A convergence analysis of the method of codifferential descent
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This paper is devoted to a detailed convergence analysis of the method of codifferential descent (MCD) developed by professor V.F. Demyanov for solving a large class of nonsmooth nonconvex optimization problems. We propose a generalization of the MCD that is more suitable for applications than the original method, and that utilizes only a part of a codifferential on every iteration, which allows one to reduce the overall complexity of the method. With the use of some general results on uniformly codifferentiable functions obtained in this paper, we prove the global convergence of the generalized MCD in the infinite dimensional case. Also, we propose and analyse a quadratic regularization of the MCD, which is the first general method for minimizing a codifferentiable function over a convex set. Apart from convergence analysis, we also discuss the robustness of the MCD with respect to computational errors, possible step size rules, and a choice of parameters of the algorithm. In the end of the paper we estimate the rate of convergence of the MCD for a class of nonsmooth nonconvex functions that arise, in particular, in cluster analysis. We prove that under some general assumptions the method converges with linear rate, and it convergence quadratically, provided a certain first order sufficient optimality condition holds true.
KeywordsNonsmooth optimization Nonconvex optimization Codifferential Quasidifferential Method of codifferential descent
Mathematics Subject Classification90C56 49J52
The author is sincerely grateful to his colleagues G.Sh. Tamasyan, A. Fominyh and T. Angelov for numerous useful discussions on the method of codifferential descent and its applications that played an important role in the preparation of this paper. Also, the author wishes to express his thanks to the anonymous referees for many thoughtful comments that helped to significantly improve the quality of the article, and to professor V.N. Malozemov for pointing out the fact that the PPP algorithm converges quadratically to a Chebyshev (Haar) point.
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