Criticality of Lagrange Multipliers in Variational Systems
04 Jun 2019-Siam Journal on Optimization (Society for Industrial and Applied Mathematics)-Vol. 29, Iss: 2, pp 1524-1557
TL;DR: Developing a novel approach, which is mainly based on advanced techniques and tools of second-order variational analysis and generalized differentiation, allows to overcome principal challenges of nonpolyhedrality and to establish complete characterizations on noncritical multipliers in such settings.
Abstract: The paper concerns the study of criticality of Lagrange multipliers in variational systems, which has been recognized in both theoretical and numerical aspects of optimization and variational analy...
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TL;DR: In this article, a comprehensive study of composite models in variational analysis and optimization is presented, with the main attention paid to the new and rather large class of fully subamenable compositions, and the underlying theme of the study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones.
Abstract: The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way we develop extended calculus rules for first-order and second-order generalized differential constructions with paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers and strong metric subregularity of KKT systems in parametric optimization, etc.
24 citations
TL;DR: The underlying theme of the study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead to significantly stronger and completely new results of first-order and second-order variational analysis and optimization.
Abstract: The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of opera...
21 citations
Posted Content•
TL;DR: This paper addresses problems of second-order cone programming important in optimization theory and applications by formulate the corresponding version ofsecond-order sufficiency and use it to establish the uniform second- order growth condition for the augmented Lagrangian.
Abstract: This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal-dual convergence of this method.
11 citations
TL;DR: In this paper , a comprehensive study of composite models in variational analysis and optimization is presented, with the main attention paid to the new and rather large class of fully subamenable compositions.
Abstract: The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way, we develop extended calculus rules for first-order and second-order generalized differential constructions while paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second-order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers, strong metric subregularity of Karush-Kuhn-Tucker systems in parametric optimization, and so on.
10 citations
TL;DR: In this article, the augmented Lagrangian method (ALM) for second-order cone programming is studied in both exact and inexact form. But the main attention is paid to the ALM for subproblems in both the exact and the inexact forms.
Abstract: This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal–dual convergence of this method.
9 citations