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

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 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. Keywords Mathematics Subject Classification

Exact makespan minimization of unrelated parallel machines

“…First, to find a feasible solution from a solution to (4a) we use a greedy destruction-reconstruction heuristic that removes machines from multiply-assigned tasks, and assign a machine to each unassigned task, always greedily minimizing the makespan. Second, we construct a fractional solution as the convex combination of all subproblem solutions x k (from the current branch-and-bound node) weighted proportionally to the step lengths t k ; this forms a so-called ergodic sequence of subproblem solutions; see [26,33] for details. To construct a binary solution each job is assigned to the machine with the highest fraction.…”

Decomposition Methods for Combinatorial Optimization

“…The literature on Lagrangian relaxation goes back to early 1970s when Held and Karp [30,31] used a Lagrangian relaxation principle on their work of a traveling salesman problem and minimum spanning trees. Many other researchers have been motivated to apply the principle in solving mathematical programming models, see for example [26,15,52,19,36,29,5,57,42,35,46].…”