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Princeton University Press eBooks

On approximation algorithms for concave quadratic programming

Stephen A. Vavasis

Published Dec 31, 1991• 18 citations
Publisher

Abstract

We consider $\epsilon$-approximation schemes for concave quadratic programming. Because the existing definition of $\epsilon$-approximation for combinatorial optimization problems is inappropriate for nonlinear optimization, we propose a new definition for $\epsilon$-approximation. We argue that such an approximation can be found in polynomial time for fixed $\epsilon$ and $k$, where $k$ denotes the number of negative eigenvalues. Our algorithm is polynomial in 1/$\epsilon$ for fixed $k$, and superexponential in $k$ for fixed $\epsilon$.

DOI
10.1515/9781400862528.3

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We consider ϵ\epsilonϵ-approximation schemes for concave quadratic programming. Because the existing definition of ϵ\epsilonϵ-approximation for combinatorial optimization problems is inappropriate for nonlinear optimization, we propose a new definition for ϵ\epsilonϵ-approximation. We argue that such an approximation can be found in polynomial time for fixed ϵ\epsilonϵ and kkk, where kkk denotes the number of negative eigenvalues. Our algorithm is polynomial in 1/ϵ\epsilonϵ for fixed kkk, and superexponential in kkk for fixed ϵ\epsilonϵ.
We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by Δ\DeltaΔ the largest absolute value of the subdeterminants of the constraint matrix. In this paper we give an algorithm that finds an ϵ\epsilonϵ-approximate solution for this problem by solving a number of integer linear programs whose constraint matrices have subdeterminants bounded by Δ\DeltaΔ in absolute value. The number of these integer linear programs is polynomial in the dimension nnn, in Δ\DeltaΔ, and in 1/ϵ1/\epsilon1/ϵ, provided that the number kkk of variables that appear nonlinearly in the objective is fixed. As a corollary, we obtain the first polynomial-time approximation algorithm for separable concave integer quadratic programming with Δ≤2\Delta \le 2Δ≤2 and kkk fixed. In the totally unimodular case Δ=1\Delta = 1Δ=1, we give an improved algorithm that only needs to solve a number of linear programs that is polynomial in 1/ϵ1/\epsilon1/ϵ and is independent of nnn, provided that kkk is fixed.
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The technique of semidefinite programming (SDP) relaxation can be used to obtain a nontrivial bound on the optimal value of a nonconvex quadratically constrained quadratic program (QCQP). We explore concave quadratic inequalities that hold for any vector in the integer lattice Zn{\bf Z}^nZn, and show that adding these inequalities to a mixed-integer nonconvex QCQP can improve the SDP-based bound on the optimal value. This scheme is tested using several numerical problem instances of the max-cut problem and the integer least squares problem.
We consider semidefinite relaxations of a quadratic optimization problem with polynomial constraints. This is an extension of quadratic problems with Boolean variables. Such combinatorial problems cannot, in general, be solved in polynomial time. Semidefinite relaxation has been proposed as a promising technique to give provable good bounds on certain Boolean quadratic problems in polynomial time. We formulate the extensions from Boolean variables to quarternary variables using (i) a polynomial relaxation or (ii) standard semidefinite relaxations of a linear transformation of Boolean variables. We compare the two different relaxation approaches analytically. The relaxations can all be expressed as semidefinite programs, which can be solved efficiently using e.g. interior point methods. Applications of our results include maximum likelihood estimation in communication systems, which we explore in simulations in order to compare the quality of the different relaxations with optimal solutions.
Quadratic optimization problems with complex-valued decision variables, in short called complex quadratic optimization problems, find many applications in engineering. In this dissertation, we study several instructive models of complex quadratic optimization, as well as its applications in combinatorial optimization. The tool that we use is a combination of semidefinite programming (SDP) relaxation and randomization technique, which has been well exploited in the last decade. Since most of the optimization models are NP-hard in nature, we shall design polynomial time approximation algorithms for a general model, or polynomial time exact algorithms for some restricted instances of a general model. To enable the analysis, we first develop two basic theoretical results: one is the probability formula for a bivariate complex normal distribution vector to be in a prescribed angular region, and the other one is the rank-one decomposition theorem for complex positive semidefinite matrices. The probability formula enables us to compute the expected value of a randomized (with a specific rounding rule) solution based on an optimal solution of the SDP relaxation problem, while the rank-one decomposition theorem provides a new proof of the complex S -lemma and leads to novel deterministic rounding procedures. With the above results in hand, we then investigate the models and applications of complex quadratic optimization via semidefinite programming in detail. We present an approximation algorithm for a convex quadratic maximization problem with discrete complex decision variables, where the approximation analysis is based on the probability formula. Besides, an approximation algorithm is proposed for a non-convex quadratic maximization problem with discrete complex decision variables. Then we study a limit of the model, i.e., a quadratic maximization problem with continuous unit module decision variables. The problem is shown to be strongly NP-hard. Approximation algorithms are described for the problem, including both convex case and non-convex case. Furthermore, if the objective matrix has a sign structure, then a stronger approximation result is shown to hold. In addition, we use the complex matrix decomposition theorem to solve complex quadratically constrained complex quadratic programming. We consider several interesting cases for which the corresponding SDP relaxation admits no gap with the true optimal value, and consequently, this yields polynomial time procedures for solving those special cases of complex quadratic optimization. Finally, as combinatorial applications of complex quadratic optimization; we consider Max 3-Cut with fixed nodes constraints, Max 3-Dicut with weight constraints, Max 3-XOR, and so on, and present corresponding bounds on the approximation ratios.