Critical multipliers in variational systems via second-order generalized differentiation

In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush–Kuhn–Tucker (KKT) systems. It has been well recognized that critical multipliers a...

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Vydáno v:Mathematical programming Ročník 169; číslo 2; s. 605 - 648
Hlavní autoři: Mordukhovich, Boris S., Sarabi, M. Ebrahim
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2018
Springer Nature B.V
Témata:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Economic models
Equivalence
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Minimax technique
Multipliers
Numerical Analysis
Optimization
Robustness (mathematics)
Theoretical
Variational systems
90C31
49J53
49J52
Composite optimization
Lipschitzian stability
Critical and noncritical multipliers
Robust isolated calmness
Generalized differentiation
Piecewise linear functions
ISSN:0025-5610, 1436-4646
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Shrnutí:In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush–Kuhn–Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal–dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1155-2