A method of linearizations for linearly constrained nonconvex nonsmooth minimization

A readily implementable algorithm is given for minimizing a (possibly nondifferentiable and nonconvex) locally Lipschitz continuous functionf subject to linear constraints. At each iteration a polyhedral approximation tof is constructed from a few previously computed subgradients and an aggregate subgradient, which accumulates the past subgradient information. This aproximation and the linear constraints generate constraints in the search direction finding subproblem that is a quadratic programming problem. Then a stepsize is found by an approximate line search. All the algorithm's accumulation points are stationary. Moreover, the algorithm converges whenf happens to be convex.

[1]  K. C. Kiwiel.  A linearization algorithm for constrained nonsmooth minimization , 1984 .

[2]  A. Bihain.  Optimization of upper semidifferentiable functions , 1984 .

[3]  Krzysztof C. Kiwiel,et al.  A Linearization Algorithm for Nonsmooth Minimization , 1985, Math. Oper. Res..

[4]  D. Mayne,et al.  On the Extension of Constrained Optimization Algorithms from Differentiable to Nondifferentiable Problems , 1983 .

[5]  C. Lemaréchal,et al.  ON A BUNDLE ALGORITHM FOR NONSMOOTH OPTIMIZATION , 1981 .

[6]  Kaj Madsen,et al.  Linearly constrained minimax optimization , 1978, Math. Program..

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

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

[9]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[10]  J. Strodiot,et al.  An algorithm for minimizing nondifferentiable convex functions under linear constraints , 1984 .

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

[12]  Jean-Jacques Strodiot,et al.  ε-Optimal solutions in nondifferentiable convex programming and some related questions , 1983, Math. Program..

[13]  Krzysztof C. Kiwiel,et al.  An aggregate subgradient method for nonsmooth convex minimization , 1983, Math. Program..

[14]  K. Kiwiel.  A Method for Solving Certain Quadratic Programming Problems Arising in Nonsmooth Optimization , 1986 .

[15]  K. Kiwiel.  A method for minimizing the sum of a convex function and a continuously differentiable function , 1986 .

[16]  K. Kiwiel.  An algorithm for linearly constrained convex nondifferentiable minimization problems , 1985 .

[17]  R. Mifflin.  A modification and an extension of Lemarechal’s algorithm for nonsmooth minimization , 1982 .