Cutting-Plane Methods for Nonconvex QP

• The paper introduces a cutting-plane framework that iteratively tightens convex relaxations through adaptive valid inequalities to eliminate non-optimal regions.
• Enhanced linearization techniques, including improved BML and SDP/DNN liftings, generate locally optimized bounds that significantly reduce the search space.
• Empirical results indicate that these frameworks achieve faster convergence and lower CPU time for large-scale NP-hard mixed-integer quadratic programs.

A cutting-plane framework for nonconvex quadratic programs is a class of global optimization methods that iteratively tighten a convex relaxation of a nonconvex quadratic objective—often over discrete or mixed-integer domains—by sequentially generating and adding valid inequalities (cuts) that exclude portions of the search space without removing any global optima. These frameworks aim to systematically refine bounds and feasible sets for problems that are known to be NP-hard and highly structured, such as quadratic 0-1 programs and box-constrained quadratic integer programs.

1. Linearization and Convex Relaxation Foundations

A central challenge in nonconvex quadratic programming is that quadratic terms $x^\top Q x$ with indefinite $Q$ create nonconvex feasible regions, complicating direct application of convex optimization methods. Linearization techniques are foundational; the classic Balas–Mazzola linearization (BML) for quadratic 0–1 programs introduces extra variables and constraints to express products $x_i x_j$ as new variables $y_{ij}$, subject to McCormick-type inequalities or strengthened relationships. In the original BML, quadratic terms are linearized using bounds $u_i$ over polyhedral relaxations of the binary set (Gharibi, 2012). For box-constrained programs, the use of auxiliary binary variables encodes integer domains, and SDPs are solved to find the minimal diagonal shift $\theta$ making $Q + \operatorname{Diag}(\theta) \succeq 0$ so that a convex relaxation is achieved (Xia et al., 2014).

Convex relaxations are also achieved via SDP or DNN (doubly nonnegative) methods, which "lift" the variable space: a matrix $X = xx^\top$ is introduced, and the rank constraint $\operatorname{rank}(X) = 1$ is relaxed to $X \succeq 0$ (SDP) and/or $X \geq 0$ (DNN) (Yildirim, 2020, Qu et al., 3 Oct 2025). Feasibility-preserving convex relaxations, especially copositive and DNN relaxations, are essential for ensuring that no feasible integer or continuous solution is erroneously cut off during the tightening process.

2. Strengthened Linearization and Modern Cutting-Plane Models

Modern frameworks replace global bounding parameters in linearizations with locally adapted ones to better reflect the effect of variable assignments on the quadratic term. For instance, the improved BML uses per-variable, per-assignment bounds $$v_i = \max \{\sum_{j \neq i} b_{ij} x_j \mid x \in \bar{X}, x_i = 0\}$$ and $$l_i = \min \{\sum_{j \neq i} b_{ij} x_j \mid x \in \bar{X}, x_i = 1\}$$, leading to a tight representation of $$x_i \sum_{j \neq i} b_{ij}x_j = \max\{\sum_{j \neq i} b_{ij}x_j + v_i x_i - v_i,\ l_i x_i\}$$ (Gharibi, 2012). This yields primal and dual LPs with strengthened constraints and tighter LP relaxations, which translate into stronger cut-generation mechanisms.

Advanced frameworks, such as those based on adaptive diagonal perturbations (Dong, 2014), iteratively construct a set of convex cutting surfaces via perturbed Hessians $Q + \operatorname{diag}(d)$ with $Q + \operatorname{diag}(d) \succeq 0$, leveraging separation oracles and customized coordinate descent algorithms to efficiently identify new diagonal perturbations generating strong violated cuts.

3. Cut Generation: Subproblem Solving and Valid Inequality Structure

At the core of any cutting-plane method is the subproblem of identifying a valid inequality that is violated by the current relaxation solution. In pure 0–1 or mixed-integer programs, the dualized linearized model produces a master problem where cuts correspond to inequalities indexed by extreme points (incidence vectors) of the binary hypercube. Given a current solution $\tilde{x}$, the associated dual subproblem can be solved analytically, yielding a cut (inequality) added to the master problem (Gharibi, 2012).

For general nonconvex QCQP and integer quadratic programs, cutting-surface or concave quadratic cuts can be constructed using either separation problems formulated as SDPs, or by exploiting, for example, that for any $a \in \mathbb{Z}^n$ and $b \in \mathbb{Z}$, $(a^\top x-b)(a^\top x-(b+1)) \geq 0$ holds for integer-valued $x$, leading after expansion to concave quadratic inequalities appended to the SDP relaxation (Park et al., 2015).

The generation of valid cuts may also leverage the structure of the convex hull of the nonconvex set. For complex QCQP, the convex hull of all $2 \times 2$ Hermitian rank-one PSD matrices under variable-wise bounds can be explicitly described with SOCP or linear constraints, and linear cuts are accordingly derived (Chen et al., 2017).

4. Algorithmic Structure and Convergence Mechanisms

A prototypical cutting-plane algorithm for a nonconvex quadratic program operates as follows:

1. Initialization: Solve a convex relaxation (e.g., LP, SDP, or DNN) of the original quadratic program.
2. Separation: Given the relaxation solution, attempt to find a cut violated by the solution but valid for all integer (or feasible) points, via structured SDPs, combinatorial criteria, or heuristic checks.
3. Cut Addition: Incorporate the violated inequality (cut) into the master relaxation.
4. Iteration: Repeat relaxation, separation, and cut addition until either the relaxation is tight (lower and upper bounds coalesce) or further progress stalls.

This generic scheme is instantiated concretely in several works: the improved BML-based cutting-plane method updates the master problem in each round with a cut defined by dual multipliers matched to the current solution, and guarantees finite convergence because each non-optimal candidate is eventually eliminated (Gharibi, 2012). In cutting-surface methods, separation problems (convex but non-smooth SDPs) are solved via primal-barrier coordinate minimization, and each new cut strictly tightens the feasible region (Dong, 2014).

Theoretical results ensure that these frameworks are not weaker than classic relaxations and often yield strictly better lower bounds. For instance, $v(R\text{-PL}_1) \leq v(R\text{-PL}_2)$ guarantees that the LP relaxation of the strengthened model dominates the basic BML relaxation (Gharibi, 2012).

5. Integration with Mixed-Binary/Lifted Convex Reformulations

For box-constrained integer quadratic programs, reformulation as a mixed-binary convex quadratic program can enable direct application of efficient convex QP solvers. This is achieved by encoding variable domains with families of binary variables and selecting diagonal shifts to convexify the Hessian, with the optimal shift parameter computed via an SDP (Xia et al., 2014). These convexified formulations are highly conducive to further tightening by valid inequalities or cutting planes, since the continuous relaxation is already tight and can be iteratively refined without sacrificing solvability. The integration of convex reformulations into a cutting-plane outer approximation loop is therefore immediate.

6. Computational Performance and Impact

Computational studies consistently show that strengthened linearizations, adaptive cut-generation via subproblem separation, and the use of strong convex relaxations yield faster convergence to global optima, reduce the number of branch-and-bound nodes, and lead to significant reductions in CPU time, especially for large-scale or hard instances (Xia et al., 2014, Dong, 2014). For instance, convexified binary quadratic formulations (MBQPe*) outperform both SDP-based and ellipsoidal relaxation-based solvers on larger domain instances, particularly due to the efficiency of convex QP solvers (Xia et al., 2014). In cutting-surface frameworks, the coordinate descent separation algorithms are at least an order of magnitude faster than interior-point approaches to SDP separation (Dong, 2014).

Although some advanced frameworks (e.g., full projected SDP+RLT) provide marginally tighter bounds, the reduction in computation time using diagonal or structured perturbation cuts justifies their use in practice. These methods are applicable to both classic quadratic assignment and knapsack problems, and their structure is extensible to larger classes of MIQCQP and general nonconvex quadratic programming applications.

7. Extensions and Outlook

Several directions for further research are outlined in recent works. The embedding of linear equality constraints into the separation strategy, expanding beyond diagonal perturbations to sparse lifting, and the development of branch-and-cut frameworks incorporating adaptive cutting-surface procedures remain of significant interest (Dong, 2014, Gharibi, 2012). Moreover, the interaction between high-quality convex relaxations and efficient cut-generation (potentially harnessing problem-specific combinatorial structure) is viewed as central to bridging the gap between tractability and tightness in large-scale nonconvex quadratic optimization.

In summary, contemporary cutting-plane frameworks for nonconvex quadratic programs combine strength in convex relaxation (strengthened linearizations, mixed-binary convex surrogates, DNN/SDP liftings) with highly structured, efficiently computed valid inequalities. Iterative cut generation and separation subproblems—often posed as low-rank SDPs or combinatorial selection heuristics—drive systematic tightening of lower bounds and progressive elimination of non-promising regions of the feasible space, with robust convergence and demonstrably superior computational efficiency across diverse problem classes.