185:). PRincipal Axis (PRAXIS) algorithm, also referred to as Brent's algorithm, is a derivative-free algorithm which assumes quadratic form of the optimized function and repeatedly updates a set of conjugate search directions. The algorithm, however, is not invariant to scaling of the objective function and may fail under its certain rank-preserving transformations (e.g., will lead to a non-quadratic shape of the objective function). A recent analysis of PRAXIS can be found in . For practical applications see, where an adaptive coordinate descent approach with step-size adaptation and local coordinate system rotation was proposed for robot-manipulator path planning in 3D space with static polygonal obstacles.
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The adaptation of an appropriate coordinate system allows adaptive coordinate descent to outperform coordinate descent on non-separable functions. The following figure illustrates the convergence of both algorithms on 2-dimensional
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The adaptive coordinate descent method reaches the target value after only 325 function evaluations (about 70 times faster than coordinate descent), that is comparable to
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Ali, U.; Kickmeier-Rust, M.D. (2008). "Implementation and
Applications of a Three-Round User Strategy for Improved Principal Axis Minimization".
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First approaches to optimization using adaptive coordinate system were proposed already in the 1960s (see, e.g.,
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Invariance with respect to monotonous transformations of the function (scaling)
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Pavlov, D. (2006). "Manipulator path planning in 3-dimensional space".
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Adaptive
Encoding: How to Render Search Coordinate System Invariant
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ACD is a MATLAB source code for
Adaptive Coordinate Descent
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