An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions

To construct an effective minimization algorithm for locally Lipschitz functions, we show how to compute a descent direction satisfying Armijo’s condition. We present a finitely terminating algorithm to construct an approximating set for the Goldstein subdifferential leading to the desired descent direction. Using this direction, we propose a minimization algorithm for locally Lipschitz functions and prove its convergence. Finally, we implement our algorithm with matrix laboratory (MATLAB) codes and report our testing results. The comparative numerical results attest to the efficiency of the proposed algorithm.

[1]  D. Mayne,et al.  Nondifferential optimization via adaptive smoothing , 1984 .

[2]  Henry P. Wynn,et al.  MODA 7 - advances in model-oriented design and analysis : proceedings of the 7th international workshop on Model-oriented design and analysis, held in Heeze, The Netherlands, June 14-18, 2004 , 2004 .

[3]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[4]  Antonio Fuduli,et al.  A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization , 2004, Optim. Methods Softw..

[5]  K. Anstreicher,et al.  A Masked Spectral Bound for Maximum-Entropy Sampling , 2004 .

[6]  Adrian S. Lewis,et al.  A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization , 2005, SIAM J. Optim..

[7]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[8]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[9]  Antonio Fuduli,et al.  Minimizing Nonconvex Nonsmooth Functions via Cutting Planes and Proximity Control , 2003, SIAM J. Optim..

[10]  A. Bagirov.  Continuous Subdifferential Approximations and Their Applications , 2003 .

[11]  A. Lewis,et al.  Robust stability and a criss‐cross algorithm for pseudospectra , 2003 .

[12]  Alexander Kaplan,et al.  Proximal Point Methods and Nonconvex Optimization , 1998, J. Glob. Optim..

[13]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[14]  P. Neittaanmäki,et al.  Nonsmooth Optimization Theory , 1992 .

[15]  Adrian S. Lewis,et al.  Approximating Subdifferentials by Random Sampling of Gradients , 2002, Math. Oper. Res..

[16]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[17]  Warren Hare,et al.  Computing proximal points of nonconvex functions , 2008, Math. Program..

[18]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[19]  A. Bagirov,et al.  Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization , 2008 .

[20]  V. N. Malozemov,et al.  Finding the Point of a Polyhedron Closest to the Origin , 1974 .

[21]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

[22]  Enrico Gorgone,et al.  Piecewise linear approximations in nonconvex nonsmooth optimization , 2009, Numerische Mathematik.

[23]  Philip Wolfe,et al.  Finding the nearest point in A polytope , 1976, Math. Program..

[24]  Robert Mifflin,et al.  An Algorithm for Constrained Optimization with Semismooth Functions , 1977, Math. Oper. Res..

[25]  Robert Mifflin,et al.  A -algorithm for convex minimization , 2005, Math. Program..

[26]  Krzysztof C. Kiwiel,et al.  Convergence of the Gradient Sampling Algorithm for Nonsmooth Nonconvex Optimization , 2007, SIAM J. Optim..

[27]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[28]  A. A. Goldstein,et al.  Optimization of lipschitz continuous functions , 1977, Math. Program..

[29]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[30]  Manlio Gaudioso,et al.  A bundle type approach to the unconstrained minimization of convex nonsmooth functions , 1982, Math. Program..

[31]  Johannes O. Royset,et al.  Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques , 2003 .

[32]  D. Bertsekas,et al.  A DESCENT NUMERICAL METHOD FOR OPTIMIZATION PROBLEMS WITH NONDIFFERENTIABLE COST FUNCTIONALS , 1973 .

[33]  Antonio Frangioni,et al.  Generalized Bundle Methods , 2002, SIAM J. Optim..

[34]  P. Wolfe.  Note on a method of conjugate subgradients for minimizing nondifferentiable functions , 1974 .