Semidefinite programming relaxations for semialgebraic problems

 A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields. Received: May 10, 2001 / Accepted May 2002 Published online: April 10, 2003 Key Words. semidefinite programming – convex optimization – sums of squares – polynomial equations – real algebraic geometry The majority of this research has been carried out while the author was with the Control & Dynamical Systems Department, California Institute of Technology, Pasadena, CA 91125, USA.     

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.     

Semidefinite programming relaxations for graph coloring and maximal clique problems

The semidefinite programming formulation of the Lovász theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number χ(G) or the clique number ω(G) of a graph, but also leads to heuristics for graph coloring and extracting large cliques. This semidefinite programming formulation can be tightened toward either χ(G) or ω(G) by adding several types of cutting planes. We explore several such strengthenings, and show that some of them can be computed with the same effort as the theta number. We also investigate computational simplifications for graphs with rich automorphism groups. Partial support by the EU project Algorithmic Discrete Optimization (ADONET), MRTN-CT-2003-504438, is gratefully acknowledged.     

LP relaxations for a class of linear semi-infinite programming problems

In this paper, we consider a subclass of linear semi-infinite programming problems whose constraint functions are polynomials in parameters and index sets are polyhedra. Based on Handelman’s representation of positive polynomials on a polyhedron, we propose two hierarchies of LP relaxations of the considered problem which respectively provide two sequences of upper and lower bounds of the optimum. These bounds converge to the optimum under some mild assumptions. Sparsity in the LP relaxations is explored for saving computational time and avoiding numerical ill behaviors.     

Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.     

Solving polynomial least squares problems via semidefinite programming relaxations

A polynomial optimization problem whose objective function is represented as a sum of positive and even powers of polynomials, called a polynomial least squares problem, is considered. Methods to transform a polynomial least square problem to polynomial semidefinite programs to reduce degrees of the polynomials are discussed. Computational efficiency of solving the original polynomial least squares problem and the transformed polynomial semidefinite programs is compared. Numerical results on selected polynomial least square problems show better computational performance of a transformed polynomial semidefinite program, especially when degrees of the polynomials are larger.     

Semidefinite relaxations of ordering problems

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element $u$ is before $v$ in the ordering. In the second case, the profit depends on whether $u$ is before $v$ and $r$ is before $s$ . The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.     

Semidefinite programming

Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in widespread use even before the development of efficient algorithms brought it into the realm of tractability. Today it is one of the basic modeling and optimization tools along with linear and quadratic programming. Our survey is an introduction to semidefinite programming, its duality and complexity theory, its applications and algorithms.     

On solving a class of fractional semi-infinite polynomial programming problems

Computational Optimization and Applications - In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex...     

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.     

Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts

In this paper, we propose to enhance Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations for polynomial programming problems by developing cutting plane strategies using concepts derived from semidefinite programming. Given an RLT relaxation, we impose positive semidefiniteness on suitable dyadic variable-product matrices, and correspondingly derive implied semidefinite cuts. In the case of polynomial programs, there are several possible variants for selecting such particular variable-product matrices on which positive semidefiniteness restrictions can be imposed in order to derive implied valid inequalities. This leads to a new class of cutting planes that we call v-semidefinite cuts. We explore various strategies for generating such cuts, and exhibit their relative effectiveness towards tightening the RLT relaxations and solving the underlying polynomial programming problems in conjunction with an RLT-based branch-and-cut scheme, using a test-bed of problems from the literature as well as randomly generated instances. Our results demonstrate that these cutting planes achieve a significant tightening of the lower bound in contrast with using RLT as a stand-alone approach, thereby enabling a more robust algorithm with an appreciable reduction in the overall computational effort, even in comparison with the commercial software BARON and the polynomial programming problem solver GloptiPoly.     

Semidefinite resolution and exactness of semidefinite relaxations for satisfiability

This paper explores new connections between the satisfiability problem and semidefinite programming. We show how the process of resolution in satisfiability is equivalent to a linear transformation between the feasible sets of the relevant semidefinite programming problems. We call this transformation semidefinite programming resolution, and we demonstrate the potential of this novel concept by using it to obtain a direct proof of the exactness of the semidefinite formulation of satisfiability without applying Lasserre’s general theory for semidefinite relaxations of 0/1 problems. In particular, our proof explicitly shows how the exactness of the semidefinite formulation for any satisfiability formula can be interpreted as the implicit application of a finite sequence of resolution steps to verify whether the empty clause can be derived from the given formula.     

Semidefinite relaxations for partitioning,assignment and ordering problems

This paper introduces polymorphic ejection chains, and applies them to the problem of repairing time assignments in high school timetables while preserving regularity. An ejection chain is a sequence of repairs, each of which removes a defect introduced by the previous repair. Just as the elements of a polymorphic list may have different types, so in a polymorphic ejection chain the individual repairs may have different types. Methods for the efficient realization of these ideas, implemented in the author’s KHE framework, are given, and some initial experiments are presented.     

Semidefinite relaxations for partitioning, assignment and ordering problems

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.     

An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem:

Interior-point algorithms for semi-infinite programming

In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.Research supported in part by NSF, AFORS and ONR through NSF grant DMS-8920550.     

AnUV-algorithm for semi-infinite multiobjective programming

In this paper, we give an application ofUV-decomposition method of convex programming to multiobjective programming, and offer a new algorithm for solving semi-infinite multiobjective programming. Finally, the superlinear convergence of the algorithm is proved.     

Semidefinite programming for the educational testing problem

Methods for solving the educational testing problem which arises from statistics are considered. The problem is to find lower bounds for the reliability of the total score on a test (or subtests) whose items are not parallel using data from a single test administration. We formulate the problem as an optimization problem with a linear objective function and semidefinite constraints. We maintain exact primal and dual feasibility during the course of the algorithm. The search direction is found using an inexact Gauss–Newton method rather than a Newton method on a symmetrized system. Computational results illustrating the robustness of the algorithm are successfully exploited. Research supported by King Fahd University of Petroleum and Minerals under Project FT/2005–2007.