strong regularity

Optimality Conditions for Constrained Minimax Optimization

  • Yu-Hong Dai
  • Liwei Zhang
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered … Read more

    Full stability of locally optimal solutions in second-order cone programming

  • Boris S. Mordukhovich
  • Jiri V. Outrata
  • Ebrahim Sarabi
    • Categories Nonsmooth Optimization Tags , , , , , ,

    The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to problems of second-order cone programming (SOCP) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sucient conditions under the corresponding constraint quali cations. We also establish close relationships between … Read more

    On metric regularity for weakly almost piecewise smooth functions and some applications in nonlinear semidefinite programming

  • Peter Fusek
    • Categories Semi-definite Programming Tags , , ,

    The one-to-one relation between the points fulfilling the KKT conditions of an optimization problem and the zeros of the corresponding Kojima function is well-known. In the present paper we study the interplay between metric regularity and strong regularity of this a priori nonsmooth function in the context of semidefinite programming. Having in mind the topological … Read more

    CHARACTERIZATIONS OF FULL STABILITY IN CONSTRAINED OPTIMIZATION

  • Boris S. Mordukhovich
  • Tyrrell Rockafellar
  • Ebrahim Sarabi
    • Categories Nonsmooth Optimization, Robust Optimization Tags , , , , , ,

    This paper is mainly devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from both viewpoints of optimization theory and its applications. Based on second- order generalized differential tools of variational analysis, we obtain … Read more

    STRONGLY REGULAR NONSMOOTH GENERALIZED EQUATIONS (REVISED)

  • Alexey F. Izmailov
    • Categories Complementarity and Variational Inequalities Tags , , , , ,

    This note suggests the implicit function theorem for generalized equations, unifying Robinson’s theorem for strongly regular generalized equations and Clarke’s implicit function theorem for equations with Lipschitz-continuous mappings. ArticleDownload View PDF

    ON REGULARITY CONDITIONS FOR COMPLEMENTARITY PROBLEMS

  • Alexey F. Izmailov
  • A. S. Kurennoy
    • Categories Complementarity and Variational Inequalities Tags , , , , , , , , , ,

    In the context of mixed complementarity problems, various concepts of solution regularity are known, each of them playing a certain role in related theoretical and algorithmic developments. In this note, we provide the complete picture of relations between the most important regularity conditions for mixed complementarity problems. A special attention is paid to the particular … Read more

    Constraint Nondegeneracy, Strong Regularity and Nonsingularity in Semidefinite Programming

  • Zi Xian Chan
  • Defeng Sun
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , ,

    It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, … Read more

    Stability and Sensitivity Analysis for Optimal Control Problems with a First-order State Constraint having (nonessential) Touch Points

  • Joseph Frédéric Bonnans
  • A. Hermant
    • Categories Infinite Dimensional Optimization Tags , , , ,

    The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. We show how to perform a sensitivity analysis that predicts which touch points will, under a small perturbation, become inactive, remain touch … Read more

    Perturbation analysis of second order programming problems

  • Joseph Frédéric Bonnans
  • Héctor Ramírez
    • Categories Linear, Cone and Semidefinite Programming, Second-Order Cone Programming Tags , , , ,

    We discuss first and second order optimality conditions for nonlinear second-order cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions. CitationResearch Report 5293 (August 2004), … Read more