Alexander Kaplan

Proximal Point Methods and Nonconvex Optimization

The goal of this paper is to discover some possibilities for applying the proximal point method to nonconvex problems. It can be proved that – for a wide class of problems – proximal regularization performed with appropriate regularization parameters ensures convexity of the auxiliary problems and each accumulation point of the method satisfies the necessary optimality conditions.

Proximal Point Approach and Approximation of Variational Inequalities

A general approach to analyze convergence of the proximal-like methods for variational inequalities with set-valued maximal monotone operators is developed. It is oriented to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and weak regularization. This approach also covers so-called multistep regularization methods, in which the number of proximal iterations in the approximated problems is controlled by a criterion characterizing these iterations as to be effective. The conditions on convergence require control of the exactness of the approximation only in a certain region of the original space. Conditions ensuring linear convergence of the methods are established.

On inexact generalized proximal methods with a weakened error tolerance criterion

Two inexact versions of a Bregman-function-based proximal method for finding a zero of a maximal monotone operator, suggested in [J. Eckstein (1998). Approximate iterations in Bregman-function-based proximal algorithms. Math. Programming, 83, 113–123; P. da Silva, J. Eckstein and C. Humes (2001). Rescaling and stepsize selection in proximal methods using separable generalized distances. SIAM J. Optim., 12, 238–261], are considered. For a wide class of Bregman functions, including the standard entropy kernel and all strongly convex Bregman functions, convergence of these methods is proved under an essentially weaker accuracy condition on the iterates than in the original papers. 1 Concerning , they coincide with A1–A3 in Section 2. Also the error criterion of a logarithmic–quadratic proximal method, developed in [A. Auslender, M. Teboulle and S. Ben-Tiba (1999). A logarithmic-quadratic proximal method for variational inequalities. Computational Optimization and Applications, 12, 31–40], is relaxed, and convergence results for the inexact version of the proximal method with entropy-like distance functions are described. For the methods mentioned, like in [R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877–898] for the classical proximal point algorithm, only summability of the sequence of error vector norms is required.

Bregman-like functions and proximal methods for variational problems with nonlinear constraints

The use of Bregman functions, entropic ϕ-divergence or logarithmic-quadratic kernels, allows to construct a series of generalized proximal methods for the stable solution of convex optimization problems and variational inequalities with maximal monotone operators. The key advantage of these methods in comparison to the classical proximal regularization is that the auxiliary problems are structurally simpler than the original ones, in particular, they result in unconstrained inclusions. But, such methods with “interior point effect” were developed so far only for linearly constrained problems. In the present article, the use of Bregman functions with a modified “convergence sensing condition” enables us to construct an interior proximal method for solving variational inequalities (with multi-valued operators) on nonpolyhedral sets. The convergence results admit a successive approximation of the multi-valued operator (by means of the concept of ε-enlargements) and an inexact calculation of proximal iterates. †Dedicated to the 65th birthday of Diethard Pallaschke.

Numerical treatment of an asset price model with non-stochastic uncertainty

In contrast to stochastic differential equation models used for the calculation of the term structure of interest rates, we develop an approach based on linear dynamical systems under non-stochastic uncertainty with perturbations. The uncertainty is described in terms of known feasible sets of varying parameters. Observations are used in order to estimate these parameters by minimizing the maximum of the absolute value of measurement errors, which leads to a linear or nonlinear semi-infinite programming problem. A regularized logarithmic barrier method for solving (ill-posed) convex semi-infinite programming problems is suggested. In this method a multi-step proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. A special deleting rule permits one to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convexC1-functions are studied. On the basis of the solutions of the semi-infinite programming problems a technical trading system for future contracts of the German DAX is suggested and developed.

Extended auxiliary problem principle using Bregman distances

An extension of the Auxiliary Problem Principle (cf., G. Cohen (1980). Auxiliary problem principle and decomposition of optimization problems. JOTA, 32, 277–305; G. Cohen (1988). Auxiliary problem principle extended to variational inequalities. JOTA, 59, 325–333.) for solving variational inequalities with maximal monotone operators is studied. Using Bregman functions to construct the symmetric components of the auxiliary operators, an “interior point effect” is provided, i.e. auxiliary problems can be treated as unconstrained ones.  For the sake of brevity we avoid here a repetition of results and facts viewed in [14,17] and refer only to the investigations which are directly connected with the main content of this article. In this general framework, classical and Bregman-function based proximal methods can be considered as particular cases. The convergence analysis allows that the auxiliary problems are solved inexactly with a sort of error summability criterion.

Prox-regularization and solution of ill-posed elliptic variational inequalities

In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem.In particular, regularization on the kernel of the differential operator and regularization with respect to a weak norm of the space are studied. These approaches are illustrated by two nonlinear problems in elasticity theory.

Proximal interior point method for convex semi-infinite programming

A regularized logarithmic Barrier method for solving (ill-posed) convex semi-infinite programming problems is considered. In this method a multi-step proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. Termination of the proximal iterations at each discretization level is controlled by means of estimates, characterizing the efficiency of these iterations. A special deleting rule permits to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convex C 1-functions are studied as well as some numerical experiments are presented.

Extended auxiliary problem principle to variational inequalities involving multi-valued operators

An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension supposes that the operator of the variational inequality is split up into the sum of a maximal monotone operator and a single-valued operator , which is linked with a sequence of non-symmetric components of auxiliary operators by a kind of pseudo Dunn property. The current auxiliary problem is constructed by fixing at the previous iterate, whereas is considered at a variable point. Using auxiliary operators of the form , the standard assumption of the strong convexity of the function h is weakened by exploiting mutual properties of and h. Convergence of the general scheme is analysed allowing that the auxiliary problems are solved approximately. Some applications are sketched briefly.

Auxiliary Problem Principle and Proximal Point Methods

AbstractAn extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension concerns the case that the operator is split into the sum of a single-valued operator