Ergodic, primal convergence in dual subgradient schemes for convex programming

Abstract

Lagrangean dualization and subgradient optimization techniques are frequently used within the eld of computational optimization for nding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational e ort.We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a tra c equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme.

Enhancing Lagrangian Dual Optimization for Linear Programs by Obviating Nondifferentiability

“…Commencing with x K as obtained after performing some K iterations using the BLR option in Phase I, we could employ an ergodicprimal-recovery strategy while executing the VTVM-GPKC algorithm in Phase II to derive such a primal solution for LP. (See Shor 1985, Sherali and Choi 1996, and Larsson et al 1999, for several ergodicprimal-recovery strategies.) We could also adopt a similar primal-recovery scheme in concert with the PT option in Phase I.…”

“…Finally, as commented in Remark 2 of §3, we experimented with the primal recovery schemes of Shor (1985) and Larsson et al (1999) for the pure subgradient strategy and Sherali and Choi (1996) for the deflected subgradient strategy. (Denote these methods by SHOR, LPS, and SC, respectively.)…”

“…If LP is a large, ill-conditioned problem, simplex or interior-point solvers can get prohibitively expensive (Adams andSherali 1993, Sherali andTuncbilek 1997, for example). Instead, Lagrangian relaxation, or Lagrangian dual optimization, along with a primal recovery scheme, could be more effective for obtaining lower bounds and selecting branching variables (Fisher 1981, Sherali and Myers 1988, Sherali and Choi 1996, Larsson et al 1999, Barahona and Anbil 2000. The Lagrangian dual of LP is…”

“…Sherali and Choi (1996) proposed an extension of such weighting strategies and established primal convergence when employing both pure and deflected subgradient algorithms. Larsson et al (1999) further extended these weighting schemes to yield primal convergence for general convex programs.…”

New approaches for optimizing over the semimetric polytope

“…In fact, some subgradient methods [20,3] also produce information that can be used to construct a primal solution, out of the previous solutions of the Lagrangian relaxations, which converges to an optimal solution of (6). On the other hand, the algorithmic parameters of some bundle methods (such as the one we used) can be chosen in such a way that the computational effort required for the master problem is potentially comparable to that required for computing the direction in a subgradient approach.…”