In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-eld of real algebraic geometry.
In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-eld of real algebraic geometry.
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 address the problem of optimizing a polynomial with real coefficients over binary variables. We show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem. The number of variables in the latter linearization is only slightly larger than in the former. If polynomial constraints are present in the original problem, then their linearized counterparts carry over to the linearized quadratic problem unchanged. If the original problem formulation does not contain any constraints, we obtain a reduction to unconstrained quadratic zero-one optimization, which is equivalent to the well-studied max-cut problem. The separation problem for general unconstrained polynomial zero-one optimization thus reduces to the separation problem for the cut polytope. This allows us to transfer the entire knowledge gained for the latter polytope by intensive research and, in particular, the sophisticated separation techniques that have been developed. We report preliminary experimental results obtained with a straightforward implementation of this approach.
Abstract. It is well known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show similar results for a wide variety of discrete optimization problems for which SDP relaxations have been derived. Based on a comprehensive study on discrete positive semidefinite matrices, we introduce a generic approach to derive mixed-integer SDP (MISDP) formulations of binary quadratically constrained quadratic programs and binary quadratic matrix programs. Applying a problem-specific approach, we derive more compact MISDP formulations of several problems, such as the quadratic assignment problem, the graph partition problem, and the integer matrix completion problem. We also show that several structured problems allow for novel compact MISDP formulations through the notion of association schemes. Complementary to the recent advances on algorithmic aspects related to MISDP, this work opens new perspectives on solution approaches for the here considered problems.
At the intersection of combinatorial and nonlinear optimization, quadratic programming (QP) plays an important role in practice and optimization theory.With wide-spread real-world applications including, but not limited to, engineering design, game theory, and economics, quadratic programming has many valuable contributions and has attracted considerable interest over the past few decades.As a more general formulation, quadratically constrained quadratic programming (QCQP) problems arise naturally from a lot of applications such as facility location, production planning, economic equilibria and Euclidean distance geometry, etc.Most global optimization approaches for nonconvex QCQPs are based on constructing a convex or linear relaxation of the problem, and embedding it within an exhaustive search mechanism, where a lower bound is calculated as the value of the objective function of this relaxation (for a minimization problem) at each iteration of the search tree.The two principle elements affecting the overall effectiveness of such exhaustive search methods are the tightness of the underlying relaxation and the efficiency of calculating the lower bound.Besides, the performance of a branch-and-bound algorithm also relies on the procedure used to select a branching node and branching variable, which leads to good quality feasible solutions or equivalently an upper bound.vii LIST OF TABLES 5.5 Computational results of 0-1 QPs with n = 40 . . . . . . . . . . . .110 5.6 Computational results of 0-1 QPs with n = 50 . . . . . . . . . . . .
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.
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.