Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems
Journal article, 2017

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.

Lagrange dual

ergodic primal sequence

homogeneous Lagrangian function

inconsistent convex program

subgradient algorithm

Author

Magnus Önnheim

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Emil Gustavsson

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Ann-Brith Strömberg

Chalmers, Mathematical Sciences, Mathematics

Michael Patriksson

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Torbjörn Larsson

Linköping University

Mathematical Programming, Series A

0025-5610 (ISSN) 1436-4646 (eISSN)

Vol. 163 1-2 57-84

Driving Forces

Sustainable development

Areas of Advance

Transport

Energy

Subject Categories

Computational Mathematics

Roots

Basic sciences

DOI

10.1007/s10107-016-1055-x

More information

Latest update

8/24/2018