6 research outputs found
Distributed Submodular Minimization over Networks: a Greedy Column Generation Approach
Submodular optimization is a special class of combinatorial optimization
arising in several machine learning problems, but also in cooperative control
of complex systems. In this paper, we consider agents in an asynchronous,
unreliable and time-varying directed network that aim at cooperatively solving
submodular minimization problems in a fully distributed way. The challenge is
that the (submodular) objective set-function is only partially known by agents,
that is, each one is able to evaluate the function only for subsets including
itself. We propose a distributed algorithm based on a proper linear programming
reformulation of the combinatorial problem. Our algorithm builds on a column
generation approach in which each agent maintains a local candidate basis and
locally generates columns with a suitable greedy inner routine. A key
interesting feature of the proposed algorithm is that the pricing problem,
which involves an exponential number of constraints, is solved by the agents
through a polynomial time greedy algorithm. We prove that the proposed
distributed algorithm converges in finite time to an optimal solution of the
submodular minimization problem and we corroborate the theoretical results by
performing numerical computations on instances of the -- minimum graph
cut problem.Comment: 12 pages, 4 figures, 57th IEEE Conference on Decision and Contro
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
Generalized Assignment for Multi-Robot Systems via Distributed Branch-And-Price
In this paper, we consider a network of agents that has to self-assign a set
of tasks while respecting resource constraints. One possible formulation is the
Generalized Assignment Problem, where the goal is to find a maximum payoff
while satisfying capability constraints. We propose a purely distributed
branch-and-price algorithm to solve this problem in a cooperative fashion.
Inspired by classical (centralized) branch-and-price schemes, in the proposed
algorithm each agent locally solves small linear programs, generates columns by
solving simple knapsack problems, and communicates to its neighbors a fixed
number of basic columns. We prove finite-time convergence of the algorithm to
an optimal solution of the problem. Then, we apply the proposed scheme to a
generalized assignment scenario in which a team of robots has to serve a set of
tasks. We implement the proposed algorithm in a ROS testbed and provide
experiments for a team of heterogeneous robots solving the assignment problem
Distributed Random Convex Programming via Constraints Consensus
This paper discusses distributed approaches for the solution of random convex
programs (RCP). RCPs are convex optimization problems with a (usually large)
number N of randomly extracted constraints; they arise in several applicative
areas, especially in the context of decision under uncertainty, see [2],[3]. We
here consider a setup in which instances of the random constraints (the
scenario) are not held by a single centralized processing unit, but are
distributed among different nodes of a network. Each node "sees" only a small
subset of the constraints, and may communicate with neighbors. The objective is
to make all nodes converge to the same solution as the centralized RCP problem.
To this end, we develop two distributed algorithms that are variants of the
constraints consensus algorithm [4],[5]: the active constraints consensus (ACC)
algorithm, and the vertex constraints consensus (VCC) algorithm. We show that
the ACC algorithm computes the overall optimal solution in finite time, and
with almost surely bounded communication at each iteration. The VCC algorithm
is instead tailored for the special case in which the constraint functions are
convex also w.r.t. the uncertain parameters, and it computes the solution in a
number of iterations bounded by the diameter of the communication graph. We
further devise a variant of the VCC algorithm, namely quantized vertex
constraints consensus (qVCC), to cope with the case in which communication
bandwidth among processors is bounded. We discuss several applications of the
proposed distributed techniques, including estimation, classification, and
random model predictive control, and we present a numerical analysis of the
performance of the proposed methods. As a complementary numerical result, we
show that the parallel computation of the scenario solution using ACC algorithm
significantly outperforms its centralized equivalent
A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization
In this paper we consider a general problem set-up for a wide class of convex
and robust distributed optimization problems in peer-to-peer networks. In this
set-up convex constraint sets are distributed to the network processors who
have to compute the optimizer of a linear cost function subject to the
constraints. We propose a novel fully distributed algorithm, named
cutting-plane consensus, to solve the problem, based on an outer polyhedral
approximation of the constraint sets. Processors running the algorithm compute
and exchange linear approximations of their locally feasible sets.
Independently of the number of processors in the network, each processor stores
only a small number of linear constraints, making the algorithm scalable to
large networks. The cutting-plane consensus algorithm is presented and analyzed
for the general framework. Specifically, we prove that all processors running
the algorithm agree on an optimizer of the global problem, and that the
algorithm is tolerant to node and link failures as long as network connectivity
is preserved. Then, the cutting plane consensus algorithm is specified to three
different classes of distributed optimization problems, namely (i) inequality
constrained problems, (ii) robust optimization problems, and (iii) almost
separable optimization problems with separable objective functions and coupling
constraints. For each one of these problem classes we solve a concrete problem
that can be expressed in that framework and present computational results. That
is, we show how to solve: position estimation in wireless sensor networks, a
distributed robust linear program and, a distributed microgrid control problem.Comment: submitted to IEEE Transactions on Automatic Contro
Locally Constrained Decision Making via Two-Stage Distributed Simplex
In this paper we propose a distributed algorithm for solving linear programs with combinations of local and global constraints in a multi-agent setup. A fully distributed and asynchronous algorithm is proposed. The computation of the local decision makers involves the solution of two distinct (local) optimization problems, namely a local copy of a global linear program and a smaller problem used to generate ”problem columns”. We show that, when running the proposed algorithm, all decision makers agree on a common optimal solution, even if the original problem has several optimal solutions, or detect unboundedness and infeasibility if necessary