N.; Palsson, Bernhard O. (May 2007). "Systems analysis of energy metabolism elucidates the affected respiratory chain complex in Leigh's syndrome".
2655:, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
3802:
2664:
996:
303:
3793:
3787:
4691:
2584:: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.
4815:
2800:(in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an
2945:
models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.
119:
5423:
1342:
2038:
is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or
6037:
1942:
studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
3961:
Cervantes-González, Juan C.; Rayas-Sánchez, José E.; López, Carlos A.; Camacho-Pérez, José R.; Brito-Brito, Zabdiel; Chávez-Hurtado, José L. (February 2016).
3180:
3185:
3071:
of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.
1095:
4341:
Wang, Yong; Joshi, Trupti; Zhang, Xiang-Sun; Xu, Dong; Chen, Luonan (2006-07-24). "Inferring gene regulatory networks from multiple microarray datasets".
136:
is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields:
3234:
4039:
Bandler, J.W.; Biernacki, R.M.; Shao Hua Chen; Hemmers, R.H.; Madsen, K. (1995). "Electromagnetic optimization exploiting aggressive space mapping".
6080:
4829:
3780:
2266:
in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see '
5309:
4004:
Bandler, J.W.; Biernacki, R.M.; Chen, Shao Hua; Grobelny, P.A.; Hemmers, R.H. (1994). "Space mapping technique for electromagnetic optimization".
539:
it suffices to solve only minimization problems. However, the opposite perspective of considering only maximization problems would be valid, too.
2223:
2310:
2246:
1592:
lie in the interval (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form
3027:. Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses
2942:
4684:
3458:
3409:
2817:
2416:, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as
5786:
3112:
has been used to analyze energy metabolism and has been applied to metabolic engineering and parameter estimation in biochemical pathways.
2046:
In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
6003:
3862:
Koziel, Slawomir; Bandler, John W. (January 2008). "Space
Mapping With Multiple Coarse Models for Optimization of Microwave Components".
5390:
4852:
3164:
2812:
Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the
6439:
5931:
5416:
4904:
4765:
3290:
2566:
4872:
896:
6030:
4615:
4591:
4569:
4546:
3649:
3174:
3154:
1985:
3691:"Factores relevantes para optimizar los servicios públicos de apoyo a los emprendedores y la tasa de supervivencia de las empresas"
2032:
Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.
1856:. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
804:{\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}
5467:
4983:
4677:
2883:
2631:
Bundle method of descent: An iterative method for small–medium-sized problems with locally
Lipschitz functions, particularly for
2571:
1903:
is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
5881:
3200:
2548:
2515:
2474:
2120:
relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker.
89:
6197:
6090:
6065:
5979:
5260:
4922:
2860:
2806:
2801:
2445:), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.
2198:; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative.
1866:
is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a
1748:
187:
of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of
6156:
6070:
5409:
5368:
4988:
3195:
3132:
3098:
2681:
2499:
2262:) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the
1881:
1556:{\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in ,\;y\in \mathbb {R} ,}
31:
5304:
5272:
4290:
Papoutsakis, Eleftherios Terry (February 1984). "Equations and calculations for fermentations of butyric acid bacteria".
3948:
2850:
means" with alternative uses. Modern optimization theory includes traditional optimization theory but also overlaps with
1067:
may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "
6444:
6418:
6023:
3732:
2875:
2433:, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses
2354:
2316:
2235:
2098:
1995:
843:
all of the function values are greater than or equal to the value at that element. Local maxima are defined similarly.
240:
6233:
6146:
5906:
5462:
5353:
4978:
2764:
176:
5299:
5255:
4857:
6095:
5477:
5148:
4877:
2999:
2915:
2510:
2191:
1969:
1014:
5038:
2648:
6243:
5863:
5500:
4481:"Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation"
3963:"Space mapping optimization of handset antennas considering EM effects of mobile phone components and human body"
3008:
2598:
529:{\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}
5223:
2551:(or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.
6258:
6136:
6131:
6075:
5936:
3606:
3476:"Satellite image recognition using ensemble neural networks and difference gradient positive-negative momentum"
3389:
3044:
3031:
to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and
2868:
2864:
2813:
2383:
2081:
1887:
1284:{\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}
699:
618:
257:
42:
35:
5085:
6105:
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5811:
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5715:
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2618:
2373:
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2066:
2014:
2006:
1979:
1072:
252:
141:
5991:
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4662:
3431:
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5816:
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5791:
5720:
5445:
5358:
5343:
5233:
5111:
4737:
4704:
4154:
3871:
3205:
3068:
3028:
2954:
2727:
2581:
2267:
2174:
2148:
2085:
2050:
2024:
1952:
1939:
1818:
1315:(the actual minimum value of that function is not what the problem asks for). In this case, the answer is
1302:
1295:
272:
206:
180:
1793:
6314:
6284:
6060:
5946:
5876:
5753:
5677:
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5601:
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5573:
5455:
5247:
5213:
5116:
5058:
4939:
4745:
4725:
4141:
Hegazy, Tarek (June 1999). "Optimization of
Resource Allocation and Leveling Using Genetic Algorithms".
3532:
Haggag, S.; Desokey, F.; Ramadan, M. (2017). "A cosmological inflationary model using optimal control".
3307:
3144:
3109:
2821:
2537:
2495:
2417:
2406:
2320:
2231:
2207:
2152:
1945:
1932:
1906:
1662:
223:
137:
73:
5962:
2190:
Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to
2628:. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.
590:
6405:
6375:
5926:
5806:
5801:
5725:
5626:
5294:
5121:
5033:
4476:
4227:
4107:
4048:
4013:
3920:
3541:
3487:
3190:
2774:
2642:
2621:
2465:
that may provide approximate solutions to some problems (although their iterates need not converge).
2336:
1973:
1894:
1693:
1670:
627:
414:
405:
200:
168:
4669:
3876:
3255:
2027:
is a programming paradigm wherein relations between variables are stated in the form of constraints.
6385:
6370:
6339:
6046:
5941:
5851:
5773:
5672:
5606:
5563:
5553:
5533:
5363:
5228:
5181:
5171:
5023:
5011:
4824:
4807:
4712:
3043:
Mathematical optimization is used in much modern controller design. High-level controllers such as
3024:
2983:
2919:
2910:
are also modeled using optimization theory, though the underlying mathematics relies on optimizing
2797:
2769:
2719:
2705:
2658:
2632:
2462:
2402:
2366:
2287:
2245:
method. The optima of problems with equality and/or inequality constraints can be found using the '
2242:
2156:
2144:
2124:
2062:
2018:
1962:
1922:
1753:
959:
867:
863:
855:
228:
210:
188:
184:
153:
1972:
is concerned with problems where the set of feasible solutions is discrete or can be reduced to a
1716:
and other researchers worked on the theoretical aspects of linear programming (like the theory of
979:
286:
6395:
6390:
6141:
6110:
5967:
5886:
5826:
5758:
5748:
5687:
5662:
5538:
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5098:
5067:
5053:
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4834:
4557:
4323:
4196:
4123:
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3761:
3615:
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3104:
3080:
2911:
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2742:
2732:
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2614:
2592:
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2438:
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1859:
1841:
1768:
1717:
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871:
651:
562:
418:
4750:
3907:
Tu, Sheng; Cheng, Qingsha S.; Zhang, Yifan; Bandler, John W.; Nikolova, Natalia K. (July 2013).
710:
In mathematics, conventional optimization problems are usually stated in terms of minimization.
2301:(1963) describes the continuity of an optimal solution as a function of underlying parameters.
1929:
values. This is not convex, and in general much more difficult than regular linear programming.
6349:
6289:
5682:
5667:
5611:
5558:
5106:
4784:
4611:
4587:
4565:
4542:
4510:
4502:
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4415:
4407:
4366:
4358:
4315:
4307:
4243:
3836:
3797:
3710:
3645:
3633:
3557:
3454:
3405:
3286:
3004:
2839:
2753:
2589:
Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):
2560:
2533:
2481:
2183:
1900:
1803:
1733:
1709:
307:
214:
172:
4605:
3637:
573:
where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.
6354:
6324:
6253:
6248:
6187:
6172:
5896:
5871:
5743:
5591:
5528:
5186:
5176:
5080:
4957:
4862:
4844:
4797:
4708:
4492:
4442:
4397:
4350:
4299:
4235:
4186:
4178:
4150:
4115:
4080:
4056:
4021:
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3928:
3881:
3828:
3806:
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3702:
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3495:
3446:
3397:
3325:
3159:
3121:
2715:
2638:
2608:
2602:
2527:
2458:
2279:
2227:
2211:
2071:
1849:
1828:
1798:
1778:
1758:
1738:
1713:
1658:
1431:{\displaystyle {\underset {x\in ,\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}
566:
145:
2401:
Constrained problems can often be transformed into unconstrained problems with the help of
2361:
of the objective function is zero (that is, the stationary points). More generally, a zero
2059:
theory is a generalization of the calculus of variations which introduces control policies.
850:
is at least as good as every feasible element. Generally, unless the objective function is
425:
below). Many real-world and theoretical problems may be modeled in this general framework.
6400:
6299:
6294:
6238:
6213:
6177:
5836:
5763:
5692:
5485:
5202:
4532:
3728:
3676:
3238:
3064:
2971:
2934:
2667:(SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.
2413:
2369:
2348:
2344:
2332:
2294:
2109:
2108:. The curve created plotting weight against stiffness of the best designs is known as the
2056:
2039:
1956:
1845:
1783:
1743:
1728:
851:
585:
95:
1982:
is used with random (noisy) function measurements or random inputs in the search process.
882:
Optimization problems are often expressed with special notation. Here are some examples:
569:, it is always necessary to continuously evaluate the quality of a data model by using a
4231:
4111:
4052:
4017:
3924:
3545:
3491:
1708:
schedules, which were the problems
Dantzig studied at that time.) Dantzig published the
703:. A feasible solution that minimizes (or maximizes) the objective function is called an
6329:
6319:
6274:
6218:
6115:
5914:
5841:
5548:
5190:
5075:
4962:
4896:
4867:
4497:
4480:
4215:
3601:
3582:
2938:
2926:
2691:
2544:
2485:
2387:
2328:
2263:
2259:
2195:
1813:
1808:
1773:
1681:
847:
84:
2918:
also uses optimization to explain trade patterns between nations. The optimization of
17:
6433:
6309:
6304:
6279:
6151:
5702:
5634:
5586:
5348:
5332:
4601:
4200:
4127:
3990:
3569:
3513:
Vereshchagin, A.F. (1989). "Modelling and control of motion of manipulation robots".
3280:
3169:
3149:
2979:
2975:
2962:
2958:
2930:
2907:
2903:
2867:
classify mathematical programming, optimization techniques, and related topics under
2747:
2675:
2441:. Usually, a global optimizer is much slower than advanced local optimizers (such as
2035:
1999:
1862:(LP), a type of convex programming, studies the case in which the objective function
1701:
1176:{\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}
676:
570:
244:
4402:
4385:
4354:
4327:
3893:
3848:
3401:
2230:, where the first derivative or the gradient of the objective function is zero (see
6344:
5644:
5639:
5543:
5286:
4792:
3737:"An Optimization-based Econometric Framework for the Evaluation of Monetary Policy"
3690:
2879:
2505:
2434:
2395:
2298:
1891:
1823:
1666:
1330:
256:, in which optimal arguments from a continuous set must be found. They can include
205:
Optimization problems can be divided into two categories, depending on whether the
3706:
3689:
Chaves Maza, Manuel; Fedriani, Eugenio M.; Ordaz Sanz, José Antonio (2018-07-01).
2961:
design, stray field reduction in superconducting magnetic energy storage systems,
1925:
studies linear programs in which some or all variables are constrained to take on
2127:
problems where the (partial) ordering is no longer given by the Pareto ordering.
1578:
pair (or pairs) that maximizes (or maximize) the value of the objective function
5846:
5510:
5433:
5373:
4755:
4641:
4446:
3499:
2851:
2779:
2625:
2442:
2430:
2362:
1910:
1875:
1788:
974:
314:
236:
161:
149:
4074:
1988:
studies the case when the set of feasible solutions is a subset of an infinite-
6334:
5831:
5710:
5505:
4239:
4119:
4084:
3832:
3553:
3060:
3056:
2761:: A popular heuristic for approximate minimization (without calling gradients)
2541:
2105:
1890:(SDP) is a subfield of convex optimization where the underlying variables are
1867:
1853:
1763:
1723:
Other notable researchers in mathematical optimization include the following:
41:"Mathematical programming" redirects here. For the peer-reviewed journal, see
4506:
4454:
4411:
4362:
4311:
4247:
3933:
3908:
3885:
3840:
3714:
3561:
3059:
parameter estimation problems. Given a set of geophysical measurements, e.g.
1884:(SOCP) is a convex program, and includes certain types of quadratic programs.
3909:"Space Mapping Optimization of Handset Antennas Exploiting Thin-Wire Models"
3474:
Abdulkadirov, R.; Lyakhov, P.; Bergerman, M.; Reznikov, D. (February 2024).
2966:
2835:
2711:
2454:
2283:
1989:
1914:
1705:
157:
48:
4462:
4419:
4370:
4319:
4182:
2925:
Since the 1970s, economists have modeled dynamic decisions over time using
2886:, are economic optimization problems. Insofar as they behave consistently,
2846:
science as the "study of human behavior as a relationship between ends and
5401:
4536:
4514:
4384:
Wang, Rui-Sheng; Wang, Yong; Zhang, Xiang-Sun; Chen, Luonan (2007-09-22).
4303:
3736:
2820:, which, while useful in many problems, has in particular been applied to
2409:
can also provide approximate solutions to difficult constrained problems.
2282:
describes how the value of an optimal solution changes when an underlying
1955:
studies the case in which some of the constraints or parameters depend on
92:. Simplex vertices are ordered by their values, with 1 having the lowest (
5735:
5654:
5581:
4775:
4649:
4386:"Inferring transcriptional regulatory networks from high-throughput data"
3450:
3306:
Du, D. Z.; Pardalos, P. M.; Wu, W. (2008). "History of
Optimization". In
2887:
2358:
2324:
2123:
Multi-objective optimization problems have been generalized further into
1909:
is a technique whereby objective and inequality constraints expressed as
1871:
1068:
2937:. A crucial distinction is between deterministic and stochastic models.
6192:
5520:
5095:
3981:
3962:
3765:
3619:
3012:
2891:
1926:
1620:
1084:
543:
232:
6015:
4191:
4060:
4025:
2595:
methods: Algorithms which update a single coordinate in each iteration
2461:
that converge to a solution (on some specified class of problems), or
6182:
3432:"A brief history of linear and mixed-integer programming computation"
2847:
2437:. Both line searches and trust regions are used in modern methods of
2353:
For unconstrained problems with twice-differentiable functions, some
581:
558:
548:
3968:
International
Journal of RF and Microwave Computer-Aided Engineering
3756:
2809:, which can also be viewed as a QP (quadratic programming) problem.
1897:. It is a generalization of linear and convex quadratic programming.
846:
While a local minimum is at least as good as any nearby elements, a
3949:“Space mapping outpaces EM optimization in handset-antenna design,”
3349:
2651:
for approximate minimization of specially structured problems with
854:
in a minimization problem, there may be several local minima. In a
144:. Optimization problems arise in all quantitative disciplines from
4632:
160:, and the development of solution methods has been of interest in
83:
47:
4265:
4216:"Modeling, Simulation, and Optimization of Traffic Flow Networks"
4079:. 2013 iREP Symposium on Bulk Power System Dynamics and Control.
2504:
Variants of the simplex algorithm that are especially suited for
264:
An optimization problem can be represented in the following way:
6223:
3604:(1969). "An Economic Interpretation of Optimal Control Theory".
2895:
30:"Optimization" and "Optimum" redirect here. For other uses, see
6019:
5405:
5330:
5146:
5009:
4937:
4723:
4673:
3023:
Another field that uses optimization techniques extensively is
2661:: Iterative methods for medium-large problems (e.g. N<1000).
2559:
Methods that evaluate
Hessians (or approximate Hessians, using
948:{\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}
2578:
problems. Some versions can handle large-dimensional problems.
2147:
problems, where multiple local extrema may be present include
221:
An optimization problem with discrete variables is known as a
2842:
that an influential definition relatedly describes economics
4921:
2922:
is an example of multi-objective optimization in economics.
2827:
This approach may be applied in cosmology and astrophysics.
2241:
Optima of equality-constrained problems can be found by the
1665:
found calculus-based formulae for identifying optima, while
3588:
2684:
methods, which have better convergence properties than the
2226:
states that optima of unconstrained problems are found at
1673:
proposed iterative methods for moving towards an optimum.
542:
Problems formulated using this technique in the fields of
27:
Study of mathematical algorithms for optimization problems
3821:
International Journal of Machine Learning and Cybernetics
2816:, and another recent and growing subset of this field is
4663:"Mathematical Optimization: Finding Minima of Functions"
2994:
Optimization has been widely used in civil engineering.
2286:
changes. The process of computing this change is called
2953:
Some common applications of optimization techniques in
3279:
Martins, Joaquim R. R. A.; Ning, Andrew (2021-10-01).
3664:
A.G. Malliaris (2008). "stochastic optimal control,"
2382:
Further, critical points can be classified using the
2078:
Mathematical programming with equilibrium constraints
1684:, although much of the theory had been introduced by
1450:
1345:
1195:
1098:
1056:
asks for the maximum value of the objective function
1017:
982:
899:
739:
723:
is defined as an element for which there exists some
593:
437:
289:
98:
4262:"New force on the political scene: the Seophonisten"
4041:
IEEE Transactions on Microwave Theory and Techniques
4006:
IEEE Transactions on Microwave Theory and Techniques
1308:
that minimizes (or minimize) the objective function
998:. The minimum value in this case is 1, occurring at
183:
values from within an allowed set and computing the
6363:
6267:
6206:
6165:
6124:
6053:
5905:
5862:
5772:
5734:
5701:
5653:
5625:
5572:
5519:
5476:
5285:
5246:
5212:
5201:
5159:
5094:
5066:
5052:
5022:
4971:
4950:
4895:
4843:
4806:
4783:
4774:
4736:
4285:
4283:
4076:
Convex relaxation of optimal power flow: A tutorial
4143:Journal of Construction Engineering and Management
3085:Nonlinear optimization methods are widely used in
2982:in 1993. Optimization techniques are also used in
2665:Simultaneous perturbation stochastic approximation
2494:Extensions of the simplex algorithm, designed for
2365:certifies that a local minimum has been found for
1555:
1430:
1329:is infeasible, that is, it does not belong to the
1283:
1175:
1045:
990:
947:
803:
608:
528:
297:
250:A problem with continuous variables is known as a
113:
6450:Mathematical and quantitative methods (economics)
2635:problems (similar to conjugate gradient methods).
2009:studies the case in which the objective function
622:, equalities or inequalities that the members of
3396:, London: Palgrave Macmillan UK, pp. 1–12,
2258:or the matrix of second derivatives (called the
1844:studies the case when the objective function is
1019:
901:
3515:Soviet Journal of Computer and Systems Sciences
2574:: A Newton-based method for small-medium scale
2457:that terminate in a finite number of steps, or
3864:IEEE Microwave and Wireless Components Letters
3055:Optimization techniques are regularly used in
2641:: An iterative method for small problems with
6031:
5417:
4685:
3913:IEEE Transactions on Antennas and Propagation
3373:Hartmann, Alexander K; Rieger, Heiko (2002).
2943:dynamic stochastic general equilibrium (DSGE)
2357:can be found by finding the points where the
2163:Classification of critical points and extrema
1294:This represents the value (or values) of the
1046:{\displaystyle \max _{x\in \mathbb {R} }\;2x}
8:
4584:A First Course in Combinatorial Optimization
3644:. Harvard University Press. pp. 57–91.
2838:is closely enough linked to optimization of
1680:" for certain optimization cases was due to
3788:numerical optimization methods in economics
3350:"Open Journal of Mathematical Optimization"
3177:(formerly Mathematical Programming Society)
1704:military to refer to proposed training and
6038:
6024:
6016:
5424:
5410:
5402:
5327:
5243:
5209:
5156:
5143:
5063:
5019:
5006:
4947:
4934:
4780:
4733:
4720:
4692:
4678:
4670:
2902:. Also, agents are often modeled as being
1538:
1510:
1504:
1488:
1481:
1412:
1397:
1250:
1244:
1224:
1153:
1036:
918:
760:
754:
692:(maximization), or, in certain fields, an
4633:"Decision Tree for Optimization Software"
4541:. Cambridge: Cambridge University Press.
4496:
4401:
4190:
3980:
3932:
3875:
3803:Arrow–Debreu model of general equilibrium
3755:
2649:Conditional gradient method (Frank–Wolfe)
1917:can be transformed into a convex program.
1852:(maximization) and the constraint set is
1546:
1545:
1505:
1463:
1453:
1451:
1449:
1405:
1404:
1358:
1348:
1346:
1344:
1245:
1229:
1208:
1198:
1196:
1194:
1158:
1111:
1101:
1099:
1097:
1030:
1029:
1022:
1016:
984:
983:
981:
928:
912:
911:
904:
898:
800:
780:
775:
766:
755:
743:
738:
600:
596:
595:
592:
515:
494:
489:
468:
450:
445:
436:
291:
290:
288:
97:
80:) = (0, 0, 4) is indicated by a blue dot.
6081:Earth systems engineering and management
4926:Optimization computes maxima and minima.
3784:(2008), 2nd Edition with Abstract links:
3781:The New Palgrave Dictionary of Economics
3666:The New Palgrave Dictionary of Economics
3394:The New Palgrave Dictionary of Economics
2234:). More generally, they may be found at
553:, speaking of the value of the function
4155:10.1061/(ASCE)0733-9364(1999)125:3(167)
3314:. Boston: Springer. pp. 1538–1542.
3256:"Mathematical Programming: An Overview"
3223:
2453:To solve problems, researchers may use
886:Minimum and maximum value of a function
3232:The Nature of Mathematical Programming
3165:Important publications in optimization
3015:management and schedule optimization.
2898:are usually assumed to maximize their
2686:Nelder–Mead heuristic (with simplices)
836:that is to say, on some region around
5122:Principal pivoting algorithm of Lemke
4171:Canadian Journal of Civil Engineering
2978:methodologies since the discovery of
2818:multidisciplinary design optimization
2449:Computational optimization techniques
2182:, is just the problem of finding any
7:
5986:
4220:SIAM Journal on Scientific Computing
3181:Mathematical optimization algorithms
3122:Machine learning § Optimization
2914:rather than on static optimization.
2906:, thereby preferring to avoid risk.
2274:Sensitivity and continuity of optima
2253:Sufficient conditions for optimality
1992:space, such as a space of functions.
5998:
3951:microwaves&rf, August 30, 2013.
2473:For a more comprehensive list, see
2219:Necessary conditions for optimality
2194:the feasibility conditions using a
4766:Successive parabolic interpolation
4637:Links to optimization source codes
4610:(2nd ed.). Berlin: Springer.
4214:Herty, M.; Klar, A. (2003-01-01).
3375:Optimization algorithms in physics
3186:Mathematical optimization software
2131:Multi-modal or global optimization
1692:in this context does not refer to
1470:
1467:
1464:
1460:
1457:
1454:
1365:
1362:
1359:
1355:
1352:
1349:
1263:
1215:
1212:
1209:
1205:
1202:
1199:
1135:
1118:
1115:
1112:
1108:
1105:
1102:
740:
557:as representing the energy of the
417:, but still in use for example in
25:
6116:Sociocultural Systems Engineering
5086:Projective algorithm of Karmarkar
4435:Molecular Genetics and Metabolism
4100:KSCE Journal of Civil Engineering
3243:Mathematical Programming Glossary
3175:Mathematical Optimization Society
3155:Deterministic global optimization
2540:differ according to whether they
2412:When the objective function is a
2367:minimization problems with convex
2080:is where the constraints include
1986:Infinite-dimensional optimization
1588:, with the added constraint that
1089:Consider the following notation:
890:Consider the following notation:
167:In the more general approach, an
5997:
5985:
5974:
5973:
5961:
5081:Ellipsoid algorithm of Khachiyan
4984:Sequential quadratic programming
4821:Broyden–Fletcher–Goldfarb–Shanno
4498:10.1093/bioinformatics/14.10.869
4292:Biotechnology and Bioengineering
3441:. Documenta Mathematica Series.
3063:, it is common to solve for the
2884:expenditure minimization problem
2759:Nelder–Mead simplicial heuristic
2617:: An iterative method for large
2572:Sequential quadratic programming
776:
767:
744:
609:{\displaystyle \mathbb {R} ^{n}}
516:
490:
469:
446:
413:(a term not directly related to
411:mathematical programming problem
403:Such a formulation is called an
5882:Computational complexity theory
4654:Course from Stanford University
4650:"EE364a: Convex Optimization I"
4535:; Vandenberghe, Lieven (2004).
3402:10.1057/978-1-349-95121-5_659-2
3282:Engineering Design Optimization
3201:Test functions for optimization
2710:Besides (finitely terminating)
2516:Quantum optimization algorithms
2475:List of optimization algorithms
2439:non-differentiable optimization
6198:Systems development life cycle
6091:Enterprise systems engineering
6066:Biological systems engineering
5039:Reduced gradient (Frank–Wolfe)
4586:. Cambridge University Press.
3480:Chaos, Solitons & Fractals
3285:. Cambridge University Press.
2890:are assumed to maximize their
2861:Journal of Economic Literature
2807:linear complementarity problem
2802:ordinary differential equation
1634:are sometimes also written as
1532:
1517:
1391:
1376:
1275:
1257:
1147:
1129:
787:
762:
616:, often specified by a set of
546:may refer to the technique as
520:
512:
500:
485:
476:
473:
465:
456:
441:
367:("minimization") or such that
108:
102:
88:Nelder-Mead minimum search of
1:
6157:System of systems engineering
6071:Cognitive systems engineering
5369:Spiral optimization algorithm
4989:Successive linear programming
4604:; Wright, Stephen J. (2006).
4403:10.1093/bioinformatics/btm465
4355:10.1093/bioinformatics/btl396
3707:10.15446/innovar.v28n69.71693
3196:Simulation-based optimization
3133:List of optimization software
3099:Computational systems biology
3093:Computational systems biology
2500:linear-fractional programming
2311:Karush–Kuhn–Tucker conditions
2247:Karush–Kuhn–Tucker conditions
2065:is the approach to solve the
2013:is constant (this is used in
1882:Second-order cone programming
428:Since the following is valid
32:Optimization (disambiguation)
5107:Simplex algorithm of Dantzig
4979:Augmented Lagrangian methods
3642:Dynamic Macroeconomic Theory
3312:Encyclopedia of Optimization
3245:, INFORMS Computing Society.
2876:utility maximization problem
2317:Critical point (mathematics)
2099:Multi-objective optimization
2093:Multi-objective optimization
1913:and equality constraints as
1696:, but comes from the use of
991:{\displaystyle \mathbb {R} }
422:
298:{\displaystyle \mathbb {R} }
52:Graph of a surface given by
6234:Quality function deployment
6147:Verification and validation
4447:10.1016/j.ymgme.2007.01.012
3500:10.1016/j.chaos.2023.114432
3326:"Mathematical optimization"
2765:Particle swarm optimization
179:by systematically choosing
6466:
6096:Health systems engineering
5932:Films about mathematicians
4564:. London: Academic Press.
3744:NBER Macroeconomics Annual
3388:Erwin Diewert, W. (2017),
3130:
3119:
3096:
3078:
3000:transportation engineering
2916:International trade theory
2854:and the study of economic
2703:
2599:Conjugate gradient methods
2536:used to solve problems of
2525:
2472:
2314:
2308:
2096:
1970:Combinatorial optimization
1082:
962:of the objective function
198:
40:
29:
6440:Mathematical optimization
6414:
6244:Systems Modeling Language
5955:
5501:Philosophy of mathematics
5441:
5386:
5339:
5326:
5310:Push–relabel maximum flow
5155:
5142:
5112:Revised simplex algorithm
5018:
5005:
4946:
4933:
4919:
4732:
4719:
4556:Gill, P. E.; Murray, W.;
4240:10.1137/S106482750241459X
4120:10.1007/s12205-017-0531-z
4085:10.1109/IREP.2013.6629391
3833:10.1007/s13042-014-0299-0
3554:10.1134/S0202289317030069
3534:Gravitation and Cosmology
3009:water resource allocation
958:This denotes the minimum
126:Mathematical optimization
6259:Work breakdown structure
6137:Functional specification
6132:Requirements engineering
6076:Configuration management
5937:Recreational mathematics
4835:Symmetric rank-one (SR1)
4816:Berndt–Hall–Hall–Hausman
3934:10.1109/TAP.2013.2254695
3886:10.1109/LMWC.2007.911969
3607:American Economic Review
3430:Bixby, Robert E (2012).
3354:ojmo.centre-mersenne.org
3045:model predictive control
2814:engineering optimization
2688:, which is listed below.
2549:computational complexity
2511:Combinatorial algorithms
2305:Calculus of optimization
2224:One of Fermat's theorems
2082:variational inequalities
1888:Semidefinite programming
1720:) around the same time.
1008:Similarly, the notation
645:, while the elements of
260:and multimodal problems.
173:maximizing or minimizing
134:mathematical programming
43:Mathematical Programming
36:Optimum (disambiguation)
6106:Reliability engineering
6101:Performance engineering
5822:Mathematical statistics
5812:Mathematical psychology
5782:Engineering mathematics
5716:Algebraic number theory
5359:Parallel metaheuristics
5167:Approximation algorithm
4878:Powell's dog leg method
4830:Davidon–Fletcher–Powell
4726:Unconstrained nonlinear
3310:; Pardalos, P. (eds.).
3211:Vehicle routing problem
3087:conformational analysis
3033:stochastic optimization
2996:Construction management
2929:. For example, dynamic
2874:In microeconomics, the
2738:Evolutionary algorithms
2469:Optimization algorithms
2149:evolutionary algorithms
2136:multi-modal optimizer.
2067:stochastic optimization
2015:artificial intelligence
2007:Constraint satisfaction
1980:Stochastic optimization
1648:argument of the maximum
1644:argument of the minimum
1079:Optimal input arguments
666:is variously called an
253:continuous optimization
142:continuous optimization
128:(alternatively spelled
6381:Industrial engineering
6086:Electrical engineering
5968:Mathematics portal
5817:Mathematical sociology
5797:Mathematical economics
5792:Mathematical chemistry
5721:Analytic number theory
5602:Differential equations
5344:Evolutionary algorithm
4927:
4607:Numerical Optimization
4562:Practical Optimization
4183:10.1139/cjce-2017-0670
3206:Calculus of variations
3029:stochastic programming
2955:electrical engineering
2949:Electrical engineering
2728:Differential evolution
2582:Interior point methods
2418:interior-point methods
2268:Second derivative test
2175:satisfiability problem
2051:Calculus of variations
2025:Constraint programming
1953:Stochastic programming
1940:Fractional programming
1819:R. Tyrrell Rockafellar
1682:George B. Dantzig
1557:
1432:
1285:
1177:
1047:
992:
949:
805:
610:
530:
299:
122:
115:
81:
18:Optimization algorithm
6315:Arthur David Hall III
6285:Benjamin S. Blanchard
6061:Aerospace engineering
5947:Mathematics education
5877:Theory of computation
5597:Hypercomplex analysis
5117:Criss-cross algorithm
4940:Constrained nonlinear
4925:
4746:Golden-section search
4642:"Global optimization"
4304:10.1002/bit.260260210
3439:Documenta Mathematica
3145:Brachistochrone curve
3110:Nonlinear programming
2935:labor-market behavior
2831:Economics and finance
2822:aerospace engineering
2626:generalized gradients
2538:nonlinear programming
2496:quadratic programming
2407:Lagrangian relaxation
2321:Differential calculus
2232:first derivative test
2208:extreme value theorem
2153:Bayesian optimization
2143:Common approaches to
1946:Nonlinear programming
1933:Quadratic programming
1907:Geometric programming
1558:
1433:
1286:
1178:
1048:
993:
950:
806:
626:have to satisfy. The
611:
531:
300:
243:must be found from a
224:discrete optimization
195:Optimization problems
138:discrete optimization
116:
90:Simionescu's function
87:
51:
6406:Software engineering
6376:Computer engineering
5927:Informal mathematics
5807:Mathematical physics
5802:Mathematical finance
5787:Mathematical biology
5726:Diophantine geometry
5034:Cutting-plane method
3790:" by Karl Schmedders
3191:Process optimization
2912:stochastic processes
2775:Stochastic tunneling
2659:Quasi-Newton methods
2506:network optimization
2403:Lagrange multipliers
2390:: If the Hessian is
2341:Rademacher's theorem
2337:Lipschitz continuity
1694:computer programming
1448:
1343:
1193:
1096:
1015:
980:
897:
737:
591:
435:
415:computer programming
406:optimization problem
287:
258:constrained problems
201:Optimization problem
169:optimization problem
114:{\displaystyle f(x)}
96:
6445:Operations research
6386:Operations research
6371:Control engineering
6340:Joseph Francis Shea
6047:Systems engineering
5942:Mathematics and art
5852:Operations research
5607:Functional analysis
5364:Simulated annealing
5182:Integer programming
5172:Dynamic programming
5012:Convex optimization
4873:Levenberg–Marquardt
4538:Convex Optimization
4479:; Kell, D. (1998).
4268:on 18 December 2014
4232:2003SJSC...25.1066H
4112:2017KSJCE..21.2226P
4053:1995ITMTT..43.2874B
4018:1994ITMTT..42.2536B
3925:2013ITAP...61.3797T
3591:, Macmillan, p. 16.
3546:2017GrCo...23..236H
3492:2024CSF...17914432A
3065:physical properties
3039:Control engineering
3025:operations research
3019:Operations research
2984:power-flow analysis
2798:rigid body dynamics
2770:Simulated annealing
2750:with random restart
2706:Heuristic algorithm
2633:convex minimization
2622:Lipschitz functions
2615:Subgradient methods
2377:Lipschitz functions
2288:comparative statics
2243:Lagrange multiplier
2180:feasibility problem
2168:Feasibility problem
2157:simulated annealing
2145:global optimization
2125:vector optimization
2063:Dynamic programming
2019:automated reasoning
1963:Robust optimization
1923:Integer programming
868:applied mathematics
864:Global optimization
652:candidate solutions
189:applied mathematics
154:operations research
72:²) + 4. The global
6396:Quality management
6391:Project management
6219:Function modelling
6142:System integration
6111:Safety engineering
5887:Numerical analysis
5496:Mathematical logic
5491:Information theory
5044:Subgradient method
4928:
4853:Conjugate gradient
4761:Nelder–Mead method
3982:10.1002/mmce.20945
3794:convex programming
3675:2017-10-18 at the
3634:Sargent, Thomas J.
3237:2014-03-05 at the
3105:Linear programming
3081:Molecular modeling
3075:Molecular modeling
3069:geometrical shapes
3061:seismic recordings
2933:are used to study
2743:Genetic algorithms
2733:Dynamic relaxation
2653:linear constraints
2593:Coordinate descent
2561:finite differences
2490:linear programming
2424:Global convergence
2178:, also called the
2017:, particularly in
1860:Linear programming
1848:(minimization) or
1842:Convex programming
1769:Narendra Karmarkar
1712:in 1947, and also
1686:Leonid Kantorovich
1678:linear programming
1553:
1486:
1428:
1410:
1281:
1222:
1173:
1151:
1043:
1035:
988:
945:
917:
872:numerical analysis
801:
672:criterion function
668:objective function
657:feasible solutions
606:
526:
419:linear programming
295:
123:
111:
82:
6427:
6426:
6350:Manuela M. Veloso
6290:Wernher von Braun
6013:
6012:
5612:Harmonic analysis
5399:
5398:
5382:
5381:
5322:
5321:
5318:
5317:
5281:
5280:
5242:
5241:
5138:
5137:
5134:
5133:
5130:
5129:
5001:
5000:
4997:
4996:
4917:
4916:
4913:
4912:
4891:
4890:
4661:Varoquaux, Gaël.
4396:(22): 3056–3064.
4349:(19): 2413–2420.
4061:10.1109/22.475649
4047:(12): 2874–2882.
4026:10.1109/22.339794
4012:(12): 2536–2544.
3798:Lawrence E. Blume
3733:Woodford, Michael
3460:978-3-936609-58-5
3411:978-1-349-95121-5
3005:resource leveling
2990:Civil engineering
2754:Memetic algorithm
2716:iterative methods
2714:and (convergent)
2603:Iterative methods
2534:iterative methods
2522:Iterative methods
2482:Simplex algorithm
2459:iterative methods
2228:stationary points
2184:feasible solution
2086:complementarities
1901:Conic programming
1804:Arkadi Nemirovski
1734:Dimitri Bertsekas
1710:Simplex algorithm
1508:
1452:
1347:
1248:
1197:
1100:
1018:
900:
866:is the branch of
758:
399:("maximization").
16:(Redirected from
6457:
6355:John N. Warfield
6325:Robert E. Machol
6254:Systems modeling
6249:Systems analysis
6188:System lifecycle
6173:Business process
6040:
6033:
6026:
6017:
6001:
6000:
5989:
5988:
5977:
5976:
5966:
5965:
5897:Computer algebra
5872:Computer science
5592:Complex analysis
5426:
5419:
5412:
5403:
5328:
5244:
5210:
5187:Branch and bound
5177:Greedy algorithm
5157:
5144:
5064:
5020:
5007:
4948:
4935:
4883:Truncated Newton
4798:Wolfe conditions
4781:
4734:
4721:
4694:
4687:
4680:
4671:
4666:
4657:
4645:
4636:
4621:
4597:
4575:
4552:
4533:Boyd, Stephen P.
4519:
4518:
4500:
4473:
4467:
4466:
4430:
4424:
4423:
4405:
4381:
4375:
4374:
4338:
4332:
4331:
4287:
4278:
4277:
4275:
4273:
4264:. Archived from
4258:
4252:
4251:
4226:(3): 1066–1087.
4211:
4205:
4204:
4194:
4165:
4159:
4158:
4138:
4132:
4131:
4106:(6): 2226–2234.
4095:
4089:
4088:
4071:
4065:
4064:
4036:
4030:
4029:
4001:
3995:
3994:
3984:
3958:
3952:
3945:
3939:
3938:
3936:
3919:(7): 3797–3807.
3904:
3898:
3897:
3879:
3859:
3853:
3852:
3816:
3810:
3807:John Geanakoplos
3776:
3770:
3769:
3759:
3741:
3729:Rotemberg, Julio
3725:
3719:
3718:
3686:
3680:
3662:
3656:
3655:
3630:
3624:
3623:
3598:
3592:
3585:(1935, 2nd ed.)
3580:
3574:
3573:
3529:
3523:
3522:
3510:
3504:
3503:
3471:
3465:
3464:
3451:10.4171/dms/6/16
3436:
3427:
3421:
3420:
3419:
3418:
3390:"Cost Functions"
3385:
3379:
3378:
3370:
3364:
3363:
3361:
3360:
3346:
3340:
3339:
3337:
3336:
3322:
3316:
3315:
3303:
3297:
3296:
3276:
3270:
3269:
3267:
3265:
3260:
3252:
3246:
3228:
3160:Goal programming
3116:Machine learning
2639:Ellipsoid method
2609:Gradient descent
2528:Iterative method
2280:envelope theorem
2264:bordered Hessian
2212:Karl Weierstrass
2072:Bellman equation
1957:random variables
1799:David Luenberger
1779:Leonid Khachiyan
1759:Ronald A. Howard
1754:Martin Grötschel
1739:Michel Bierlaire
1714:John von Neumann
1642:, and stand for
1641:
1637:
1633:
1629:
1619:ranges over all
1618:
1614:
1612:
1602:
1600:
1591:
1587:
1577:
1562:
1560:
1559:
1554:
1549:
1509:
1506:
1487:
1485:
1473:
1441:or equivalently
1437:
1435:
1434:
1429:
1411:
1409:
1408:
1368:
1328:
1321:
1314:
1307:
1300:
1290:
1288:
1287:
1282:
1249:
1246:
1234:
1233:
1223:
1218:
1186:or equivalently
1182:
1180:
1179:
1174:
1163:
1162:
1152:
1150:
1121:
1066:
1062:
1052:
1050:
1049:
1044:
1034:
1033:
1004:
997:
995:
994:
989:
987:
973:from the set of
972:
969:, when choosing
968:
954:
952:
951:
946:
944:
940:
933:
932:
916:
915:
842:
832:
810:
808:
807:
802:
790:
786:
785:
784:
779:
770:
759:
756:
747:
729:
722:
705:optimal solution
690:fitness function
686:utility function
684:(minimization),
665:
648:
636:
632:
625:
615:
613:
612:
607:
605:
604:
599:
579:
567:machine learning
556:
535:
533:
532:
527:
519:
499:
498:
493:
472:
455:
454:
449:
398:
388:
366:
356:
334:
312:
305:
304:
302:
301:
296:
294:
146:computer science
120:
118:
117:
112:
21:
6465:
6464:
6460:
6459:
6458:
6456:
6455:
6454:
6430:
6429:
6428:
6423:
6410:
6401:Risk management
6359:
6300:Harold Chestnut
6295:Kathleen Carley
6263:
6239:System dynamics
6214:Decision-making
6202:
6178:Fault tolerance
6161:
6120:
6049:
6044:
6014:
6009:
5960:
5951:
5901:
5858:
5837:Systems science
5768:
5764:Homotopy theory
5730:
5697:
5649:
5621:
5568:
5515:
5486:Category theory
5472:
5437:
5430:
5400:
5395:
5378:
5335:
5314:
5277:
5238:
5215:
5204:
5197:
5151:
5126:
5090:
5057:
5048:
5025:
5014:
4993:
4967:
4963:Penalty methods
4958:Barrier methods
4942:
4929:
4909:
4905:Newton's method
4887:
4839:
4802:
4770:
4751:Powell's method
4728:
4715:
4698:
4660:
4648:
4640:
4631:
4628:
4618:
4600:
4594:
4578:
4572:
4555:
4549:
4531:
4528:
4526:Further reading
4523:
4522:
4491:(10): 869–883.
4475:
4474:
4470:
4432:
4431:
4427:
4383:
4382:
4378:
4340:
4339:
4335:
4289:
4288:
4281:
4271:
4269:
4260:
4259:
4255:
4213:
4212:
4208:
4167:
4166:
4162:
4140:
4139:
4135:
4097:
4096:
4092:
4073:
4072:
4068:
4038:
4037:
4033:
4003:
4002:
3998:
3960:
3959:
3955:
3946:
3942:
3906:
3905:
3901:
3877:10.1.1.147.5407
3861:
3860:
3856:
3818:
3817:
3813:
3800:
3791:
3785:
3777:
3773:
3757:10.2307/3585236
3739:
3727:
3726:
3722:
3688:
3687:
3683:
3677:Wayback Machine
3668:, 2nd Edition.
3663:
3659:
3652:
3632:
3631:
3627:
3602:Dorfman, Robert
3600:
3599:
3595:
3581:
3577:
3531:
3530:
3526:
3512:
3511:
3507:
3473:
3472:
3468:
3461:
3434:
3429:
3428:
3424:
3416:
3414:
3412:
3387:
3386:
3382:
3372:
3371:
3367:
3358:
3356:
3348:
3347:
3343:
3334:
3332:
3324:
3323:
3319:
3305:
3304:
3300:
3293:
3278:
3277:
3273:
3263:
3261:
3258:
3254:
3253:
3249:
3239:Wayback Machine
3229:
3225:
3220:
3215:
3140:
3135:
3129:
3124:
3118:
3101:
3095:
3083:
3077:
3053:
3041:
3021:
2992:
2972:surrogate model
2951:
2939:Macroeconomists
2833:
2794:
2789:
2784:
2708:
2702:
2567:Newton's method
2530:
2524:
2488:, designed for
2478:
2471:
2451:
2426:
2414:convex function
2355:critical points
2351:
2349:Convex analysis
2345:Convex function
2333:Definite matrix
2313:
2307:
2295:maximum theorem
2276:
2255:
2236:critical points
2221:
2204:
2170:
2165:
2133:
2110:Pareto frontier
2101:
2095:
2057:Optimal control
2040:surrogate model
1838:
1836:Major subfields
1833:
1784:Bernard Koopman
1744:Stephen P. Boyd
1729:Richard Bellman
1656:
1639:
1635:
1631:
1627:
1616:
1610:
1604:
1598:
1593:
1589:
1579:
1567:
1566:represents the
1474:
1446:
1445:
1369:
1341:
1340:
1323:
1316:
1309:
1305:
1298:
1225:
1191:
1190:
1154:
1122:
1094:
1093:
1087:
1081:
1064:
1057:
1013:
1012:
999:
978:
977:
970:
963:
924:
923:
919:
895:
894:
888:
880:
837:
815:
814:the expression
774:
765:
761:
735:
734:
724:
717:
694:energy function
663:
646:
634:
630:
623:
594:
589:
588:
586:Euclidean space
577:
554:
488:
444:
433:
432:
390:
378:
368:
358:
346:
336:
329:
323:
310:
285:
284:
275:
203:
197:
164:for centuries.
94:
93:
46:
39:
28:
23:
22:
15:
12:
11:
5:
6463:
6461:
6453:
6452:
6447:
6442:
6432:
6431:
6425:
6424:
6422:
6421:
6415:
6412:
6411:
6409:
6408:
6403:
6398:
6393:
6388:
6383:
6378:
6373:
6367:
6365:
6364:Related fields
6361:
6360:
6358:
6357:
6352:
6347:
6342:
6337:
6332:
6330:Radhika Nagpal
6327:
6322:
6320:Derek Hitchins
6317:
6312:
6307:
6302:
6297:
6292:
6287:
6282:
6277:
6275:James S. Albus
6271:
6269:
6265:
6264:
6262:
6261:
6256:
6251:
6246:
6241:
6236:
6231:
6226:
6221:
6216:
6210:
6208:
6204:
6203:
6201:
6200:
6195:
6190:
6185:
6180:
6175:
6169:
6167:
6163:
6162:
6160:
6159:
6154:
6149:
6144:
6139:
6134:
6128:
6126:
6122:
6121:
6119:
6118:
6113:
6108:
6103:
6098:
6093:
6088:
6083:
6078:
6073:
6068:
6063:
6057:
6055:
6051:
6050:
6045:
6043:
6042:
6035:
6028:
6020:
6011:
6010:
6008:
6007:
5995:
5983:
5971:
5956:
5953:
5952:
5950:
5949:
5944:
5939:
5934:
5929:
5924:
5923:
5922:
5915:Mathematicians
5911:
5909:
5907:Related topics
5903:
5902:
5900:
5899:
5894:
5889:
5884:
5879:
5874:
5868:
5866:
5860:
5859:
5857:
5856:
5855:
5854:
5849:
5844:
5842:Control theory
5834:
5829:
5824:
5819:
5814:
5809:
5804:
5799:
5794:
5789:
5784:
5778:
5776:
5770:
5769:
5767:
5766:
5761:
5756:
5751:
5746:
5740:
5738:
5732:
5731:
5729:
5728:
5723:
5718:
5713:
5707:
5705:
5699:
5698:
5696:
5695:
5690:
5685:
5680:
5675:
5670:
5665:
5659:
5657:
5651:
5650:
5648:
5647:
5642:
5637:
5631:
5629:
5623:
5622:
5620:
5619:
5617:Measure theory
5614:
5609:
5604:
5599:
5594:
5589:
5584:
5578:
5576:
5570:
5569:
5567:
5566:
5561:
5556:
5551:
5546:
5541:
5536:
5531:
5525:
5523:
5517:
5516:
5514:
5513:
5508:
5503:
5498:
5493:
5488:
5482:
5480:
5474:
5473:
5471:
5470:
5465:
5460:
5459:
5458:
5453:
5442:
5439:
5438:
5431:
5429:
5428:
5421:
5414:
5406:
5397:
5396:
5394:
5393:
5387:
5384:
5383:
5380:
5379:
5377:
5376:
5371:
5366:
5361:
5356:
5351:
5346:
5340:
5337:
5336:
5333:Metaheuristics
5331:
5324:
5323:
5320:
5319:
5316:
5315:
5313:
5312:
5307:
5305:Ford–Fulkerson
5302:
5297:
5291:
5289:
5283:
5282:
5279:
5278:
5276:
5275:
5273:Floyd–Warshall
5270:
5265:
5264:
5263:
5252:
5250:
5240:
5239:
5237:
5236:
5231:
5226:
5220:
5218:
5207:
5199:
5198:
5196:
5195:
5194:
5193:
5179:
5174:
5169:
5163:
5161:
5153:
5152:
5147:
5140:
5139:
5136:
5135:
5132:
5131:
5128:
5127:
5125:
5124:
5119:
5114:
5109:
5103:
5101:
5092:
5091:
5089:
5088:
5083:
5078:
5076:Affine scaling
5072:
5070:
5068:Interior point
5061:
5050:
5049:
5047:
5046:
5041:
5036:
5030:
5028:
5016:
5015:
5010:
5003:
5002:
4999:
4998:
4995:
4994:
4992:
4991:
4986:
4981:
4975:
4973:
4972:Differentiable
4969:
4968:
4966:
4965:
4960:
4954:
4952:
4944:
4943:
4938:
4931:
4930:
4920:
4918:
4915:
4914:
4911:
4910:
4908:
4907:
4901:
4899:
4893:
4892:
4889:
4888:
4886:
4885:
4880:
4875:
4870:
4865:
4860:
4855:
4849:
4847:
4841:
4840:
4838:
4837:
4832:
4827:
4818:
4812:
4810:
4804:
4803:
4801:
4800:
4795:
4789:
4787:
4778:
4772:
4771:
4769:
4768:
4763:
4758:
4753:
4748:
4742:
4740:
4730:
4729:
4724:
4717:
4716:
4699:
4697:
4696:
4689:
4682:
4674:
4668:
4667:
4658:
4646:
4638:
4627:
4626:External links
4624:
4623:
4622:
4616:
4602:Nocedal, Jorge
4598:
4592:
4576:
4570:
4553:
4547:
4527:
4524:
4521:
4520:
4485:Bioinformatics
4468:
4425:
4390:Bioinformatics
4376:
4343:Bioinformatics
4333:
4298:(2): 174–187.
4279:
4253:
4206:
4160:
4149:(3): 167–175.
4133:
4090:
4066:
4031:
3996:
3975:(2): 121–128.
3953:
3947:N. Friedrich,
3940:
3899:
3854:
3827:(4): 621–636.
3811:
3771:
3720:
3681:
3657:
3650:
3625:
3614:(5): 817–831.
3593:
3583:Lionel Robbins
3575:
3540:(3): 236–239.
3524:
3505:
3466:
3459:
3422:
3410:
3380:
3365:
3341:
3317:
3298:
3292:978-1108833417
3291:
3271:
3247:
3222:
3221:
3219:
3216:
3214:
3213:
3208:
3203:
3198:
3193:
3188:
3183:
3178:
3172:
3167:
3162:
3157:
3152:
3147:
3141:
3139:
3136:
3131:Main article:
3128:
3125:
3120:Main article:
3117:
3114:
3097:Main article:
3094:
3091:
3079:Main article:
3076:
3073:
3052:
3049:
3040:
3037:
3020:
3017:
2991:
2988:
2950:
2947:
2927:control theory
2832:
2829:
2793:
2790:
2788:
2785:
2783:
2782:
2777:
2772:
2767:
2762:
2756:
2751:
2745:
2740:
2735:
2730:
2724:
2704:Main article:
2701:
2698:
2697:
2696:
2695:
2694:
2692:Mirror descent
2689:
2682:Pattern search
2679:
2670:
2669:
2668:
2662:
2656:
2646:
2636:
2629:
2612:
2606:
2596:
2587:
2586:
2585:
2579:
2569:
2526:Main article:
2523:
2520:
2519:
2518:
2513:
2508:
2502:
2492:
2486:George Dantzig
2470:
2467:
2450:
2447:
2425:
2422:
2388:Hessian matrix
2329:Hessian matrix
2309:Main article:
2306:
2303:
2275:
2272:
2260:Hessian matrix
2254:
2251:
2220:
2217:
2203:
2200:
2196:slack variable
2169:
2166:
2164:
2161:
2132:
2129:
2097:Main article:
2094:
2091:
2090:
2089:
2075:
2060:
2054:
2044:
2043:
2033:
2030:
2029:
2028:
2004:
2000:metaheuristics
1993:
1983:
1977:
1967:
1960:
1950:
1943:
1937:
1930:
1920:
1919:
1918:
1904:
1898:
1885:
1879:
1837:
1834:
1832:
1831:
1826:
1821:
1816:
1814:Lev Pontryagin
1811:
1809:Yurii Nesterov
1806:
1801:
1796:
1791:
1786:
1781:
1776:
1774:William Karush
1771:
1766:
1761:
1756:
1751:
1749:Roger Fletcher
1746:
1741:
1736:
1731:
1725:
1655:
1652:
1564:
1563:
1552:
1548:
1544:
1541:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1503:
1500:
1497:
1494:
1491:
1484:
1480:
1477:
1472:
1469:
1466:
1462:
1459:
1456:
1439:
1438:
1427:
1424:
1421:
1418:
1415:
1407:
1403:
1400:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1367:
1364:
1361:
1357:
1354:
1351:
1292:
1291:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1243:
1240:
1237:
1232:
1228:
1221:
1217:
1214:
1211:
1207:
1204:
1201:
1184:
1183:
1172:
1169:
1166:
1161:
1157:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1120:
1117:
1114:
1110:
1107:
1104:
1083:Main article:
1080:
1077:
1054:
1053:
1042:
1039:
1032:
1028:
1025:
1021:
986:
956:
955:
943:
939:
936:
931:
927:
922:
914:
910:
907:
903:
887:
884:
879:
876:
856:convex problem
848:global minimum
812:
811:
799:
796:
793:
789:
783:
778:
773:
769:
764:
753:
750:
746:
742:
637:is called the
603:
598:
537:
536:
525:
522:
518:
514:
511:
508:
505:
502:
497:
492:
487:
484:
481:
478:
475:
471:
467:
464:
461:
458:
453:
448:
443:
440:
401:
400:
376:
344:
327:
317:
293:
262:
261:
248:
227:, in which an
199:Main article:
196:
193:
110:
107:
104:
101:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6462:
6451:
6448:
6446:
6443:
6441:
6438:
6437:
6435:
6420:
6417:
6416:
6413:
6407:
6404:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6382:
6379:
6377:
6374:
6372:
6369:
6368:
6366:
6362:
6356:
6353:
6351:
6348:
6346:
6343:
6341:
6338:
6336:
6333:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6310:Barbara Grosz
6308:
6306:
6305:Wolt Fabrycky
6303:
6301:
6298:
6296:
6293:
6291:
6288:
6286:
6283:
6281:
6280:Ruzena Bajcsy
6278:
6276:
6273:
6272:
6270:
6266:
6260:
6257:
6255:
6252:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6220:
6217:
6215:
6212:
6211:
6209:
6205:
6199:
6196:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6174:
6171:
6170:
6168:
6164:
6158:
6155:
6153:
6152:Design review
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6133:
6130:
6129:
6127:
6123:
6117:
6114:
6112:
6109:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6062:
6059:
6058:
6056:
6052:
6048:
6041:
6036:
6034:
6029:
6027:
6022:
6021:
6018:
6006:
6005:
5996:
5994:
5993:
5984:
5982:
5981:
5972:
5970:
5969:
5964:
5958:
5957:
5954:
5948:
5945:
5943:
5940:
5938:
5935:
5933:
5930:
5928:
5925:
5921:
5918:
5917:
5916:
5913:
5912:
5910:
5908:
5904:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5878:
5875:
5873:
5870:
5869:
5867:
5865:
5864:Computational
5861:
5853:
5850:
5848:
5845:
5843:
5840:
5839:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5779:
5777:
5775:
5771:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5741:
5739:
5737:
5733:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5708:
5706:
5704:
5703:Number theory
5700:
5694:
5691:
5689:
5686:
5684:
5681:
5679:
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5660:
5658:
5656:
5652:
5646:
5643:
5641:
5638:
5636:
5635:Combinatorics
5633:
5632:
5630:
5628:
5624:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5600:
5598:
5595:
5593:
5590:
5588:
5587:Real analysis
5585:
5583:
5580:
5579:
5577:
5575:
5571:
5565:
5562:
5560:
5557:
5555:
5552:
5550:
5547:
5545:
5542:
5540:
5537:
5535:
5532:
5530:
5527:
5526:
5524:
5522:
5518:
5512:
5509:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5489:
5487:
5484:
5483:
5481:
5479:
5475:
5469:
5466:
5464:
5461:
5457:
5454:
5452:
5449:
5448:
5447:
5444:
5443:
5440:
5435:
5427:
5422:
5420:
5415:
5413:
5408:
5407:
5404:
5392:
5389:
5388:
5385:
5375:
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5355:
5352:
5350:
5349:Hill climbing
5347:
5345:
5342:
5341:
5338:
5334:
5329:
5325:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5292:
5290:
5288:
5287:Network flows
5284:
5274:
5271:
5269:
5266:
5262:
5259:
5258:
5257:
5254:
5253:
5251:
5249:
5248:Shortest path
5245:
5235:
5232:
5230:
5227:
5225:
5222:
5221:
5219:
5217:
5216:spanning tree
5211:
5208:
5206:
5200:
5192:
5188:
5185:
5184:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5164:
5162:
5158:
5154:
5150:
5149:Combinatorial
5145:
5141:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5104:
5102:
5100:
5097:
5093:
5087:
5084:
5082:
5079:
5077:
5074:
5073:
5071:
5069:
5065:
5062:
5060:
5055:
5051:
5045:
5042:
5040:
5037:
5035:
5032:
5031:
5029:
5027:
5021:
5017:
5013:
5008:
5004:
4990:
4987:
4985:
4982:
4980:
4977:
4976:
4974:
4970:
4964:
4961:
4959:
4956:
4955:
4953:
4949:
4945:
4941:
4936:
4932:
4924:
4906:
4903:
4902:
4900:
4898:
4894:
4884:
4881:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4859:
4856:
4854:
4851:
4850:
4848:
4846:
4845:Other methods
4842:
4836:
4833:
4831:
4828:
4826:
4822:
4819:
4817:
4814:
4813:
4811:
4809:
4805:
4799:
4796:
4794:
4791:
4790:
4788:
4786:
4782:
4779:
4777:
4773:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4743:
4741:
4739:
4735:
4731:
4727:
4722:
4718:
4714:
4710:
4706:
4702:
4695:
4690:
4688:
4683:
4681:
4676:
4675:
4672:
4664:
4659:
4655:
4651:
4647:
4643:
4639:
4634:
4630:
4629:
4625:
4619:
4617:0-387-30303-0
4613:
4609:
4608:
4603:
4599:
4595:
4593:0-521-01012-8
4589:
4585:
4581:
4577:
4573:
4571:0-12-283952-8
4567:
4563:
4559:
4558:Wright, M. H.
4554:
4550:
4548:0-521-83378-7
4544:
4540:
4539:
4534:
4530:
4529:
4525:
4516:
4512:
4508:
4504:
4499:
4494:
4490:
4486:
4482:
4478:
4472:
4469:
4464:
4460:
4456:
4452:
4448:
4444:
4440:
4436:
4429:
4426:
4421:
4417:
4413:
4409:
4404:
4399:
4395:
4391:
4387:
4380:
4377:
4372:
4368:
4364:
4360:
4356:
4352:
4348:
4344:
4337:
4334:
4329:
4325:
4321:
4317:
4313:
4309:
4305:
4301:
4297:
4293:
4286:
4284:
4280:
4267:
4263:
4257:
4254:
4249:
4245:
4241:
4237:
4233:
4229:
4225:
4221:
4217:
4210:
4207:
4202:
4198:
4193:
4188:
4184:
4180:
4176:
4172:
4164:
4161:
4156:
4152:
4148:
4144:
4137:
4134:
4129:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4094:
4091:
4086:
4082:
4078:
4077:
4070:
4067:
4062:
4058:
4054:
4050:
4046:
4042:
4035:
4032:
4027:
4023:
4019:
4015:
4011:
4007:
4000:
3997:
3992:
3988:
3983:
3978:
3974:
3970:
3969:
3964:
3957:
3954:
3950:
3944:
3941:
3935:
3930:
3926:
3922:
3918:
3914:
3910:
3903:
3900:
3895:
3891:
3887:
3883:
3878:
3873:
3869:
3865:
3858:
3855:
3850:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3815:
3812:
3808:
3804:
3799:
3795:
3789:
3783:
3782:
3775:
3772:
3767:
3763:
3758:
3753:
3749:
3745:
3738:
3734:
3730:
3724:
3721:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3685:
3682:
3678:
3674:
3671:
3667:
3661:
3658:
3653:
3651:9780674043084
3647:
3643:
3639:
3635:
3629:
3626:
3621:
3617:
3613:
3609:
3608:
3603:
3597:
3594:
3590:
3589:
3584:
3579:
3576:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3539:
3535:
3528:
3525:
3520:
3516:
3509:
3506:
3501:
3497:
3493:
3489:
3485:
3481:
3477:
3470:
3467:
3462:
3456:
3452:
3448:
3444:
3440:
3433:
3426:
3423:
3413:
3407:
3403:
3399:
3395:
3391:
3384:
3381:
3376:
3369:
3366:
3355:
3351:
3345:
3342:
3331:
3327:
3321:
3318:
3313:
3309:
3302:
3299:
3294:
3288:
3284:
3283:
3275:
3272:
3257:
3251:
3248:
3244:
3240:
3236:
3233:
3227:
3224:
3217:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3176:
3173:
3171:
3170:Least squares
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3150:Curve fitting
3148:
3146:
3143:
3142:
3137:
3134:
3126:
3123:
3115:
3113:
3111:
3106:
3100:
3092:
3090:
3088:
3082:
3074:
3072:
3070:
3066:
3062:
3058:
3050:
3048:
3046:
3038:
3036:
3034:
3030:
3026:
3018:
3016:
3014:
3010:
3006:
3001:
2997:
2989:
2987:
2985:
2981:
2980:space mapping
2977:
2976:space mapping
2973:
2968:
2964:
2963:space mapping
2960:
2959:active filter
2956:
2948:
2946:
2944:
2940:
2936:
2932:
2931:search models
2928:
2923:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2889:
2885:
2881:
2877:
2872:
2870:
2866:
2863:
2862:
2857:
2853:
2849:
2845:
2841:
2837:
2830:
2828:
2825:
2823:
2819:
2815:
2810:
2808:
2803:
2799:
2791:
2786:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2760:
2757:
2755:
2752:
2749:
2748:Hill climbing
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2725:
2723:
2721:
2717:
2713:
2707:
2699:
2693:
2690:
2687:
2683:
2680:
2677:
2676:Interpolation
2674:
2673:
2671:
2666:
2663:
2660:
2657:
2654:
2650:
2647:
2644:
2640:
2637:
2634:
2630:
2627:
2623:
2620:
2616:
2613:
2610:
2607:
2604:
2600:
2597:
2594:
2591:
2590:
2588:
2583:
2580:
2577:
2573:
2570:
2568:
2565:
2564:
2562:
2558:
2557:
2556:
2552:
2550:
2546:
2543:
2539:
2535:
2529:
2521:
2517:
2514:
2512:
2509:
2507:
2503:
2501:
2497:
2493:
2491:
2487:
2483:
2480:
2479:
2476:
2468:
2466:
2464:
2460:
2456:
2448:
2446:
2444:
2440:
2436:
2435:trust regions
2432:
2431:line searches
2423:
2421:
2419:
2415:
2410:
2408:
2404:
2399:
2397:
2393:
2389:
2385:
2380:
2378:
2375:
2371:
2368:
2364:
2360:
2356:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2312:
2304:
2302:
2300:
2296:
2291:
2289:
2285:
2281:
2273:
2271:
2269:
2265:
2261:
2252:
2250:
2248:
2244:
2239:
2237:
2233:
2229:
2225:
2218:
2216:
2213:
2209:
2201:
2199:
2197:
2193:
2188:
2185:
2181:
2177:
2176:
2167:
2162:
2160:
2158:
2154:
2150:
2146:
2141:
2137:
2130:
2128:
2126:
2121:
2117:
2113:
2111:
2107:
2100:
2092:
2087:
2083:
2079:
2076:
2073:
2068:
2064:
2061:
2058:
2055:
2052:
2049:
2048:
2047:
2041:
2037:
2036:Space mapping
2034:
2031:
2026:
2023:
2022:
2020:
2016:
2012:
2008:
2005:
2001:
1997:
1994:
1991:
1987:
1984:
1981:
1978:
1975:
1971:
1968:
1964:
1961:
1958:
1954:
1951:
1947:
1944:
1941:
1938:
1934:
1931:
1928:
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1794:László Lovász
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715:local minimum
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683:
682:cost function
679:
678:
677:loss function
673:
669:
662:The function
660:
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571:cost function
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249:
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245:countable set
242:
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190:
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177:real function
174:
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6345:Katia Sycara
6229:Optimization
6228:
6002:
5990:
5978:
5959:
5892:Optimization
5891:
5754:Differential
5678:Differential
5645:Order theory
5640:Graph theory
5544:Group theory
5354:Local search
5300:Edmonds–Karp
5256:Bellman–Ford
5026:minimization
4858:Gauss–Newton
4808:Quasi–Newton
4793:Trust region
4701:Optimization
4700:
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4272:14 September
4270:. Retrieved
4266:the original
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3250:
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2924:
2908:Asset prices
2880:dual problem
2873:
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2796:Problems in
2795:
2787:Applications
2718:, there are
2709:
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2531:
2452:
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2400:
2396:saddle point
2391:
2384:definiteness
2381:
2352:
2299:Claude Berge
2292:
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1936:programming.
1892:semidefinite
1863:
1824:Naum Z. Shor
1722:
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1331:feasible set
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975:real numbers
964:
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315:real numbers
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130:optimisation
129:
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121:best) value.
77:
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53:
6004:WikiProject
5847:Game theory
5827:Probability
5564:Homological
5554:Multilinear
5534:Commutative
5511:Type theory
5478:Foundations
5434:mathematics
5374:Tabu search
4785:Convergence
4756:Line search
3750:: 297–346.
3521:(5): 29–38.
3445:: 107–121.
3377:. Citeseer.
3308:Floudas, C.
3057:geophysical
2904:risk-averse
2869:JEL:C61-C63
2852:game theory
2780:Tabu search
2643:quasiconvex
2576:constrained
2363:subgradient
1990:dimensional
1911:posynomials
1789:Harold Kuhn
1690:Programming
1507:subject to:
1336:Similarly,
1247:subject to:
649:are called
619:constraints
576:Typically,
322:an element
237:permutation
231:such as an
162:mathematics
150:engineering
6434:Categories
6335:Simon Ramo
5832:Statistics
5711:Arithmetic
5673:Arithmetic
5539:Elementary
5506:Set theory
5205:algorithms
4713:heuristics
4705:Algorithms
4477:Mendes, P.
4192:1807/93364
3870:(1): 1–3.
3486:: 114432.
3417:2024-08-18
3359:2024-08-24
3335:2024-08-24
3051:Geophysics
2965:design of
2920:portfolios
2856:equilibria
2824:problems.
2805:solving a
2720:heuristics
2712:algorithms
2700:Heuristics
2463:heuristics
2455:algorithms
2372:and other
2315:See also:
2106:Pareto set
1996:Heuristics
1868:polyhedron
1764:Fritz John
1688:in 1939. (
1676:The term "
1626:Operators
730:such that
700:functional
643:choice set
335:such that
306:from some
211:continuous
6125:Processes
6054:Subfields
5759:Geometric
5749:Algebraic
5688:Euclidean
5663:Algebraic
5559:Universal
5160:Paradigms
5059:quadratic
4776:Gradients
4738:Functions
4507:1367-4803
4455:1096-7192
4412:1460-2059
4363:1460-2059
4312:0006-3592
4248:1064-8275
4201:116480238
4177:: 81–86.
4128:113616284
3991:110195165
3872:CiteSeerX
3841:1868-8071
3715:2248-6968
3570:125980981
3562:1995-0721
3035:methods.
2967:microwave
2888:consumers
2836:Economics
2792:Mechanics
2370:functions
2284:parameter
2202:Existence
2003:problems.
1915:monomials
1874:if it is
1706:logistics
1543:∈
1521:−
1515:∈
1496:
1420:
1402:∈
1380:−
1374:∈
1270:−
1264:∞
1261:−
1255:∈
1142:−
1136:∞
1133:−
1127:∈
1073:undefined
1027:∈
909:∈
862:problem.
795:δ
792:≤
782:∗
772:−
749:∈
741:∀
507:−
504:≤
480:−
477:⇔
460:≥
207:variables
158:economics
6419:Category
6166:Concepts
5980:Category
5736:Topology
5683:Discrete
5668:Analytic
5655:Geometry
5627:Discrete
5582:Calculus
5574:Analysis
5529:Abstract
5468:Glossary
5451:Timeline
5391:Software
5268:Dijkstra
5099:exchange
4897:Hessians
4863:Gradient
4582:(2004).
4580:Lee, Jon
4560:(1982).
4463:17336115
4420:17890736
4371:16864593
4328:25023799
4320:18551704
3894:11086218
3849:13071135
3735:(1997).
3673:Archived
3670:Abstract
3638:"Search"
3636:(1987).
3264:26 April
3235:Archived
3138:See also
2957:include
2894:, while
2878:and its
2545:Hessians
2542:evaluate
2498:and for
2392:positive
2359:gradient
2325:Gradient
1974:discrete
1895:matrices
1872:polytope
1663:Lagrange
1621:integers
1615:, where
1322:, since
1303:interval
1296:argument
1069:infinity
1063:, where
878:Notation
788:‖
763:‖
580:is some
389:for all
357:for all
279: :
273:function
215:discrete
6193:V-Model
5992:Commons
5774:Applied
5744:General
5521:Algebra
5446:History
5234:Kruskal
5224:Borůvka
5214:Minimum
4951:General
4709:methods
4515:9927716
4228:Bibcode
4108:Bibcode
4049:Bibcode
4014:Bibcode
3921:Bibcode
3766:3585236
3695:Innovar
3620:1810679
3542:Bibcode
3488:Bibcode
3127:Solvers
3013:traffic
2892:utility
2678:methods
2619:locally
2386:of the
2374:locally
1927:integer
1876:bounded
1850:concave
1718:duality
1700:by the
1698:program
1654:History
1632:arg max
1628:arg min
1605:{−5, (2
1306:(−∞,−1]
1301:in the
1085:Arg max
833:holds;
698:energy
641:or the
584:of the
563:modeled
544:physics
423:History
320:Sought:
313:to the
233:integer
78:x, y, z
74:maximum
6268:People
6183:System
5693:Finite
5549:Linear
5456:Future
5432:Major
5096:Basis-
5054:Linear
5024:Convex
4868:Mirror
4825:L-BFGS
4711:, and
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2858:. The
2848:scarce
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2624:using
2347:, and
1854:convex
1846:convex
1667:Newton
1659:Fermat
1640:argmax
1636:argmin
1071:" or "
852:convex
728:> 0
628:domain
582:subset
561:being
559:system
549:energy
421:– see
269:Given:
229:object
64:) = −(
6207:Tools
5920:lists
5463:Lists
5436:areas
5295:Dinic
5203:Graph
4324:S2CID
4197:S2CID
4124:S2CID
3987:S2CID
3890:S2CID
3845:S2CID
3805:" by
3796:" by
3778:From
3762:JSTOR
3740:(PDF)
3616:JSTOR
3566:S2CID
3435:(PDF)
3259:(PDF)
3218:Notes
2896:firms
2865:codes
2192:relax
1870:or a
1671:Gauss
1594:{5, 2
960:value
823:*) ≤
757:where
565:. In
409:or a
241:graph
185:value
181:input
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6224:IDEF
5261:SPFA
5229:Prim
4823:and
4612:ISBN
4588:ISBN
4566:ISBN
4543:ISBN
4511:PMID
4503:ISSN
4459:PMID
4451:ISSN
4416:PMID
4408:ISSN
4367:PMID
4359:ISSN
4316:PMID
4308:ISSN
4274:2013
4244:ISSN
3837:ISSN
3711:ISSN
3646:ISBN
3558:ISSN
3455:ISBN
3443:2012
3406:ISBN
3287:ISBN
3266:2024
3067:and
2998:and
2974:and
2532:The
2443:BFGS
2293:The
2278:The
2206:The
2172:The
2155:and
1998:and
1976:one.
1966:set.
1669:and
1661:and
1646:and
1638:and
1630:and
1609:+ 1)
1603:and
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148:and
140:and
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1200:a
1171:,
1168:1
1165:+
1160:2
1156:x
1148:]
1145:1
1139:,
1130:(
1124:x
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1109:g
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1103:a
1065:x
1060:x
1058:2
1041:x
1038:2
1031:R
1024:x
1001:x
985:R
971:x
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942:)
938:1
935:+
930:2
926:x
921:(
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906:x
841:*
839:x
831:)
829:x
827:(
825:f
821:x
819:(
817:f
798:,
777:x
768:x
752:A
745:x
726:δ
721:*
719:x
664:f
647:A
635:f
631:A
624:A
602:n
597:R
578:A
555:f
524:,
521:)
517:x
513:(
510:f
501:)
496:0
491:x
486:(
483:f
474:)
470:x
466:(
463:f
457:)
452:0
447:x
442:(
439:f
396:A
392:x
387:)
385:x
383:(
381:f
377:0
374:x
372:(
370:f
364:A
360:x
355:)
353:x
351:(
349:f
345:0
342:x
340:(
338:f
332:A
328:0
325:x
311:A
292:R
281:A
277:f
247:.
109:)
106:x
103:(
100:f
70:y
66:x
62:y
58:x
54:z
45:.
38:.
20:)
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