Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality

Second-order sufficient conditions for local optimality have long been central to designing solution algorithms and justifying claims about their convergence. Here a far-reaching extension of such conditions, called variational sufficiency, is explored in territory beyond just classical nonlinear pr...

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Bibliographic Details
Published in:Mathematical programming Vol. 198; no. 1; pp. 159 - 194
Main Author: Rockafellar, R. Tyrrell
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023
Springer
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
ALM
Second-order variational analysis
Local duality
Generalized nonlinear programming
Second-order cone programming
Local maximal monotonicity
90C46
Sufficient conditions for local optimality
Generalized augmented Lagrangians
Nonsmooth optimization
Tilt stability
Metric regularity
Variational convexity
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:Second-order sufficient conditions for local optimality have long been central to designing solution algorithms and justifying claims about their convergence. Here a far-reaching extension of such conditions, called variational sufficiency, is explored in territory beyond just classical nonlinear programming. Variational sufficiency is already known to support multiplier methods that are able, even without convexity, to achieve problem decomposition, but further insight has been needed into how it coordinates with other sufficient conditions. In the framework of this paper, it is shown to characterize local optimality in terms of having a convex–concave-type local saddle point of an augmented Lagrangian function. A stronger version of variational sufficiency is tied in turn to local strong convexity in the primal argument of that function and a property of augmented tilt stability that offers crucial aid to Lagrange multiplier methods at a fundamental level of analysis. Moreover, that strong version is translated here through second-order variational analysis into statements that can readily be compared to existing sufficient conditions in nonlinear programming, second-order cone programming, and other problem formulations which can incorporate nonsmooth objectives and regularization terms.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01768-w