We introduce a cutting-plane framework for nonconvex quadratic programs (QPs) that progressively tightens convex relaxations. Our approach leverages the doubly nonnegative (DNN) relaxation to compute strong lower bounds and generate separating cuts, which are iteratively added to improve the relaxation. We establish that, at any Karush-Kuhn-Tucker (KKT) point satisfying a second-order sufficient condition, a valid cut can be obtained by solving a linear semidefinite program (SDP), and we devise a finite-termination local search procedure to identify such points. Extensive computational experiments on both benchmark and synthetic instances demonstrate that our approach yields tighter bounds and consistently outperforms leading commercial and academic solvers in terms of efficiency, robustness, and scalability. Notably, on a standard desktop, our algorithm reduces the relative optimality gap to 0.01% on 138 out of 140 instances of dimension 100 within one hour, without resorting to branch-and-bound.
My work focuses on cutting planes technology in Mixed Integer Programming. I explore novel classes of valid linear inequalities to strengthen linear relaxations of both Linear and Nonlinear Mixed Integer problems. My dissertation consists of three chapters that investigate theoretically and computationally the families of cuts considered. The first chapter is based on joint work with Prof. Pietro Belotti and Prof. Francois Margot. We study linear relaxations of Quadratically Constrained Quadratic Programs. The proposed relaxations are models with both semi-definite constraints (PSD) and linear constraints given by the Reformulation Linearization Technique (RLT). It is known from the literature (Anstreicher, 2007) that PSD and RLT used together yield better bounds than each technique used separately. We adopt a linear outer-approximation of the PSD cone, and we use exclusively linear programming tools to enforce the PSD condition via a cutting plane approach in the lifted space containing the Yij = xixj variables. We study new classes of valid linear inequalities and we test their effectiveness empirically. These include sparse PSD cuts and cuts derived from principal minors. Computational results based on instances from the GLOBALLib and Boxed Constrained Quadratic Programs show that this approach yields better bounds than using solely the standard PSD cuts on top of the RLT inequalities. The C++ code developed for this study has been included in Coin-OR as part of the Couenne project (exact solver for MINLPs). In the second chapter I present a work closely related to the recent developments in the area of \cuts from multiple rows of the simplex tableau" (Andersen et al., 2007). This chapter is based on joint work with Prof. Egon Balas. We generate intersection cuts from lattice-free convex sets as lift-and-project cuts from multiple-term disjunctions. We use the concept of \Disjunctive Hull" defined for a Mixed Integer Program at a fractional vertex v of its linear relaxation P as the convex hull of points in P satisfying all basic disjunctions that cut o v but no integer point. We examine the relationship between the Disjunctive Hull and the Integer Hull and we give procedures to generate inequalities for the Integer Hull derived from the Cut Generating Linear Program associated to the Disjunctive Hull. Strengthening techniques based on coefficient modularization and monoidal strengthening are also discussed. In this chapter we also analyze the case of 0-1 programming which has not been covered in the literature. Our framework applies to this setting with minor changes and produces valid families of cuts for the 0-1 case but invalid for general integer programs. These cuts include the triangle, quadrilateral, split cuts for the MIP case, and cuts from cones and truncated cones for the 0-1 setting. Moreover we run an experiment in which we separate two families of non facet defining cuts: we consider cuts from fixed shape triangles of type 1 and conic disjunctions having the apex at one vertex of the unit cube and two extreme rays, each containing an additional vertex of the cube. In practice, in order to match the strength of the non facet defining inequalities, a large number of facet defining inequalities is typically needed. Triangles of type 1 produce a consistent improvement over the Gomory cuts, as shown in experiments on the MIPLIB 3 0-1 instances. In the third chapter I present a theoretical result on strengthening valid inequalities for Mixed Integer Linear Programs. This chapter is also based on joint work with Prof. Egon Balas. There are two distinct strengthening methods for disjunctive cuts; they differ in the way they modularize the coefficients associated to integer constrained variables. We introduce a new variant of one of these methods, the monoidal cut strengthening procedure (Balas and Jeroslow, 1980), based on the paradox that sometimes weakening a disjunction helps the strengthening procedure and results in sharper cuts. We first derive a general result that applies to cuts from disjunctions with any number of terms. It defines the coefficients of the cut in a way that takes advantage of the option of adding new terms to the disjunction. We then specialize this result to the case of split cuts, in particular Gomory Mixed Integer cuts, and to intersection cuts from multiple rows of a simplex tableau. In both instances we give conditions for the new cuts to have stronger coefficients than the cuts obtained by either of the two currently known strengthening procedures.
The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P∗). This equivalent problem (P∗) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y=xxT, where x are the initial variables of problem (P). We then propose to solve (P∗) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO.
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.
Solving nonconvex quadratic optimization problems to global optimality is a challenging task. For this problem class Gurobi takes a branch-and-bound approach, so the building blocks for the solver are: A relaxation of the problem to solve at the nodes, cutting planes to reduce the feasible region of the relaxation, and heuristics to compute primal feasible solutions. Here we discuss the latter two building blocks in more detail, focussing on so-called PSD cuts, and a heuristic based on an interior point method.
Binary Quadratic Program with Variable Partitioning Constraints The binary quadratic program with variable partitioning constraints is a very general class of optimization problems that is very difficult to solve because of the nonconvexity and integrality of the variables and is ubiquitous, among others, in network design, computer vision, and transportation and logistics. In their article, “A convex reformulation and an outer approximation for a large class of binary quadratic programs,” Rostami et al. show how to transform such a nonconvex challenging problem into a convex bilinear program with decomposable structure. The authors develop a branch-and-cut algorithm based on outer approximation cuts, in which the cuts are generated on the fly by efficiently solving separation subproblems. The results of their computational experiments on different problems confirm the efficacy of the solution methods in solving large-scale problem instances.