698 research outputs found
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
Technical Report: Distributed Asynchronous Large-Scale Mixed-Integer Linear Programming via Saddle Point Computation
We solve large-scale mixed-integer linear programs (MILPs) via distributed
asynchronous saddle point computation. This is motivated by the MILPs being
able to model problems in multi-agent autonomy, e.g., task assignment problems
and trajectory planning with collision avoidance constraints in multi-robot
systems. To solve a MILP, we relax it with a nonlinear program approximation
whose accuracy tightens as the number of agents increases relative to the
number of coupled constraints. Next, we form an equivalent Lagrangian saddle
point problem, and then regularize the Lagrangian in both the primal and dual
spaces to create a regularized Lagrangian that is
strongly-convex-strongly-concave. We then develop a parallelized algorithm to
compute saddle points of the regularized Lagrangian. This algorithm partitions
problems into blocks, which are either scalars or sub-vectors of the primal or
dual decision variables, and it is shown to tolerate asynchrony in the
computations and communications of primal and dual variables. Suboptimality
bounds and convergence rates are presented for convergence to a saddle point.
The suboptimality bound includes (i) the regularization error induced by
regularizing the Lagrangian and (ii) the suboptimality gap between solutions to
the original MILP and its relaxed form. Simulation results illustrate these
theoretical developments in practice, and show that relaxation and
regularization together have only a mild impact on the quality of solution
obtained.Comment: 14 pages, 2 figure
AC OPF in Radial Distribution Networks - Parts I,II
The optimal power-flow problem (OPF) has played a key role in the planning
and operation of power systems. Due to the non-linear nature of the AC
power-flow equations, the OPF problem is known to be non-convex, therefore hard
to solve. Most proposed methods for solving the OPF rely on approximations that
render the problem convex, but that may yield inexact solutions. Recently,
Farivar and Low proposed a method that is claimed to be exact for radial
distribution systems, despite no apparent approximations. In our work, we show
that it is, in fact, not exact. On one hand, there is a misinterpretation of
the physical network model related to the ampacity constraint of the lines'
current flows. On the other hand, the proof of the exactness of the proposed
relaxation requires unrealistic assumptions related to the unboundedness of
specific control variables. We also show that the extension of this approach to
account for exact line models might provide physically infeasible solutions.
Recently, several contributions have proposed OPF algorithms that rely on the
use of the alternating-direction method of multipliers (ADMM). However, as we
show in this work, there are cases for which the ADMM-based solution of the
non-relaxed OPF problem fails to converge. To overcome the aforementioned
limitations, we propose an algorithm for the solution of a non-approximated,
non-convex OPF problem in radial distribution systems that is based on the
method of multipliers, and on a primal decomposition of the OPF. This work is
divided in two parts. In Part I, we specifically discuss the limitations of BFM
and ADMM to solve the OPF problem. In Part II, we provide a centralized version
and a distributed asynchronous version of the proposed OPF algorithm and we
evaluate its performances using both small-scale electrical networks, as well
as a modified IEEE 13-node test feeder
Lagrangian Relaxation for Mixed-Integer Linear Programming: Importance, Challenges, Recent Advancements, and Opportunities
Operations in areas of importance to society are frequently modeled as
Mixed-Integer Linear Programming (MILP) problems. While MILP problems suffer
from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope
to resolve the associated difficulties through decomposition. Due to the
non-smooth nature of Lagrangian dual functions, the coordination aspect of the
method has posed serious challenges. This paper presents several significant
historical milestones (beginning with Polyak's pioneering work in 1967) toward
improving Lagrangian Relaxation coordination through improved optimization of
non-smooth functionals. Finally, this paper presents the most recent
developments in Lagrangian Relaxation for fast resolution of MILP problems. The
paper also briefly discusses the opportunities that Lagrangian Relaxation can
provide at this point in time
A Tutorial on Distributed Optimization for Cooperative Robotics: from Setups and Algorithms to Toolboxes and Research Directions
Several interesting problems in multi-robot systems can be cast in the
framework of distributed optimization. Examples include multi-robot task
allocation, vehicle routing, target protection and surveillance. While the
theoretical analysis of distributed optimization algorithms has received
significant attention, its application to cooperative robotics has not been
investigated in detail. In this paper, we show how notable scenarios in
cooperative robotics can be addressed by suitable distributed optimization
setups. Specifically, after a brief introduction on the widely investigated
consensus optimization (most suited for data analytics) and on the
partition-based setup (matching the graph structure in the optimization), we
focus on two distributed settings modeling several scenarios in cooperative
robotics, i.e., the so-called constraint-coupled and aggregative optimization
frameworks. For each one, we consider use-case applications, and we discuss
tailored distributed algorithms with their convergence properties. Then, we
revise state-of-the-art toolboxes allowing for the implementation of
distributed schemes on real networks of robots without central coordinators.
For each use case, we discuss their implementation in these toolboxes and
provide simulations and real experiments on networks of heterogeneous robots
Randomized Constraints Consensus for Distributed Robust Mixed-Integer Programming
In this paper, we consider a network of processors aiming at cooperatively
solving mixed-integer convex programs subject to uncertainty. Each node only
knows a common cost function and its local uncertain constraint set. We propose
a randomized, distributed algorithm working under asynchronous, unreliable and
directed communication. The algorithm is based on a local computation and
communication paradigm. At each communication round, nodes perform two updates:
(i) a verification in which they check---in a randomized fashion---the robust
feasibility of a candidate optimal point, and (ii) an optimization step in
which they exchange their candidate basis (the minimal set of constraints
defining a solution) with neighbors and locally solve an optimization problem.
As main result, we show that processors can stop the algorithm after a finite
number of communication rounds (either because verification has been successful
for a sufficient number of rounds or because a given threshold has been
reached), so that candidate optimal solutions are consensual. The common
solution is proven to be---with high confidence---feasible and hence optimal
for the entire set of uncertainty except a subset having an arbitrary small
probability measure. We show the effectiveness of the proposed distributed
algorithm using two examples: a random, uncertain mixed-integer linear program
and a distributed localization in wireless sensor networks. The distributed
algorithm is implemented on a multi-core platform in which the nodes
communicate asynchronously.Comment: Submitted for publication. arXiv admin note: text overlap with
arXiv:1706.0048
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