525:
55:
46:
517:
71:
38:
1179:, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number of iterations grows like a polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated); however, such theoretically "efficient" methods use "divergent-series"
962:
424:
1068:
734:
1320:
2357:
2139:
Generalized concavity in optimization and economics: Proceedings of the NATO Advanced Study
Institute held at the University of British Columbia, Vancouver, B.C., August 4–15, 1980
49:
A function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex
1485:
1374:
628:
1521:
1410:
2137:
Di Guglielmo, F. (1981). "Estimates of the duality gap for discrete and quasiconvex optimization problems". In
Schaible, Siegfried; Ziemba, William T. (eds.).
2026:
1677:
2350:
1117:
228:
1806:
801:
1977:
1922:
318:
150:
2288:. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997. xxii+491 pp.
2343:
1553:
1442:
1857:
1735:
280:
763:
156:. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be
1591:
827:. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does.
559:
449:
is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then
1187:. Classical subgradient methods using divergent-series rules are much slower than modern methods of convex minimization, such as subgradient projection methods,
1611:
856:
821:
507:
487:
467:
447:
248:
1171:, whose biduals provide quasiconvex closures of the primal problem, which therefore provide tighter bounds than do the convex closures provided by Lagrangian
2547:
1986:
is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare
864:
326:
973:
639:
2447:
31:
2539:
2275:
1260:
2552:
2094:
2293:
2146:
2048:
2572:
1447:
1336:
2489:
1176:
564:
2796:
78:
1490:
1379:
1240:
2791:
59:
1163:
that converge to a minimum (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization of
2806:
2557:
2315:
1224:
2587:
1925:
1228:
1136:
2577:
1999:
176:
functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple
2786:
2562:
2404:
2170:
Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization".
1616:
1924:) need not be quasiconvex. Such functions are called "additively decomposed" in economics and "separable" in
2647:
2624:
2518:
2442:
1322:) is quasiconvex. Similarly, maximum of strict quasiconvex functions is strict quasiconvex. Similarly, the
1220:
1094:
2801:
2765:
2726:
2642:
2567:
2494:
2479:
2432:
177:
102:
98:
199:
2499:
2329:
2059:
1740:
1156:
1132:
1330:
functions is quasiconcave, and the minimum of strictly-quasiconcave functions is strictly-quasiconcave.
768:
2256:
Crouzeix, J.-P. (2008). "Quasi-concavity". In
Durlauf, Steven N.; Blume, Lawrence E (eds.).
1944:
1862:
2370:
2437:
2427:
2422:
1192:
181:
63:
2257:
2227:
285:
120:
2666:
2384:
2203:
2111:
1983:
1212:
1184:
1180:
1164:
2325:
1526:
1415:
1818:
1696:
524:
2484:
2289:
2271:
2187:
2142:
1078:
1074:
253:
742:
528:
The graph of a function that is both concave and quasiconvex on the nonnegative real numbers.
2637:
2582:
2473:
2468:
2263:
2179:
2103:
2043:
1561:
1236:
1208:
1172:
1160:
75:
54:
2335:
2310:
2199:
2156:
2123:
509:, and the point directly between them, can be points on a line or more generally points in
2657:
2628:
2602:
2597:
2592:
2523:
2508:
2417:
2389:
2366:
2195:
2152:
2119:
2069:
2053:
2038:
1232:
535:
164:
45:
2744:
2652:
2513:
2412:
2213:
2064:
1994:
1596:
1204:
841:
806:
492:
472:
452:
432:
233:
516:
70:
30:
For the unrelated generalization of convexity used in the calculus of variations, see
17:
2780:
2711:
2703:
2699:
2695:
2691:
2687:
2528:
1188:
114:
2207:
37:
2749:
2216:
first established that quasiconvex minimization problems can be solved efficiently.
1168:
110:
2092:
Di Guglielmo, F. (1977). "Nonconvex duality in multiobjective optimization".
2305:
2739:
2734:
2618:
1987:
1216:
1140:
1120:
957:{\displaystyle f(\lambda x+(1-\lambda )y)\geq \min {\big \{}f(x),f(y){\big \}}.}
419:{\displaystyle f(\lambda x+(1-\lambda )y)\leq \max {\big \{}f(x),f(y){\big \}}.}
185:
173:
94:
86:
2267:
2662:
2632:
2394:
2028:
is an example of a quasiconvex function that is neither convex nor continuous.
1063:{\displaystyle f(\lambda x+(1-\lambda )y)>\min {\big \{}f(x),f(y){\big \}}}
838:
is a function whose negative is strictly quasiconvex. Equivalently a function
729:{\displaystyle f(\lambda x+(1-\lambda )y)<\max {\big \{}f(x),f(y){\big \}}}
169:
153:
106:
2191:
2306:
SION, M., "On general minimax theorems", Pacific J. Math. 8 (1958), 171-176.
1144:
532:
An alternative way (see introduction) of defining a quasi-convex function
2671:
2107:
2452:
2229:
Parameter
Optimization for Equilibrium Solutions of Mass Action Systems
2183:
2115:
2056:
in the sense of several complex variables (not generalized convexity)
1315:{\displaystyle f=\max \left\lbrace f_{1},\ldots ,f_{n}\right\rbrace }
163:
Quasiconvexity is a more general property than convexity in that all
167:
are also quasiconvex, but not all quasiconvex functions are convex.
523:
515:
69:
44:
36:
1167:. Quasiconvex programming is used in the solution of "surrogate"
1077:, while a (strictly) quasiconcave function has (strictly) convex
2339:
2141:. New York: Academic Press, Inc. . pp. 281–298.
2319:
1199:
Economics and partial differential equations: Minimax theorems
520:
A quasilinear function is both quasiconvex and quasiconcave.
1480:{\displaystyle g:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
1369:{\displaystyle g:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
2262:(Second ed.). Palgrave Macmillan. pp. 815–816.
623:{\displaystyle S_{\alpha }(f)=\{x\mid f(x)\leq \alpha \}}
188:
sublevel sets can be unimodal without being quasiconvex.
1084:
A function that is both quasiconvex and quasiconcave is
1073:
A (strictly) quasiconvex function has (strictly) convex
2248:
Avriel, M., Diewert, W.E., Schaible, S. and Zang, I.,
1516:{\displaystyle h:\mathbb {R} \rightarrow \mathbb {R} }
1405:{\displaystyle h:\mathbb {R} \rightarrow \mathbb {R} }
2002:
1947:
1865:
1821:
1743:
1699:
1619:
1599:
1564:
1529:
1493:
1450:
1418:
1382:
1339:
1263:
1097:
976:
867:
844:
809:
771:
745:
642:
567:
538:
495:
475:
455:
435:
329:
288:
256:
236:
202:
123:
1941:
A concave function can be quasiconvex. For example,
2758:
2725:
2680:
2611:
2537:
2461:
2403:
2377:
834:is a function whose negative is quasiconvex, and a
184:is unimodal but not quasiconvex and functions with
2020:
1971:
1916:
1851:
1800:
1729:
1671:
1605:
1585:
1547:
1515:
1479:
1436:
1404:
1368:
1333:composition with a non-decreasing function :
1314:
1111:
1062:
956:
850:
815:
795:
757:
728:
622:
553:
501:
481:
461:
441:
418:
312:
274:
242:
222:
144:
27:Mathematical function with convex lower level sets
250:of a real vector space is quasiconvex if for all
1636:
1270:
1016:
907:
682:
369:
1239:, Sion's theorem is also used in the theory of
1215:. Quasiconvex functions are important also in
2351:
2259:The New Palgrave Dictionary of Economics
1123:, in which there is a locally maximal value.
1055:
1021:
946:
912:
721:
687:
408:
374:
8:
2226:Johansson, Edvard; Petersson, David (2016).
2087:
2015:
2009:
1811:The sum of quasiconvex functions defined on
1693:need not be quasiconvex: In other words, if
1689:The sum of quasiconvex functions defined on
617:
590:
1183:, which were first developed for classical
1131:Quasiconvex functions have applications in
2358:
2344:
2336:
2021:{\displaystyle x\mapsto \lfloor x\rfloor }
2178:(1). Berlin, Heidelberg: Springer: 1–25.
2001:
1946:
1864:
1820:
1742:
1698:
1672:{\displaystyle h(x)=\inf _{y\in C}f(x,y)}
1639:
1618:
1598:
1563:
1528:
1509:
1508:
1501:
1500:
1492:
1473:
1472:
1463:
1459:
1458:
1449:
1417:
1398:
1397:
1390:
1389:
1381:
1362:
1361:
1352:
1348:
1347:
1338:
1301:
1282:
1262:
1105:
1104:
1096:
1091:A particular case of quasi-concavity, if
1054:
1053:
1020:
1019:
975:
945:
944:
911:
910:
866:
843:
808:
770:
744:
720:
719:
686:
685:
641:
572:
566:
537:
494:
474:
454:
434:
407:
406:
373:
372:
328:
287:
255:
235:
216:
215:
201:
122:
41:A quasiconvex function that is not convex
1684:Operations not preserving quasiconvexity
53:
2448:Locally convex topological vector space
2080:
1257:maximum of quasiconvex functions (i.e.
32:Quasiconvexity (calculus of variations)
1938:Every convex function is quasiconvex.
1112:{\displaystyle S\subset \mathbb {R} }
561:is to require that each sublevel set
469:is quasiconvex. Note that the points
7:
1252:Operations preserving quasiconvexity
2316:Concave and Quasi-Concave Functions
1227:, particularly for applications of
223:{\displaystyle f:S\to \mathbb {R} }
2172:Mathematical Programming, Series A
2095:Mathematics of Operations Research
1801:{\displaystyle (f+g)(x)=f(x)+g(x)}
1159:, quasiconvex programming studies
130:
25:
2326:Quasiconcavity and quasiconvexity
2311:Mathematical programming glossary
796:{\displaystyle \lambda \in (0,1)}
180:. For example, the 2-dimensional
2049:Logarithmically concave function
1979:is both concave and quasiconvex.
1972:{\displaystyle x\mapsto \log(x)}
1917:{\displaystyle h(x,y)=f(x)+g(y)}
66:is quasiconcave but not concave.
2553:Ekeland's variational principle
2006:
1966:
1960:
1951:
1911:
1905:
1896:
1890:
1881:
1869:
1846:
1840:
1831:
1825:
1795:
1789:
1780:
1774:
1765:
1759:
1756:
1744:
1724:
1718:
1709:
1703:
1666:
1654:
1629:
1623:
1580:
1568:
1505:
1469:
1444:is quasiconvex. Similarly, if
1394:
1358:
1247:Preservation of quasiconvexity
1241:partial differential equations
1050:
1044:
1035:
1029:
1010:
1004:
992:
980:
941:
935:
926:
920:
901:
895:
883:
871:
836:strictly quasiconcave function
790:
778:
716:
710:
701:
695:
676:
670:
658:
646:
608:
602:
584:
578:
548:
542:
403:
397:
388:
382:
363:
357:
345:
333:
307:
295:
212:
139:
124:
1:
967:and strictly quasiconcave if
2232:(MSc thesis). pp. 13–14
313:{\displaystyle \lambda \in }
145:{\displaystyle (-\infty ,a)}
60:probability density function
2573:Hermite–Hadamard inequality
230:defined on a convex subset
2823:
2268:10.1057/9780230226203.1375
1548:{\displaystyle f=h\circ g}
1437:{\displaystyle f=h\circ g}
1225:general equilibrium theory
1211:imply that consumers have
1191:of descent, and nonsmooth
29:
2212:Kiwiel acknowledges that
1926:mathematical optimization
1852:{\displaystyle f(x),g(y)}
1730:{\displaystyle f(x),g(x)}
1151:Mathematical optimization
1137:mathematical optimization
192:Definition and properties
2759:Applications and related
2563:Fenchel-Young inequality
2328:- by Martin J. Osborne,
2286:Abstract convex analysis
1808:need not be quasiconvex.
275:{\displaystyle x,y\in S}
2519:Legendre transformation
2443:Legendre transformation
2332:Department of Economics
2322:Department of Economics
2088:Di Guglielmo (1977
1221:industrial organization
758:{\displaystyle x\neq y}
117:of any set of the form
2766:Convexity in economics
2700:(lower) ideally convex
2558:Fenchel–Moreau theorem
2548:Carathéodory's theorem
2022:
1973:
1918:
1853:
1802:
1737:are quasiconvex, then
1731:
1673:
1607:
1587:
1586:{\displaystyle f(x,y)}
1549:
1517:
1481:
1438:
1406:
1370:
1316:
1229:Sion's minimax theorem
1157:nonlinear optimization
1113:
1064:
958:
852:
817:
797:
759:
730:
624:
555:
529:
521:
503:
483:
463:
443:
420:
314:
276:
244:
224:
146:
82:
67:
51:
42:
18:Quasi-concave function
2797:Generalized convexity
2688:Convex series related
2588:Shapley–Folkman lemma
2330:University of Toronto
2318:- by Charles Wilson,
2252:, Plenum Press, 1988.
2250:Generalized Concavity
2090:, pp. 287–288):
2060:Pseudoconvex function
2023:
1974:
1919:
1854:
1803:
1732:
1674:
1608:
1588:
1550:
1523:non-decreasing, then
1518:
1482:
1439:
1412:non-decreasing, then
1407:
1371:
1317:
1133:mathematical analysis
1114:
1065:
959:
853:
832:quasiconcave function
818:
798:
760:
731:
625:
556:
527:
519:
504:
484:
464:
444:
421:
315:
277:
245:
225:
147:
73:
57:
48:
40:
2578:Krein–Milman theorem
2371:variational analysis
2108:10.1287/moor.2.3.285
2000:
1945:
1863:
1819:
1741:
1697:
1617:
1597:
1562:
1527:
1491:
1448:
1416:
1380:
1337:
1261:
1095:
974:
865:
842:
825:strictly quasiconvex
807:
769:
743:
640:
565:
554:{\displaystyle f(x)}
536:
513:-dimensional space.
493:
473:
453:
433:
327:
286:
254:
234:
200:
121:
91:quasiconvex function
2792:Convex optimization
2568:Jensen's inequality
2438:Lagrange multiplier
2428:Convex optimization
2423:Convex metric space
1558:minimization (i.e.
1185:subgradient methods
858:is quasiconcave if
182:Rosenbrock function
64:normal distribution
2807:Types of functions
2696:(cs, bcs)-complete
2667:Algebraic interior
2385:Convex combination
2184:10.1007/PL00011414
2018:
1984:monotonic function
1969:
1914:
1849:
1798:
1727:
1669:
1650:
1603:
1583:
1545:
1513:
1477:
1434:
1402:
1366:
1312:
1213:convex preferences
1165:convex programming
1109:
1079:upper contour sets
1075:lower contour sets
1060:
954:
848:
813:
793:
755:
726:
620:
551:
530:
522:
499:
479:
459:
439:
416:
310:
272:
240:
220:
142:
83:
68:
52:
43:
2774:
2773:
2277:978-0-333-78676-5
1859:are quasiconvex,
1815:domains (i.e. if
1635:
1613:convex set, then
1606:{\displaystyle C}
1231:. Generalizing a
1209:utility functions
1161:iterative methods
851:{\displaystyle f}
816:{\displaystyle f}
630:is a convex set.
502:{\displaystyle y}
482:{\displaystyle x}
462:{\displaystyle f}
442:{\displaystyle f}
243:{\displaystyle S}
16:(Redirected from
2814:
2692:(cs, lcs)-closed
2638:Effective domain
2593:Robinson–Ursescu
2469:Convex conjugate
2360:
2353:
2346:
2337:
2281:
2242:
2241:
2239:
2237:
2223:
2217:
2211:
2167:
2161:
2160:
2134:
2128:
2127:
2085:
2044:Concave function
2027:
2025:
2024:
2019:
1978:
1976:
1975:
1970:
1923:
1921:
1920:
1915:
1858:
1856:
1855:
1850:
1807:
1805:
1804:
1799:
1736:
1734:
1733:
1728:
1678:
1676:
1675:
1670:
1649:
1612:
1610:
1609:
1604:
1592:
1590:
1589:
1584:
1555:is quasiconcave.
1554:
1552:
1551:
1546:
1522:
1520:
1519:
1514:
1512:
1504:
1486:
1484:
1483:
1478:
1476:
1468:
1467:
1462:
1443:
1441:
1440:
1435:
1411:
1409:
1408:
1403:
1401:
1393:
1375:
1373:
1372:
1367:
1365:
1357:
1356:
1351:
1321:
1319:
1318:
1313:
1311:
1307:
1306:
1305:
1287:
1286:
1237:John von Neumann
1118:
1116:
1115:
1110:
1108:
1069:
1067:
1066:
1061:
1059:
1058:
1025:
1024:
963:
961:
960:
955:
950:
949:
916:
915:
857:
855:
854:
849:
822:
820:
819:
814:
802:
800:
799:
794:
764:
762:
761:
756:
735:
733:
732:
727:
725:
724:
691:
690:
629:
627:
626:
621:
577:
576:
560:
558:
557:
552:
508:
506:
505:
500:
488:
486:
485:
480:
468:
466:
465:
460:
448:
446:
445:
440:
425:
423:
422:
417:
412:
411:
378:
377:
319:
317:
316:
311:
281:
279:
278:
273:
249:
247:
246:
241:
229:
227:
226:
221:
219:
165:convex functions
151:
149:
148:
143:
81:is quasiconcave.
76:bivariate normal
21:
2822:
2821:
2817:
2816:
2815:
2813:
2812:
2811:
2787:Convex analysis
2777:
2776:
2775:
2770:
2754:
2721:
2676:
2607:
2533:
2524:Semi-continuity
2509:Convex function
2490:Logarithmically
2457:
2418:Convex geometry
2399:
2390:Convex function
2373:
2367:Convex analysis
2364:
2302:
2278:
2255:
2245:
2235:
2233:
2225:
2224:
2220:
2169:
2168:
2164:
2149:
2136:
2135:
2131:
2091:
2086:
2082:
2078:
2070:Concavification
2054:Pseudoconvexity
2039:Convex function
2035:
1998:
1997:
1943:
1942:
1935:
1861:
1860:
1817:
1816:
1739:
1738:
1695:
1694:
1691:the same domain
1686:
1679:is quasiconvex)
1615:
1614:
1595:
1594:
1560:
1559:
1525:
1524:
1489:
1488:
1457:
1446:
1445:
1414:
1413:
1378:
1377:
1346:
1335:
1334:
1297:
1278:
1277:
1273:
1259:
1258:
1254:
1249:
1233:minimax theorem
1207:, quasiconcave
1201:
1181:step size rules
1153:
1129:
1093:
1092:
972:
971:
863:
862:
840:
839:
805:
804:
767:
766:
741:
740:
638:
637:
633:If furthermore
568:
563:
562:
534:
533:
491:
490:
471:
470:
451:
450:
431:
430:
325:
324:
284:
283:
252:
251:
232:
231:
198:
197:
194:
119:
118:
35:
28:
23:
22:
15:
12:
11:
5:
2820:
2818:
2810:
2809:
2804:
2799:
2794:
2789:
2779:
2778:
2772:
2771:
2769:
2768:
2762:
2760:
2756:
2755:
2753:
2752:
2747:
2745:Strong duality
2742:
2737:
2731:
2729:
2723:
2722:
2720:
2719:
2684:
2682:
2678:
2677:
2675:
2674:
2669:
2660:
2655:
2653:John ellipsoid
2650:
2645:
2640:
2635:
2621:
2615:
2613:
2609:
2608:
2606:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2550:
2544:
2542:
2540:results (list)
2535:
2534:
2532:
2531:
2526:
2521:
2516:
2514:Invex function
2511:
2502:
2497:
2492:
2487:
2482:
2476:
2471:
2465:
2463:
2459:
2458:
2456:
2455:
2450:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2413:Choquet theory
2409:
2407:
2401:
2400:
2398:
2397:
2392:
2387:
2381:
2379:
2378:Basic concepts
2375:
2374:
2365:
2363:
2362:
2355:
2348:
2340:
2334:
2333:
2323:
2313:
2308:
2301:
2300:External links
2298:
2297:
2296:
2282:
2276:
2253:
2244:
2243:
2218:
2162:
2147:
2129:
2102:(3): 285–291.
2079:
2077:
2074:
2073:
2072:
2067:
2065:Invex function
2062:
2057:
2051:
2046:
2041:
2034:
2031:
2030:
2029:
2017:
2014:
2011:
2008:
2005:
1995:floor function
1991:
1980:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1939:
1934:
1931:
1930:
1929:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1809:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1685:
1682:
1681:
1680:
1668:
1665:
1662:
1659:
1656:
1653:
1648:
1645:
1642:
1638:
1634:
1631:
1628:
1625:
1622:
1602:
1582:
1579:
1576:
1573:
1570:
1567:
1556:
1544:
1541:
1538:
1535:
1532:
1511:
1507:
1503:
1499:
1496:
1487:quasiconcave,
1475:
1471:
1466:
1461:
1456:
1453:
1433:
1430:
1427:
1424:
1421:
1400:
1396:
1392:
1388:
1385:
1364:
1360:
1355:
1350:
1345:
1342:
1331:
1310:
1304:
1300:
1296:
1293:
1290:
1285:
1281:
1276:
1272:
1269:
1266:
1253:
1250:
1248:
1245:
1205:microeconomics
1200:
1197:
1193:filter methods
1189:bundle methods
1152:
1149:
1128:
1125:
1107:
1103:
1100:
1071:
1070:
1057:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1023:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
965:
964:
953:
948:
943:
940:
937:
934:
931:
928:
925:
922:
919:
914:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
847:
812:
792:
789:
786:
783:
780:
777:
774:
754:
751:
748:
737:
736:
723:
718:
715:
712:
709:
706:
703:
700:
697:
694:
689:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
575:
571:
550:
547:
544:
541:
498:
478:
458:
438:
427:
426:
415:
410:
405:
402:
399:
396:
393:
390:
387:
384:
381:
376:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
309:
306:
303:
300:
297:
294:
291:
271:
268:
265:
262:
259:
239:
218:
214:
211:
208:
205:
193:
190:
141:
138:
135:
132:
129:
126:
113:such that the
101:defined on an
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2819:
2808:
2805:
2803:
2802:Real analysis
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2784:
2782:
2767:
2764:
2763:
2761:
2757:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2732:
2730:
2728:
2724:
2717:
2715:
2709:
2707:
2701:
2697:
2693:
2689:
2686:
2685:
2683:
2679:
2673:
2670:
2668:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2630:
2626:
2622:
2620:
2617:
2616:
2614:
2610:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2583:Mazur's lemma
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2545:
2543:
2541:
2536:
2530:
2529:Subderivative
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2477:
2475:
2472:
2470:
2467:
2466:
2464:
2460:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2410:
2408:
2406:
2405:Topics (list)
2402:
2396:
2393:
2391:
2388:
2386:
2383:
2382:
2380:
2376:
2372:
2368:
2361:
2356:
2354:
2349:
2347:
2342:
2341:
2338:
2331:
2327:
2324:
2321:
2317:
2314:
2312:
2309:
2307:
2304:
2303:
2299:
2295:
2294:0-471-16015-6
2291:
2287:
2284:Singer, Ivan
2283:
2279:
2273:
2269:
2265:
2261:
2260:
2254:
2251:
2247:
2246:
2231:
2230:
2222:
2219:
2215:
2214:Yuri Nesterov
2209:
2205:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2166:
2163:
2158:
2154:
2150:
2148:0-12-621120-5
2144:
2140:
2133:
2130:
2125:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2096:
2089:
2084:
2081:
2075:
2071:
2068:
2066:
2063:
2061:
2058:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2036:
2032:
2012:
2003:
1996:
1992:
1989:
1985:
1981:
1963:
1957:
1954:
1948:
1940:
1937:
1936:
1932:
1927:
1908:
1902:
1899:
1893:
1887:
1884:
1878:
1875:
1872:
1866:
1843:
1837:
1834:
1828:
1822:
1814:
1810:
1792:
1786:
1783:
1777:
1771:
1768:
1762:
1753:
1750:
1747:
1721:
1715:
1712:
1706:
1700:
1692:
1688:
1687:
1683:
1663:
1660:
1657:
1651:
1646:
1643:
1640:
1632:
1626:
1620:
1600:
1593:quasiconvex,
1577:
1574:
1571:
1565:
1557:
1542:
1539:
1536:
1533:
1530:
1497:
1494:
1464:
1454:
1451:
1431:
1428:
1425:
1422:
1419:
1386:
1383:
1376:quasiconvex,
1353:
1343:
1340:
1332:
1329:
1325:
1308:
1302:
1298:
1294:
1291:
1288:
1283:
1279:
1274:
1267:
1264:
1256:
1255:
1251:
1246:
1244:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1198:
1196:
1194:
1190:
1186:
1182:
1178:
1174:
1173:dual problems
1170:
1169:dual problems
1166:
1162:
1158:
1150:
1148:
1146:
1142:
1138:
1134:
1126:
1124:
1122:
1101:
1098:
1089:
1087:
1082:
1080:
1076:
1047:
1041:
1038:
1032:
1026:
1013:
1007:
1001:
998:
995:
989:
986:
983:
977:
970:
969:
968:
951:
938:
932:
929:
923:
917:
904:
898:
892:
889:
886:
880:
877:
874:
868:
861:
860:
859:
845:
837:
833:
828:
826:
810:
787:
784:
781:
775:
772:
752:
749:
746:
713:
707:
704:
698:
692:
679:
673:
667:
664:
661:
655:
652:
649:
643:
636:
635:
634:
631:
614:
611:
605:
599:
596:
593:
587:
581:
573:
569:
545:
539:
526:
518:
514:
512:
496:
476:
456:
436:
429:In words, if
413:
400:
394:
391:
385:
379:
366:
360:
354:
351:
348:
342:
339:
336:
330:
323:
322:
321:
304:
301:
298:
292:
289:
269:
266:
263:
260:
257:
237:
209:
206:
203:
191:
189:
187:
183:
179:
175:
172:
171:
166:
161:
159:
155:
136:
133:
127:
116:
115:inverse image
112:
108:
107:convex subset
104:
100:
96:
92:
88:
80:
79:joint density
77:
72:
65:
61:
56:
47:
39:
33:
19:
2750:Weak duality
2713:
2705:
2625:Orthogonally
2504:
2285:
2258:
2249:
2234:. Retrieved
2228:
2221:
2175:
2171:
2165:
2138:
2132:
2099:
2093:
2083:
1812:
1690:
1328:quasiconcave
1327:
1323:
1202:
1154:
1130:
1127:Applications
1090:
1085:
1083:
1072:
966:
835:
831:
829:
824:
738:
632:
531:
510:
428:
195:
168:
162:
158:quasiconcave
157:
111:vector space
90:
84:
2740:Duality gap
2735:Dual system
2619:Convex hull
1988:unimodality
1217:game theory
1141:game theory
1121:unimodality
1086:quasilinear
196:A function
186:star-convex
87:mathematics
2781:Categories
2663:Radial set
2633:Convex set
2395:Convex set
2236:26 October
2076:References
170:Univariate
154:convex set
109:of a real
2648:Hypograph
2192:0025-5610
2016:⌋
2010:⌊
2007:↦
1958:
1952:↦
1813:different
1644:∈
1540:∘
1506:→
1470:→
1429:∘
1395:→
1359:→
1292:…
1145:economics
1139:, and in
1102:⊂
1002:λ
999:−
984:λ
905:≥
893:λ
890:−
875:λ
776:∈
773:λ
750:≠
668:λ
665:−
650:λ
615:α
612:≤
597:∣
574:α
367:≤
355:λ
352:−
337:λ
293:∈
290:λ
267:∈
213:→
178:arguments
131:∞
128:−
2672:Zonotope
2643:Epigraph
2208:10043417
2033:See also
1933:Examples
739:for all
320:we have
174:unimodal
105:or on a
103:interval
99:function
97:-valued
2727:Duality
2629:Pseudo-
2603:Ursescu
2500:Pseudo-
2474:Concave
2453:Simplex
2433:Duality
2200:1819784
2157:0652702
2124:0484418
2116:3689518
1324:minimum
1223:, and
803:, then
62:of the
2710:, and
2681:Series
2598:Simons
2505:Quasi-
2495:Proper
2480:Closed
2292:
2274:
2206:
2198:
2190:
2155:
2145:
2122:
2114:
1177:theory
2538:Main
2204:S2CID
2112:JSTOR
1175:. In
1135:, in
1119:, is
152:is a
93:is a
2658:Lens
2612:Sets
2462:Maps
2369:and
2290:ISBN
2272:ISBN
2238:2016
2188:ISSN
2143:ISBN
1993:The
1982:Any
1143:and
1014:>
765:and
680:<
489:and
282:and
95:real
89:, a
74:The
58:The
50:set.
2712:(Hw
2320:NYU
2264:doi
2180:doi
2104:doi
1955:log
1637:inf
1326:of
1271:max
1243:.
1235:of
1203:In
1155:In
1017:min
908:min
823:is
683:max
370:max
85:In
2783::
2704:(H
2702:,
2698:,
2694:,
2631:)
2627:,
2507:)
2485:K-
2270:.
2202:.
2196:MR
2194:.
2186:.
2176:90
2174:.
2153:MR
2151:.
2120:MR
2118:.
2110:.
2098:.
1990:).
1219:,
1195:.
1147:.
1088:.
1081:.
830:A
160:.
2718:)
2716:)
2714:x
2708:)
2706:x
2690:(
2665:/
2623:(
2478:(
2359:e
2352:t
2345:v
2280:.
2266::
2240:.
2210:.
2182::
2159:.
2126:.
2106::
2100:2
2013:x
2004:x
1967:)
1964:x
1961:(
1949:x
1928:.
1912:)
1909:y
1906:(
1903:g
1900:+
1897:)
1894:x
1891:(
1888:f
1885:=
1882:)
1879:y
1876:,
1873:x
1870:(
1867:h
1847:)
1844:y
1841:(
1838:g
1835:,
1832:)
1829:x
1826:(
1823:f
1796:)
1793:x
1790:(
1787:g
1784:+
1781:)
1778:x
1775:(
1772:f
1769:=
1766:)
1763:x
1760:(
1757:)
1754:g
1751:+
1748:f
1745:(
1725:)
1722:x
1719:(
1716:g
1713:,
1710:)
1707:x
1704:(
1701:f
1667:)
1664:y
1661:,
1658:x
1655:(
1652:f
1647:C
1641:y
1633:=
1630:)
1627:x
1624:(
1621:h
1601:C
1581:)
1578:y
1575:,
1572:x
1569:(
1566:f
1543:g
1537:h
1534:=
1531:f
1510:R
1502:R
1498::
1495:h
1474:R
1465:n
1460:R
1455::
1452:g
1432:g
1426:h
1423:=
1420:f
1399:R
1391:R
1387::
1384:h
1363:R
1354:n
1349:R
1344::
1341:g
1309:}
1303:n
1299:f
1295:,
1289:,
1284:1
1280:f
1275:{
1268:=
1265:f
1106:R
1099:S
1056:}
1051:)
1048:y
1045:(
1042:f
1039:,
1036:)
1033:x
1030:(
1027:f
1022:{
1011:)
1008:y
1005:)
996:1
993:(
990:+
987:x
981:(
978:f
952:.
947:}
942:)
939:y
936:(
933:f
930:,
927:)
924:x
921:(
918:f
913:{
902:)
899:y
896:)
887:1
884:(
881:+
878:x
872:(
869:f
846:f
811:f
791:)
788:1
785:,
782:0
779:(
753:y
747:x
722:}
717:)
714:y
711:(
708:f
705:,
702:)
699:x
696:(
693:f
688:{
677:)
674:y
671:)
662:1
659:(
656:+
653:x
647:(
644:f
618:}
609:)
606:x
603:(
600:f
594:x
591:{
588:=
585:)
582:f
579:(
570:S
549:)
546:x
543:(
540:f
511:n
497:y
477:x
457:f
437:f
414:.
409:}
404:)
401:y
398:(
395:f
392:,
389:)
386:x
383:(
380:f
375:{
364:)
361:y
358:)
349:1
346:(
343:+
340:x
334:(
331:f
308:]
305:1
302:,
299:0
296:[
270:S
264:y
261:,
258:x
238:S
217:R
210:S
207::
204:f
140:)
137:a
134:,
125:(
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
