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

Journal article
Authors Magnus Önnheim
Emil Gustavsson
Ann-Brith Strömberg
Michael Patriksson
Torbjörn Larsson
Published in Mathematical programming
Volume 163
Issue 1
Pages 57-84
ISSN 0025-5610
Publication year 2017
Published at Department of Mathematical Sciences
Pages 57-84
Language en
Links dx.doi.org/10.1007/s10107-016-1055-...
Keywords inconsistent convex program, Lagrange dual, homogeneous Lagrangian function, subgradient algorithm, ergodic primal sequence
Subject categories Optimization, systems theory

Abstract

Consider the utilization of a Lagrangian dual method which is convergent for consistent 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 ε-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.

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