Comparing SOS and SDP relaxations of sensor network localization

We investigate the relationships between various sum of squares (SOS) and semidefinite programming (SDP) relaxations for the sensor network localization problem. In particular, we show that Biswas and Ye’s SDP relaxation is equivalent to the degree one SOS relaxation of Kim et al. We also show that Nie’s sparse-SOS relaxation is stronger than the edge-based semidefinite programming (ESDP) relaxation, and that the trace test for accuracy, which is very useful for SDP and ESDP relaxations, can be extended to the sparse-SOS relaxation.     

多项式分裂可行问题

本文研究多项式分裂可行问题,即由多项式不等式定义的分裂可行问题,包括凸与非凸、可行与不可行的问题;给出多项式分裂可行问题解集的半定松弛表示;研究其半定松弛化问题的性质;并基于这些性质建立求解多项式分裂可行问题的半定松弛算法.本文在较为一般的条件下证明了,如果分裂可行问题有解,则可通过本文建立的算法求得一个解点;如果问题无解,则该算法能够判别问题不可行.最后通过数值实验对算法进行验证.     

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.     

Relaxing the optimality conditions of box QP

We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the first- and second-order necessary optimality conditions, and establish theoretical relationships between the new relaxations and a basic semidefinite relaxation due to Shor. We compare these relaxations in the context of branch-and-bound to determine a global optimal solution, where it is shown empirically that the new relaxations are significantly stronger than Shor’s. An effective branching strategy is also developed.     

LP and SDP branch-and-cut algorithms for the minimum graph bisection problem: a computational comparison

While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These are also applicable for semidefinite relaxations via the spectral bundle method, which allows to exploit structural properties like sparsity. In order to evaluate the relative strengths of linear and semidefinite approaches for large sparse instances, we set up a common branch-and-cut framework for linear and semidefinite relaxations of the minimum graph bisection problem. It incorporates separation algorithms for valid inequalities of the bisection cut polytope described in a recent study by the authors. While the problem specific cuts help to strengthen the linear relaxation significantly, the semidefinite bound profits much more from separating the cycle inequalities of the cut polytope on a slightly enlarged support. Extensive numerical experiments show that this semidefinite branch-and-cut approach without problem specific cuts is a superior choice to the classical simplex approach exploiting bisection specific inequalities on a clear majority of our large sparse test instances from VLSI design and numerical optimization.     

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates (BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.     

An evaluation of semidefinite programming based approaches for discrete lot-sizing problems

The present work is intended as a first step towards applying semidefinite programming models and tools to discrete lot-sizing problems including sequence-dependent changeover costs and times. Such problems can be formulated as quadratically constrained quadratic binary programs. We investigate several semidefinite relaxations by combining known reformulation techniques recently proposed for generic quadratic binary problems with problem-specific strengthening procedures developed for lot-sizing problems. Our computational results show that the semidefinite relaxations consistently provide lower bounds of significantly improved quality as compared with those provided by the best previously published linear relaxations. In particular, the gap between the semidefinite relaxation and the optimal integer solution value can be closed for a significant proportion of the small-size instances, thus avoiding to resort to a tree search procedure. The reported computation times are significant. However improvements in SDP technology can still be expected in the future, making SDP based approaches to discrete lot-sizing more competitive.     

Contribution of copositive formulations to the graph partitioning problem

This article provides analysis of several copositive formulations of the graph partitioning problem and semidefinite relaxations based on them. We prove that the copositive formulations based on results from Burer [S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479–495] and the author of the paper [J. Povh, Semidefinite approximations for quadratic programs over orthogonal matrices. J. Global Optim. 48 (2010), pp. 447–463] are equivalent and that they both imply semidefinite relaxations which are stronger than the Donath–Hoffman eigenvalue lower bound [W.E. Donath and A.J. Hoffman, Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 (1973), pp. 420–425] and the projected semidefinite lower bound from Wolkowicz and Zhao [H. Wolkowicz and Q. Zhao, Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97 (1999), pp. 461–479].     

Graph partitioning using linear and semidefinite programming

 Graph partition is used in the telecommunication industry to subdivide a transmission network into small clusters. We consider both linear and semidefinite relaxations for the equipartition problem and present numerical results on real data from France Telecom networks with up 900 nodes, and also on randomly generated problems. Received: August 8, 2001 / Accepted: November 9, 2001 Published online: December 9, 2002 RID="★★" ID="★★" This research was carried out while this author was working at France Telecom R & D, 38–40 rue du Général Leclerc, F-92794 Issy-Les-Moulineaux Cedex 9, France. RID="★" ID="★" This author greatfully acknowledges financial support from the Austrian Science Foundation FWF Project P12660-MAT. Key words. graph partitioning – semidefinite programming     

Approximating Pareto curves using semidefinite relaxations

We approximate as closely as desired the Pareto curve associated with bicriteria polynomial optimization problems. We use three formulations (including the weighted sum approach and the Chebyshev approximation) and each of them is viewed as a parametric polynomial optimization problem. For each case is associated a hierarchy of semidefinite relaxations and from an optimal solution of each relaxation one approximates the Pareto curve by solving an inverse problem (first two cases) or by building a polynomial underestimator (third case).     

Approximation algorithms from inexact solutions to semidefinite programming relaxations of combinatorial optimization problems

Semidefinite relaxations of certain combinatorial optimization problems lead to approximation algorithms with performance guarantees. For large-scale problems, it may not be computationally feasible to solve the semidefinite relaxations to optimality. In this paper, we investigate the effect on the performance guarantees of an approximate solution to the semidefinite relaxation for MaxCut, Max2Sat, and Max3Sat. We show that it is possible to make simple modifications to the approximate solutions and obtain performance guarantees that depend linearly on the most negative eigenvalue of the approximate solution, the size of the problem, and the duality gap. In every case, we recover the original performance guarantees in the limit as the solution approaches the optimal solution to the semidefinite relaxation.     

Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition

We propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities. Our approach is based on Lagrangian duality, where the inequalities are dualized, and only a basic set of constraints is maintained explicitly. This leads to function evaluations requiring to solve a relatively simple semidefinite program. Our approach provides accurate solutions to semidefinite relaxations of the Max-Cut and the Equipartition problem, which are not achievable by direct approaches based only on interior-point methods. Received: April, 2004 The last author gratefully acknowledges the support from the Austrian Science Foundation FWF Project P12660-MAT.     

A note on set-semidefinite relaxations of nonconvex quadratic programs

We consider semidefinite, copositive, and more general, set-semidefinite programming relaxations of general nonconvex quadratic problems. For the semidefinite case a comparison between the feasible set of the original program and the feasible set of the relaxation has been given by Kojima and Tunçel (SIAM J Optim 10(3):750–778, 2000). In this paper the comparison is presented for set-positive relaxations which contain copositive relaxations as a special case.     

Stochastic nuclear outages semidefinite relaxations

This paper deals with stochastic scheduling of nuclear power plant outages. Focusing on the main constraints of the problem, we propose a stochastic formulation with a discrete distribution for random variables, that leads to a mixed 0/1 quadratically constrained quadratic program. Then we investigate semidefinite relaxations for solving this hard problem. Numerical results on several instances of the problem show the efficiency of this approach, i.e., the gap between the optimal solution and the continuous relaxation is on average equal to 53.35 % whereas the semidefinite relaxation yields an average gap of 2.76 %. A feasible solution is then obtained with a randomized rounding procedure.     

Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.     

Tightening a copositive relaxation for standard quadratic optimization problems

We focus in this paper the problem of improving the semidefinite programming (SDP) relaxations for the standard quadratic optimization problem (standard QP in short) that concerns with minimizing a quadratic form over a simplex. We first analyze the duality gap between the standard QP and one of its SDP relaxations known as “strengthened Shor’s relaxation”. To estimate the duality gap, we utilize the duality information of the SDP relaxation to construct a graph G ^?. The estimation can be then reduced to a two-phase problem of enumerating first all the minimal vertex covers of G ^? and solving next a family of second-order cone programming problems. When there is a nonzero duality gap, this duality gap estimation can lead to a strictly tighter lower bound than the strengthened Shor’s SDP bound. With the duality gap estimation improving scheme, we develop further a heuristic algorithm for obtaining a good approximate solution for standard QP.     

Solving quadratic (0,1)-problems by semidefinite programs and cutting planes

We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Large parts of this paper were prepared while the author was working at the Christian Doppler Laboratory for Discrete Optimization at Technische Universität Graz.     

Semidefinite relaxations for non-convex quadratic mixed-integer programming

We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.