A Primal Decomposition Method with Suboptimality Bounds for Distributed Mixed-Integer Linear Programming

In this paper we deal with a network of agents seeking to solve in a distributed way Mixed-Integer Linear Programs (MILPs) with a coupling constraint (modeling a limited shared resource) and local constraints. MILPs are NP-hard problems and several challenges arise in a distributed framework, so that looking for suboptimal solutions is of interest. To achieve this goal, the presence of a linear coupling calls for tailored decomposition approaches. We propose a fully distributed algorithm based on a primal decomposition approach and a suitable tightening of the coupling constraints. Agents repeatedly update local allocation vectors, which converge to an optimal resource allocation of an approximate version of the original problem. Based on such allocation vectors, agents are able to (locally) compute a mixed-integer solution, which is guaranteed to be feasible after a sufficiently large time. Asymptotic and finite-time suboptimality bounds are established for the computed solution. Numerical simulations highlight the efficacy of the proposed methodology.Comment: 57th IEEE Conference on Decision and Contro

A Consensus-ADMM Approach for Strategic Generation Investment in Electricity Markets

This paper addresses a multi-stage generation investment problem for a strategic (price-maker) power producer in electricity markets. This problem is exposed to different sources of uncertainty, including short-term operational (e.g., rivals' offering strategies) and long-term macro (e.g., demand growth) uncertainties. This problem is formulated as a stochastic bilevel optimization problem, which eventually recasts as a large-scale stochastic mixed-integer linear programming (MILP) problem with limited computational tractability. To cope with computational issues, we propose a consensus version of alternating direction method of multipliers (ADMM), which decomposes the original problem by both short- and long-term scenarios. Although the convergence of ADMM to the global solution cannot be generally guaranteed for MILP problems, we introduce two bounds on the optimal solution, allowing for the evaluation of the solution quality over iterations. Our numerical findings show that there is a trade-off between computational time and solution quality

A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming

Many problems of interest for cyber-physical network systems can be formulated as Mixed Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithm to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. In the proposed algorithm, at each communication round, agents solve locally a small LP, generate suitable cutting planes, namely intersection cuts and cost-based cuts, and communicate a fixed number of active constraints, i.e., a candidate optimal basis. We prove that, if the cost is integer, the algorithm converges to the lexicographically minimal optimal solution in a finite number of communication rounds. Finally, through numerical computations, we analyze the algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure

Primal decomposition and constraint generation for asynchronous distributed mixed-integer linear programming

In this paper, we deal with large-scale Mixed Integer Linear Programs (MILPs) with coupling constraints that must be solved by processors over networks. We propose a finite-time distributed algorithm that computes a feasible solution with suboptimality bounds over asynchronous and unreliable networks. As shown in a previous work of ours, a feasible solution of the considered MILP can be computed by resorting to a primal decomposition of a suitable problem convexification. In this paper we reformulate the primal decomposition resource allocation problem as a linear program with an exponential number of unknown constraints. Then we design a distributed protocol that allows agents to compute an optimal allocation by generating and exchanging only few of the unknown constraints. Each allocation is iteratively used to compute a candidate feasible solution of the original MILP. We establish finite-time convergence of the proposed algorithm under very general assumptions on the communication network. A numerical example corroborates the theoretical results

Linear programs for resource sharing among heterogeneous agents: the effect of random agent arrivals

We consider a multi-agent resource sharing problem that can be represented by a linear program. The amount of resource to be shared is fixed, and each agent adds to the linear cost and constraint a term that depends on some randomly extracted parameters, thus modelling heterogeneity among agents. We study the probability that the arrival of a new agent does not affect the optimal value and the resource share of the other agents, which means that the system cannot accommodate the request of a further agent and has reached its saturation limit. In particular, we determine the maximum number of requests for the shared resource that the system can accommodate in a probabilistic sense. This result is proven by first formulating the dual of the resource sharing linear program, and then showing that this is a random linear program. Using results from the scenario theory for randomized optimization, we bound the probability of constraint violation for the dual optimal solution, and prove that this is equivalent with the primal optimal value and resource share remaining unchanged upon arrival of a new agent. We discuss how this can be thought of as probabilistic sensitivity analysis and offer an interpretation of this setting in an electric vehicle charging control problem

Tracking-ADMM for Distributed Constraint-Coupled Optimization

We consider constraint-coupled optimization problems in which agents of a network aim to cooperatively minimize the sum of local objective functions subject to individual constraints and a common linear coupling constraint. We propose a novel optimization algorithm that embeds a dynamic average consensus protocol in the parallel Alternating Direction Method of Multipliers (ADMM) to design a fully distributed scheme for the considered set-up. The dynamic average mechanism allows agents to track the time-varying coupling constraint violation (at the current solution estimates). The tracked version of the constraint violation is then used to update local dual variables in a consensus-based scheme mimicking a parallel ADMM step. Under convexity, we prove that all limit points of the agents' primal solution estimates form an optimal solution of the constraint-coupled (primal) problem. The result is proved by means of a Lyapunov-based analysis simultaneously showing consensus of the dual estimates to a dual optimal solution, convergence of the tracking scheme and asymptotic optimality of primal iterates. A numerical study on optimal charging schedule of plug-in electric vehicles corroborates the theoretical results.Comment: 14 pages, 2 figures, submitted to Automatic