NDLI: Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems

Content Provider	Springer Nature Link
Author	Önnheim, Magnus Strömberg, Ann Brith Larsson, Torbjörn Gustavsson, Emil Patriksson, Michael
Copyright Year	2016
Abstract	Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal–dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which—in the inconsistent case—the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional $$\varepsilon $$ -subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal–dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved.
Ending Page	84
Page Count	28
Starting Page	57
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1-2
Volume Number	163
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-07-28
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Optimality conditions, duality Homogeneous Lagrangian function Theoretical, Mathematical and Computational Physics Subgradient algorithm Convex programming Ergodic primal sequence Mathematical Methods in Physics Calculus of Variations and Optimal Control; Optimization Mathematics of Computing Numerical Analysis Lagrange dual Nonlinear programming Combinatorics Inconsistent convex program Minimax problems
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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