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

SEMIDEFINITE PROGRAMMING BASED ALGORITHMS FOR THE SPARSEST CUT PROBLEM

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all tested instances. The semidefinite relaxation gives a lower bound C/W and each heuristic produces a cut S with a ratio c(S)/omega(S) where either cs is at most a factor of C or omega(S) is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a. known semidefinite programming solver.45275100Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Estudo prático de regularidade de problemas de programação semidefinida

Mestrado em Matemática e Aplicações - Matemática Empresarial e TecnológicaUm problema linear de Programa c~ao Semide nida (SDP) consiste na minimiza c~ao de uma fun c~ao linear sujeita a condi c~ao de que a fun c~ao matricial linear seja semide nida. Um problema de SDP considera-se regular se certas condi c~oes est~ao satisfeitas. H a diferentes caracteriza c~oes de regularidade de um problema, sendo uma delas a veri ca c~ao da condi c~ao de Slater. Os problemas regulares de SDP t^em sido estudados e as condi c~oes de optimalidade para estes problemas t^em a forma de teoremas cl assicos do tipo Karush-Kuhn-Tucker, e s~ao facilmente veri cadas. Na pr atica, e frequente encontrar problemas n~ao regulares. O estudo destes problemas e bem mais complicado. Por isso, tem surgido o interesse em estudar e testar a regularidade dos problemas de SDP e deduzir condi c~oes de optimalidade e m etodos de resolu c~ao dos problemas n~ao regulares. Em Kostyukova e Tchemisova [32] e proposto um algoritmo, chamado Algoritmo DIIS (Algorithm of Determination of the Immobile Index Subspace), que permite veri car se as restri c~oes de um dado problema de SDP satisfazem a condi c~ao de Slater. A teoria que serve de base a constru c~ao deste algoritmo assenta nas no c~oes de ndices e subespa co de ndices im oveis, originalmente usadas em Programa c~ao Semi-In nita (SIP), e transpostas em [32] para SDP. Este algoritmo constr oi uma matriz b asica do subespa co de ndices im oveis, caso a condi c~ao de Slater n~ao seja veri cada. A dimens~ao desta matriz caracteriza o grau de n~ao regularidade do problema. O objectivo deste trabalho e estudar o Algoritmo DIIS, implement a- -lo e test a-lo usando v arios problemas de teste de diferentes bases de dados de problemas de SDP. O Algoritmo DIIS foi implementado e executado a partir do MatLab e os testes num ericos efectuados permitiram concluir que o programa constru do veri ca com sucesso a maioria dos problemas teste. Al em disso, o algoritmo permite caracterizar o grau de n~ao regularidade dos problemas de SDP e pode ser usado para constru c~ao de algoritmos de resolu c~ao dos problemas de SDP n~ao regulares.A linear problem of Semide nite Programming (SDP) consists of minimizing a linear function subject to the constraint that the linear matrix function is semide nite. An SDP problem is considered regular if certain conditions are satis- ed. There are several characterizations of a problem regularity, one of which is checking the Slater condition. The regular SDP problems have been studied and the optimality conditions for these problems have the form of Karush-Kuhn-Tucker type theorems, and are easily veri ed. In practice it is common to nd problems that are not regular. The study of these problems is far more complicated. Therefore, there has been interest in studying and testing the regularity of the problems and deduct the SDP optimality conditions and methods for solving non-regular problems. In Kostyukova e Tchemisova [32] is proposed an algorithm, called Algorithm DIIS (Algorithm of Determination of the Immobile Index Subspace), which allows to check if the constraints of a given SDP problem satisfy the Slater condition. The theory that underlies the construction of this algorithm is based on the notions of subspace of immobile indices and immobile indices properties, originally used in Semi-In nite Programming (SIP), and implemented in [32] for SDP. This algorithm constructs a basic matrix of the subspace of immobile indices if the Slater condition is not veri ed. The size of this matrix characterizes the degree of non-regularity of the problem. The purpose of this work is to study the DIIS algorithm, implement and test it using several test problems of di erent databases of SDP problems. The DIIS algorithm was implemented and executed from MatLab and numerical tests carried out showed that the program checks successfully the majority of test problems. Moreover, the algorithm allows to characterize the degree of non-regular problems of SDP and can be used to construct algorithms for solving non-regular SDP problems