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Optimization

A note on approximating quadratic programming with rank constraint

Shenglong Hu, Zheng‐Hai Huang

Published Nov 12, 2010• 0 citations
Publisher

Abstract

The quadratic maximization problem with rank constraint means where n and m are two given positive integers with n ≤ m, Q is a given real symmetric matrix, is the unit sphere in and stands for the Euclidean inner product. In this note, motivated by Briët, Filho and Vallentin's and Ye's excellent works [J. Briët, F. Filho, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.; Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226], we first derive the relative approximation ratio for the above problem with arbitrary symmetric matrix Q. While Q was supposed to be positive semidefinite in [J. Briët, F. Filhom, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.]. Secondly, we extend such a relative approximation ratio to a generalization of the above optimization problem. Finally, a slightly improved relative approximation ratio for the optimization problem considered in [Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226] is presented.

DOI
10.1080/02331934.2010.523111

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The quadratic maximization problem with rank constraint means where n and m are two given positive integers with n ≤ m, Q is a given real symmetric matrix, is the unit sphere in and stands for the Euclidean inner product. In this note, motivated by Briët, Filho and Vallentin's and Ye's excellent works [J. Briët, F. Filho, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.; Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226], we first derive the relative approximation ratio for the above problem with arbitrary symmetric matrix Q. While Q was supposed to be positive semidefinite in [J. Briët, F. Filhom, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.]. Secondly, we extend such a relative approximation ratio to a generalization of the above optimization problem. Finally, a slightly improved relative approximation ratio for the optimization problem considered in [Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226] is presented.
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.
This paper considers the following type of quadratic programming problem. Given an arbitrary matrix A, whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ such that x/sup T/Ax is maximized. Our approximation algorithm for this problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in /spl Omega/(1/ logn). This quadratic programming problem can be seen as an extension to that of maximizing x/sup T/Ay (where y's components are also /spl plusmn/1). Grothendieck's inequality states that the ratio of the optimum value of the latter problem to the optimum of its canonical semidefinite relaxation is bounded below by a constant. The study of this type of quadratic program arose from a desire to approximate the maximum correlation in correlation clustering. Nothing substantive was known about this problem; we present an /spl Omega/ (1/logn) approximation, based on our quadratic programming algorithm. We can also guarantee that our quadratic programming algorithm returns a solution to the MAXCUT problem that has a significant advantage over a random assignment.
We consider the problem of approximating Quadratic O-1 Integer Programs with bounded number of constraints and non-negative constraint matrix entries, which we term as PIQP. We describe and analyze a randomized algorithm based on a program with hyperbolic constraints (a Second-Order Cone Programming -SOCP- formulation) that achieves an approximation ratio of O(amaxnβ(n))O(a_{max} \frac{n}{β(n)})O(amax​β(n)n​), where amaxa_{max}amax​ is the maximum size of an entry in the constraint matrix and $β(n) \leq \min_i{W_i} $, where WiW_iWi​ are the constant terms that define the constraint inequalities. We note that by appropriately choosing β(n)β(n)β(n) the randomized algorithm, when combined with other algorithms that achieve good approximations for smaller values of $ W_i$, allows better algorithms for the complete range of WiW_iWi​. This, together with a greedy algorithm, provides a O∗(amaxn1/2)O^*(a_{max} n^{1/2} )O∗(amax​n1/2) factor approximation, where O∗O^*O∗ hides logarithmic terms. Our solution is achieved by a randomization of the optimal solution to the relaxed version of the hyperbolic program. We show that this solution provides the approximation bounds using concentration bounds provided by Chernoff-Hoeffding and Kim-Vu.
Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constraint is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 111 and we give two applications: approximating ground states in the nnn-vector model in statistical mechanics and XOR games in quantum information theory.
Every quadratic programming problem with a mix of continuous and binary variables can be equivalently reformulated as a completely positive optimization problem, that is, a linear optimization problem over the convex but computationally intractable cone of completely positive matrices. In this paper, we focus on general inner approximations of the cone of completely positive matrices on instances of completely positive optimization problems that arise from the reformulation of mixed binary quadratic programming problems. We provide a characterization of the feasibility of such an inner approximation as well as the optimal value of a feasible inner approximation. In particular, our results imply that polyhedral inner approximations are equivalent to a finite discretization of the feasible region of the original completely positive optimization problem. Our characterization yields, as a byproduct, an upper bound on the gap between the optimal value of an inner approximation and that of the original instance. We discuss the implications of this error bound for standard and box-constrained quadratic programs as well as general mixed binary quadratic programs with a bounded feasible region.
This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ that maximizes x/sup T/Mx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 - /spl epsi/ for all /spl epsi/ > 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log/sup /spl gamma// n)for some /spl gamma/ > 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be /spl Theta/(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is /spl Omega/ (log n/log log n), essentially answering one of the open problems of Alon et al. [AMMN].