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Dimensionality reduction for binary quadratic programming problems

Rabih Battikh, Bilal Hammoud, Hassan Alabboud, Danny El Kass, Adnan Yassine

Published Aug 5, 2025• 0 citations
Publisher

Abstract

Abstract This paper addresses the constrained binary quadratic problem (QP). This problem consists in minimizing a quadratic function a binary variables (0-1 variables) with linear constraints. Accordingly, in this work, We have generalized reliable fixing criteria (F0) and (F1) for the (QP) problem. The purpose of these fixing criteria is to facilitate the resolution of the initial problem (QP) where through these criteria we can decrease the dimension of the (QP) problem when it is possible and sometimes we can solve this problem completely by using a repeat loop based on these criteria (QPFA algorithm). Numerical results are presented to consolidate the demonstrated theoretical results and prove efficiency and performance in terms of speed, quality and robustness of our work. Mathematics Subject Classification: 90C20 , 90C27 , 90C10

DOI
10.21203/rs.3.rs-7273186/v1

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Abstract This paper addresses the constrained binary quadratic problem (QP). This problem consists in minimizing a quadratic function a binary variables (0-1 variables) with linear constraints. Accordingly, in this work, We have generalized reliable fixing criteria (F0) and (F1) for the (QP) problem. The purpose of these fixing criteria is to facilitate the resolution of the initial problem (QP) where through these criteria we can decrease the dimension of the (QP) problem when it is possible and sometimes we can solve this problem completely by using a repeat loop based on these criteria (QPFA algorithm). Numerical results are presented to consolidate the demonstrated theoretical results and prove efficiency and performance in terms of speed, quality and robustness of our work. Mathematics Subject Classification: 90C20 , 90C27 , 90C10
The unconstrained binary quadratic programming problem (QP) is a classical non-linear problem of optimizing a quadratic objective by a suitable choice of binary decision variables. This paper proposes new Lagrangean decompositions to find bounds for QP. The methods presented treat a mixed binary linear version (LQP) of QP with constraints represented by a graph. This graph is partitioned into clusters of vertices forming a dual problem that is solved by a subgradient algorithm. The subproblems formed by the generated subgraphs are solved by CPLEX. Computational experiments consider a data set formed by several difficult instances with different features. The results show the efficiency of the proposed methods over traditional Lagrangean relaxations and other methods found in the literature.
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.
We consider binary quadratic programs (QP) having a quadratic objectivefunction, linear constraints, and binary variables. Many classical solution methods ofthese problems are based on exact reformulation of QP into anequivalent mixed integer linear program. Several linearization methods werestudied in the literature. More recent solution methods alsobuild an exact reformulation but into a problem which objective function isquadratic and convex. A common point of the two approaches is that thecontinuous relaxation of the reformulated problem is a convex optimizationproblem that can be solved in polynomial time. This makes it possible to use a general branch-and-bound framework to solve the reformulated problem and even to rely on the strongness of standard solvers. In this paper, we show that several quadratic convex reformulation methods, as well as classical linearization, can be viewed within a unified framework. This shows the non-surprising result that linearization is a particular quadratic convex reformulation on the one hand. On the other hand, it allows to compare these methods from a theoretical point of view.
-Let (QP) be a binary quadratic program that consists in minimizing a quadratic function subject to linear constraints. To solve (QP) we reformulate it into an equivalent program with a convex objective function. Our reformulation, that we call EQCR (Extended Quadratic Convex Reformulation), is optimal from the continuous relaxation bound point of view. We show that this best reformulation can be deduced from the solution of a semidefinite relaxation of (QP) and that EQCR outperforms QCR. We carry out computational experiments on Max-Cut and on the k-cluster problem.
Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.
To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of p (integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches.