Multiple-response agents: Fast, feasible, approximate primal recovery for dual optimization methods

TL;DR

This work tackles distributed convex optimization with block-separable objectives and affine coupling constraints by augmenting dual-based methods with a novel primal-recovery step. The proposed Multiple-Response Agents (MRA) generate multiple ε-suboptimal primal responses per price and form a convex combination to rapidly reduce primal infeasibility while tolerating some suboptimality; the parallel nature of response computation avoids extra wall-clock time relative to the underlying dual algorithm. The approach is compatible with both price-localization and dual-subgradient methods and is shown, through extensive numerics on resource allocation, assignment, multi-commodity flow, and shipment problems, to yield fast convergence to feasible, near-optimal solutions; history and suboptimality-mixing provide tunable speed–quality trade-offs. The framework extends to nonconvex problems via convex relaxations and MILP heuristics, broadening practical applicability for large-scale distributed optimization.

Abstract

We consider the problem of minimizing the sum of agent functions subject to affine coupling constraints. Dual methods are attractive for such problems because they allow the agent-level subproblems to be solved in parallel. However, achieving primal feasibility with dual methods is a challenge; it can take many iterations to find sufficiently precise prices that recover a primal feasible solution, and even with exact prices primal feasibility is not guaranteed, unless special conditions like strict convexity hold. This behavior can limit the usefulness of dual decomposition methods. To overcome this limitation, we propose a novel primal recovery method, multiple-response agents (MRA), that is able to rapidly reduce primal infeasibility, tolerating some degree of suboptimality, and can be used with any dual algorithm. Rather than returning a single primal response to each price query, MRA requires agents to generate multiple primal responses, each of which has bounded suboptimality. These multiple responses can be computed in parallel, so there is no increase in the wall clock time of the underlying dual algorithm. Then a convex combination of the multiple responses is formed by minimizing the sum of the primal and complementary slackness residuals. We test MRA using both a price localization method and a dual subgradient method and show that it typically converges to a feasible, approximate solution in a few tens of iterations. Moreover, hyperparameters can be tuned to control the trade-off among speed, computational budget, and degree of suboptimality of the feasible solutions returned.

Figures (21)

• Figure 1: Price discovery method followed by MRA.
• Figure 2: Function value suboptimality (top left), primal violations (top right), relative residuals (bottom left), and best-to-date suboptimality of the most recent feasible point (bottom right) versus localization method iterations for the resource allocation problem.
• Figure 3: Function value suboptimality (top left), primal violations (top right), relative residuals (bottom left), and best-to-date suboptimality of the most recent feasible point (bottom right) versus dual subgradient method iterations for the resource allocation problem.
• Figure 4: Function value suboptimality (top left), primal violations (top right), relative residuals (bottom left), and best-to-date suboptimality of the most recent feasible point (bottom right) versus localization method iterations for convex relaxation of the assignment problem.
• Figure 5: Function value suboptimality (top left), primal violations (top right), relative residuals (bottom left), and best-to-date suboptimality of the most recent feasible point (bottom right) versus dual subgradient method iterations for convex relaxation of the assignment problem.

Theorems & Definitions (1)

• Remark 1