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Aubin property and uniqueness of solutions in cone constrained optimization

MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2013年第3期77卷 291-304页

作者： Klatte, Diethard Kummer, Bernd Univ Zurich Inst Betriebswirtschaftslehre CH-8044 Zurich Switzerland Humboldt Univ Inst Math D-10099 Berlin Germany

We discuss conditions for the Aubin property of solutions to perturbed cone constrained programs, by using and refining results given in Klatte and Kummer (Nonsmooth equations in optimization. Kluwer, Dordrecht, 2002). In particular, we show that constraint nondegeneracy and hence uniqueness of the multiplier is necessary for the Aubin property of the critical point map. Moreover, we give conditions under which the critical point map has the Aubin property if and only if it is locally single-valued and Lipschitz.

关键词： cone constrained optimization Aubin property Critical points Constraint nondegeneracy Locally single-valued solutions

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Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

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JOURNAL OF GLOBAL optimization 2018年第2期71卷 237-296页

作者： Dolgopolik, M. V. St Petersburg State Univ St Petersburg Russia Russian Acad Sci Inst Problems Mech Engn St Petersburg Russia

In this article we present a general theory of augmented Lagrangian functions for cone constrained optimization problems that allows one to study almost all known augmented Lagrangians for these problems within a unified framework. We develop a new general method for proving the existence of global saddle points of augmented Lagrangian functions, called the localization principle. The localization principle unifies, generalizes and sharpens most of the known results on the existence of global saddle points, and, in essence, reduces the problem of the existence of global saddle points to a local analysis of optimality conditions. With the use of the localization principle we obtain first necessary and sufficient conditions for the existence of a global saddle point of an augmented Lagrangian for cone constrained minimax problems via both second and first order optimality conditions. In the second part of the paper, we present a general approach to the construction of globally exact augmented Lagrangian functions. The general approach developed in this paper allowed us not only to sharpen most of the existing results on globally exact augmented Lagrangians, but also to construct first globally exact augmented Lagrangian functions for equality constrained optimization problems, for nonlinear second order cone programs and for nonlinear semidefinite programs. These globally exact augmented Lagrangians can be utilized in order to design new superlinearly (or even quadratically) convergent optimization methods for cone constrained optimization problems.

关键词： Augmented Lagrangian cone constrained optimization Localization principle Saddle point Second order cone Semidefinite programming Semi-infinite programming

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DC semidefinite programming and cone constrained DC optimization II: local search methods

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COMPUTATIONAL optimization AND APPLICATIONS 2023年第3期85卷 993-1031页

作者： Dolgopolik, M. V. Russian Acad Sci Inst Problems Mech Engn St Petersburg Russia

The second part of our study is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian.

关键词： DC optimization DCA Semidefinite programming cone constrained optimization Compressed modes Sphere packing Grassmannian

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DC Semidefinite programming and cone constrained DC optimization I: theory

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COMPUTATIONAL optimization AND APPLICATIONS 2022年第3期82卷 649-671页

作者： Dolgopolik, M., V Russian Acad Sci Inst Problems Mech Engn St Petersburg Russia

In this two-part study, we discuss possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear cone constrained optimization problems. In the first paper, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued mappings and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function. This DC decomposition can be used to reformulate DC semidefinite constraints as DC inequality constrains. Finally, we study local optimality conditions for general cone constrained DC optimization problems.

关键词： DC optimization Semidefinite programming DC decomposition cone constrained optimization

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