Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.02948

Progressive Bound Strengthening via Doubly Nonnegative Cutting Planes for Nonconvex Quadratic Programs

Qu, Zheng, Defeng Sun, Jintao Xu

Published Oct 3, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2510.02948

Related papers

Solving Quadratic Programming by Cutting Planes

Pierre Bonami, Andrea Lodi, Jonas Schweiger +1 more

We propose new cutting planes for strengthening the linear relaxations that appear in the solution of nonconvex quadratic problems with linear constraints. By a famous result of Motzkin and Straus, these problems are connected to the clique number of a graph. Our cuts are derived in the context of a spatial branch-and-bound algorithm where linearization variables are introduced to represent products. Their validity is based on the result of Motzkin and Straus, in that it depends on the clique number of certain graphs. For convenience, we derive our cuts for the special case of the so-called standard quadratic programs, where the only (linear) constraint imposes that variables must belong to a simplex. We specifically consider cuts that correspond to carefully constructed complete bipartite graphs. We study the relation between these cuts and the classical ones obtained at the first level of the reformulation-linearization technique. By studying this relation, we derive a new type of valid inequalities that generalizes both types of cuts, and thus leading to a large family of cutting planes to strengthen the linear relaxation. We present extensive computational results using the different cutting planes we propose within the spatial branch and bound implemented by the commercial solver CPLEX. We show that our cuts allow us to obtain a significantly better bound than reformulation-linearization cuts and reduce computing times for global optimality. Finally, we formally establish the scaling necessary for using the proposed cuts to solve nonconvex quadratic problems with linear constraints, e.g., quadratic knapsack problems.

SIAM Journal on Optimization•Jan 1, 2019•22 citations

The equivalence between doubly nonnegative relaxation and semidefinite relaxation for binary quadratic programming problems

Chuan-Hao Guo, Yanqin Bai, Tang, Li-Ping

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.

arXiv:1211.5406•Nov 23, 2012•0 citations

Cutting Planes for Mixed Integer Programming

Andrea Qualizza

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.

•Jan 1, 2011•2 citations

Enhancing convexification techniques for solving nonconvex quadratic programming problems

Xiaofei Qi

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 . . . . . . . . . . . .

•Jan 1, 2017•0 citations

Concave Quadratic Cuts for Mixed-Integer Quadratic Problems

Jaehyun Park, Stephen Boyd

The technique of semidefinite programming (SDP) relaxation can be used to obtain a nontrivial bound on the optimal value of a nonconvex quadratically constrained quadratic program (QCQP). We explore concave quadratic inequalities that hold for any vector in the integer lattice ${\bf Z}^n$, and show that adding these inequalities to a mixed-integer nonconvex QCQP can improve the SDP-based bound on the optimal value. This scheme is tested using several numerical problem instances of the max-cut problem and the integer least squares problem.

arXiv:1510.06421•Oct 21, 2015•3 citations

Cuts and heuristics for nonconvex quadratic optimization

Luce, Robert

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.

•Feb 23, 2022•0 citations
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.
We propose new cutting planes for strengthening the linear relaxations that appear in the solution of nonconvex quadratic problems with linear constraints. By a famous result of Motzkin and Straus, these problems are connected to the clique number of a graph. Our cuts are derived in the context of a spatial branch-and-bound algorithm where linearization variables are introduced to represent products. Their validity is based on the result of Motzkin and Straus, in that it depends on the clique number of certain graphs. For convenience, we derive our cuts for the special case of the so-called standard quadratic programs, where the only (linear) constraint imposes that variables must belong to a simplex. We specifically consider cuts that correspond to carefully constructed complete bipartite graphs. We study the relation between these cuts and the classical ones obtained at the first level of the reformulation-linearization technique. By studying this relation, we derive a new type of valid inequalities that generalizes both types of cuts, and thus leading to a large family of cutting planes to strengthen the linear relaxation. We present extensive computational results using the different cutting planes we propose within the spatial branch and bound implemented by the commercial solver CPLEX. We show that our cuts allow us to obtain a significantly better bound than reformulation-linearization cuts and reduce computing times for global optimality. Finally, we formally establish the scaling necessary for using the proposed cuts to solve nonconvex quadratic problems with linear constraints, e.g., quadratic knapsack 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.
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.
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 . . . . . . . . . . . .
The technique of semidefinite programming (SDP) relaxation can be used to obtain a nontrivial bound on the optimal value of a nonconvex quadratically constrained quadratic program (QCQP). We explore concave quadratic inequalities that hold for any vector in the integer lattice Zn{\bf Z}^nZn, and show that adding these inequalities to a mixed-integer nonconvex QCQP can improve the SDP-based bound on the optimal value. This scheme is tested using several numerical problem instances of the max-cut problem and the integer least squares problem.
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.