Binary quadratic programming (BQP) problem was an NP-hard problem and had a large number of applications.In this paper, a new relaxation method, that was doubly nonnegative relaxation, was proposed for solving BQP problem.Moreover, we prove that the doubly nonnegative relaxation for BQP is equivalent to a new tighter semidifinite relaxation.When BQP problem reduces to densest k-subgraph problem, the doubly nonnegative relaxation is equivalent to a tighter semidifinite relaxation.Finally, some comparative numerical results are reported to show that the efficiency of the doubly nonnegative relaxation is more promising than that of semidefinite relaxation for solving some specific BQP problems.
Binary quadratic programming (BQP) problem was an NP-hard problem and had a large number of applications.In this paper, a new relaxation method, that was doubly nonnegative relaxation, was proposed for solving BQP problem.Moreover, we prove that the doubly nonnegative relaxation for BQP is equivalent to a new tighter semidifinite relaxation.When BQP problem reduces to densest k-subgraph problem, the doubly nonnegative relaxation is equivalent to a tighter semidifinite relaxation.Finally, some comparative numerical results are reported to show that the efficiency of the doubly nonnegative relaxation is more promising than that of semidefinite relaxation for solving some specific BQP problems.
It has recently been shown (Burer, Math. Program Ser. A 120:479-495, 2009) that a large class of NP-hard nonconvex quadratic programming problems can be modeled as so called completely positive programming problems, which are convex but still NP-hard in general. A basic tractable relaxation is gotten by doubly nonnegative relaxation, resulting in a doubly nonnegative programming. In this paper, we prove that doubly nonnegative relaxation for binary quadratic programming (BQP) problem is equivalent to a tighter semidifinite relaxation for it. When problem (BQP) reduces to max-cut (MC) problem, doubly nonnegative relaxation for it is equivalent to the standard semidifinite relaxation. Furthermore, some compared numerical results are reported.
Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spec-tral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high for large scale prob-lems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, com-pared with conventional SDP methods, the new SDP for-mulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmen-tation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs. 1.
In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.
In this thesis, we consider a special class of binary quadratic programming problem (BQP) where the number of nonzero elements is fixed. Such problems arise frequently from various applications and have been proved to be NP-hard. After a brief review of the quadratic programming problem, several optimization algorithms are presented. In Chapter 3, we propose a new simple second order conic relaxation of the BQP problem. We derive some additional constraints based on the information from the data matrix. The algorithm will be compared with the existing SDP relaxation algorithm in terms of their numerical performances. In Chapter 4, we use the convex quadratic relaxation as a geometric embedding tool to reformulate the underlying BQP as a clustering problem, where the target is to find a single cluster of fixed size. This connection allows us to employ many effective clustering algorithm developed in the data mining field. A 2-approximation algorithm for the clustering problem is presented. Numerical results based on the new relaxation model and the proposed algorithm are reported. The last Chapter mainly discusses some theoretical results we put forward on the derived clustering problem. Core-set technique is used to derive a new algorithm, which can provide a (1+\epsilon) approximation ratio to the reformulated clustering problem.
Binary Quadratic Programs (BQPs) are a class of NP-hard problems that arise in a wide range of applications, including finance, machine learning, and logistics. These problems are challenging to solve due to the combinatorial search space and nonlinearity. In fact, this class of optimization problems is so challenging that, in many instances, standard algorithms struggle to identify feasible solutions within a reasonable time. Primal heuristic algorithms have been developed to quickly identify feasible solutions to BQPs. In this paper, we propose Cover-Relax-Search, an efficient primal heuristic for BQPs. This approach is inspired by multiple local search algorithms, including Undercover. We evaluate the \emph{Cover-Relax-Search} algorithm on multiple BQP benchmarks and show that our proposed heuristic identifies high-quality solutions at a faster speed and significantly reduces the primal integral compared to state-of-the-art solvers and other local search baselines.
We consider the problem of handling binary constraints in optimization problems. We review methods for solving binary quadratic programs (BQPs), such as the spectral method and semidefinite programming relaxations. We then discuss two new methods for handling these constraints. The first involves the introduction of an extra unconstrained variable along with the alternating direction method of multipliers (ADMM), which has now also appeared in other recent literature. This allows the effect of the binary variables to be decoupled when they are minimized over. The second involves rewarding the one-norm while restricting the infinity-norm, based on a reformulation of the original problem. The piecewise linearity of the negative penalty results in the problem being convex until it hits a critical point, at which point the parameters of this linear term can be changed. These two methods can be applied to any problems which are convex except for binary constraints. In addition to testing them on BQPs, we show the efficacy of these approaches on point segmentation and image segmentation problems.