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.
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.
Let
(QP) be an integer quadratic program that consists in minimizing a quadratic function subject to linear constraints. A such problem belongs to the class of
NP-Hard problems. Standard solvers that use a Branch and Bound algorithm can efficiently solve
(QP) in the specific case where its objective function is convex. Thus, to solve
(QP), we choose to reformulate it into an equivalent problem with a convex objective function. Two reformulations are possible: either we reformulate
(QP) into a linear program, or we reformulate it into a convex quadratic program. \par In the first part of this dissertation, we present several linearizations of
(QP), i.e. reformulations into an equivalent program with a linear objective function. Many linearization methods for the quadratic binary programs are known. A natural approach when considering
(QP) is therefore to reformulate it into a quadratic binary program. This can be done by the binary decomposition of each integer variable and then the linearization of each new product of two binary variables. However, this method, that we denote by \texttt{BBL (Binary Binary Linearization)}, leads to a linear program with a large number of variables and constraints. We then present a new approach, \texttt{BIL (Binary Integer Linearization)}, that consists in reformulating
(QP) into a particular quadratic integer program where each quadratic term is the product of an integer variable by a binary variable. As in the \texttt{BBL} approach, the binary variables come from the binary decomposition of the initial integer variables. Then, we linearize the obtained program by replacing each quadratic term by a new real variable and a set of inequalities. As the number of quadratic terms is lower than in the \texttt{BBL} approach, the number of additional variables is reduced. Hence, the obtained integer linear program is significantly smaller in the \texttt{BIL} approach. Moreover, contrary to what one might think, the \texttt{BIL} approach provides a better bound obtained by continuous relaxation than the \texttt{BBL} approach. Each reformulation leads to an integer linear program that is equivalent to
(QP) and that we improve by adding valid inequalities. \par In a second part, we present several quadratic convex reformulations of
(QP), i.e. we reformulate
(QP) into an equivalent program, with a quadratic convex objective function. We first introduce a simple approach to convexify
(QP) that consists in expressing linearly the squares of integer variables using their unary decompositions, and then to convexify with the smallest eigenvalue of the Hessian matrix of
(QP). We call this approach \texttt{NC (Naive Convexification)}. Then, we introduce a new convex reformulation scheme that perturbs the objective function of
(QP) with the linear expression of the products of integer variables, and the equality constraints of
(QP). Then, we show that we can compute, within this scheme, an optimal reformulation of
(QP) in terms of bound obtained by continuous relaxation : the \texttt{IQCR (Integer Quadratic Convex Reformulation)} approach. This reformulation is based on the optimal dual solution of a semi-definite relaxation of
(QP). Moreover, we show that the method \texttt{IQCR} is easily adaptable to mixed-integer programming. This adaptation, that we call \texttt{IQCRs}, also allows us to integrate the inequality constraints of
(QP) into the perturbation of the objective function of our convex reformulation scheme. Then, we present an interesting restriction of the method \texttt{IQCR}, called \texttt{CQCR (Compact Quadratic Convex Reformulation)}. The difference between this last approach and \texttt{IQCR} is that \texttt{CQCR} only uses the linear expression of the integer variable squares to perturb the objective function, while \texttt{IQCR} uses all the products. The interest is that \texttt{CQCR} produces a reformulated problem with a reduced size in comparison with the \texttt{IQCR} approach, what could be profitable experimentally. Finally, we apply these
3 methods \texttt{NC}, \texttt{CQCR} and \texttt{IQCR} to binary quadratic programming. We show that \texttt{NC} and \texttt{CQCR} are equivalent to existing binary convex reformulations. An interesting result is that \texttt{IQCR} is an improvement of existing convexifications for binary quadratic programming.\par In a third part, we design a specific Branch and Bound algorithm based on a projection property of the \texttt{IQCR} approach.\par Finally, we compare the four obtained linearizations and the three obtained quadratic convex reformulations from the computational point of view. For integer programming, computational experiences are carried out with two classes of instances of
(QP), the first having one equality constraint, and the other having one inequality constraint. The results show that most of the considered instances with up to
40 variables can be solved in one hour of CPU time by the \texttt{IQCR} and \texttt{CQCR} approaches. We then test \texttt{IQCR} on binary quadratic programming. The results corroborate that our approach \texttt{IQCR} improves existing convexifications.
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.
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.
We consider problem (QP) of minimizing a quadratic function subject to linear or quadratic constraints. Variables are integer and bounded. This very general problem can model many classical problems in Combinatorial Optimization.A major difference between (QP) and integer linear programs lies in the fact that, in general, its continuous relaxation is an NP-hard optimization problem. To overcome this difficulty, the Convex Quadratic Reformulation approach transforms (QP) into a problem (QP'), equivalent to (QP), but whose continuous relaxation is a convex problem. To compute a global optimal solution to (QP) it becomes possible to solve (QP') by an implicit enumeration algorithm based on continuous convex optimization.We make an overview of recent developments in this approach. We show in particular how positive semidefinite relaxations can be used to build the most interesting equivalent problems (QP'). In the case of binary variables, we give a vision of classical linearization as a special case of Convex Quadratic Reformulation.Finally, we illustrate the generality and the computational efficiency of the Quadratic Convex Reformulation approach for several combinatorial optimization problems.