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Convex Analysisの二,三の進展について

General Topology-I

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.

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Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods

Convex Analysis: (pms-28)

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.

Transversality and Alternating Projections for Nonconvex Sets

The word “tame” is used in the title in the same context as in expressions like “convex optimization,” “nonsmooth optimization,” etc.—as a reference to the class of objects involved in the formulation of optimization problems. Definable and tame functions and mappings associated with various o-minimal structures (e.g. semilinear, semialgebraic, globally subanalytic, and others) have a number of remarkable properties which make them an attractive domain for various applications. This relates both to the power of results that can be obtained and the power of available analytic techniques. The paper surveys certain ideas and recent results, some new, which have been or (hopefully) can be productively used in studies relating to variational analysis and nonsmooth optimization.

An Invitation to Tame Optimization

The concluding result of the paper states that variational problems are generically solvable (and even well-posed in a strong sense) without the convexity and growth conditions always present in individual existence theorems. This and some other generic well-posedness theorems are obtained as realizations of a general variational principle extending the variational principle of Deville--Godefroy--Zizler.

Variational Principles and Well-Posedness in Optimization and Calculus of Variations

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives consid- ered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gateaux differentiable spaces, Frechet and limiting Frechet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions.

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Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings

It is shown that a Lagrange multiplier rule involving the Michel-Penot subdifferentials is valid for the problem: minimizef0(x) subject tofi(x) ⩽ 0,i = 1, ⋯,m;fi(x) = 0,i = m + 1,⋯,n;x ∈Q where all functionsf are Lipschitz continuous andQ is a closed convex set. The proof is based on the theory of fans.

A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints

Alexander D. Ioffe

Papers

Transversality and Alternating Projections for Nonconvex Sets

An Invitation to Tame Optimization

Variational Principles and Well-Posedness in Optimization and Calculus of Variations

Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings

A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints