Multiple Approximate-Response Agents (MARA): Fast Near-Optimal Primal Recovery for Distributed Optimization

Abstract

Dual methods are useful for distributed optimization because they allow 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 prices that recover primal feasibility, and even with optimal dual prices primal feasibility is not guaranteed unless special conditions like strict convexity hold. To address this limitation, we propose a simple primal recovery method, multiple approximate-response agents (MARA), that is able to rapidly reduce primal infeasibility, tolerating some degree of suboptimality. The method is agnostic to how dual prices are computed, so MARA can be applied to enhance any dual algorithm. Rather than returning a single primal response to each price query, MARA requires agents to generate multiple primal responses, each of which has bounded suboptimality. Because these multiple responses can be computed in parallel, there is no increase in the wall-clock time of the underlying dual algorithm. MARA then constructs a convex combination of the multiple responses by minimizing the sum of the primal and complementary slackness residuals to produce a high-quality primal solution. Tests of MARA using both a price localization method and a dual subgradient method show that it typically converges to a feasible, near-optimal solution in a few tens of iterations. Moreover, hyperparameters of MARA can be flexibly tuned to control the trade-off among speed, computational budget, and degree of suboptimality of the feasible solutions.