Mixed-Binary Quadratic Programs

• Mixed-binary quadratic programs are mathematical optimization models that incorporate continuous and binary variables with quadratic objectives and constraints.
• They facilitate addressing complex decision-making in areas such as finance, energy, communications, and scheduling by uniting logical decisions and continuous resource allocation.
• Recent advances include convex reformulation, semidefinite and DNN relaxations, and quantum-classical hybrid algorithms that enhance tractability and solution speed.

Mixed-binary quadratic programs (MBQPs) are mathematical optimization models featuring both continuous and binary variables, with a quadratic objective and/or constraints. These problems lie at the intersection of discrete optimization, convex and nonconvex quadratic programming, semidefinite programming, and combinatorial algorithms. MBQPs are ubiquitous in applications requiring logical decisions coupled with continuous resource allocation, including finance, energy, communications, machine learning, and scheduling.

1. Formulations and Problem Classes

A standard MBQP can be written as

$$\begin{aligned} \min_{x\in\mathbb{R}^n,\;z\in\{0,1\}^p}\hspace{1mm} & x^\top Q x + 2c^\top x + d^\top z \ \text{s.t.}\quad\, & A x + B z \le b \ & x_i \in [\ell_i,u_i] \;\;(i\in C) \ & z_j \in\{0,1\} \;\;(j\in B) \end{aligned}$$

where $Q$ may be indefinite, and additional quadratic or linear (in)equalities and cardinality constraints may be present (Naik et al., 2021, Xia et al., 2014, Hou et al., 19 Feb 2025). More general MBQPs allow quadratic constraints, separable or nonseparable, and mixed-integer constraints on subsets of variables. Structured MBQPs such as box-constrained QPs, knapsack-type, assignment, or max-cut variants often motivate specialized methods.

MBQPs are generically NP-hard—even strongly NP-hard—when $Q$ is not positive semidefinite or the discrete set is nontrivial (Xia et al., 2014, Hou et al., 19 Feb 2025). Many real-world MBQP instances are highly ill-conditioned, high-dimensional, or have nonconvex feasible regions.

2. Reformulation and Convexification Techniques

Significant progress in tractable MBQP solution methods derives from problem reformulations that expose convex substructure or enable efficient relaxation.

Mixed-binary convex reformulation via diagonal shift: For box-constrained, nonconvex quadratic integer programs,

• Each integer $x_i \in \{\ell_i,\ldots,u_i\}$ is encoded using a binary sign-selector $z_i$ and a collection of binaries $y_{ik}$ forming a binary expansion for the distance from the midpoint (Xia et al., 2014).
• A diagonal shift $\theta^*$ is introduced so that $Q + \operatorname{Diag}(\theta^*)\succeq 0$, making the node relaxations in branch-and-bound convex; $\theta^*$ is optimized by solving an SDP that tightens the continuous relaxation (Xia et al., 2014).
• The reformulated MBQP is then a convex QP in $(x, y, z)$ with polyhedral and integrality constraints, efficiently tractable by branch-and-cut.

Step	Description	Reference
Diagonal shift	Ensures convex node relaxations by making $Q$ PSD via $\theta^*$	(Xia et al., 2014)
Binary expansion	Efficient encoding of discrete $x_i$ values with sign and magnitude binaries	(Xia et al., 2014)
Big-M constraints	Enforce one-to-one correspondence between binary codes and nominal integer values	(Xia et al., 2014)

Completely positive and doubly nonnegative (DNN) relaxations: MBQPs can be equivalently reformulated in lifted matrix space:

• Completely positive programming (CPP) expresses MBQPs via $$Y = \begin{pmatrix} 1 & x^\top \ x & xx^\top \end{pmatrix}$$ in the CP cone, but practically, this is relaxed to doubly nonnegative (symmetric PS and elementwise nonnegative) constraints (Cifuentes et al., 2023, Hou et al., 19 Feb 2025, Hou et al., 18 Jul 2025).
• DNN relaxations are computationally tractable SDPs and, after reformulation-linearization (SDP-RLT), are equivalent in terms of relaxation bound to the DNN lifted model, but operate with reduced matrix dimensions (Hou et al., 18 Jul 2025).

Model	Cone Constraint	Matrix Dimension	Reference
CPP	CP	$n+1$	(Cifuentes et al., 2023)
DNN	PSD and $Y\ge0$	$n+l+1$ (w/ slacks)	(Hou et al., 18 Jul 2025)
SDP-RLT	PSD	$n+1$	(Hou et al., 18 Jul 2025)

Discretization-based MIP relaxations: For MIQCQPs with dense nonconvex quadratic terms, discretizations such as the hybrid separable ("HybS") and sawtooth univariate relaxations provide sharp convex MIP outer-approximations (Beach et al., 2022). HybS is more compact than classical separable schemes, requiring only $O(n L)$ binaries rather than $O(n^2 L)$ per quadratic term.

3. Solvers and Algorithmic Paradigms

Branch-and-Bound with Embedded QP Solvers

In exact algorithms, branch-and-bound is the dominant global solution paradigm for MBQPs and MIQPs. The performance bottleneck is typically the solution of the QP relaxations at each node:

• Accelerated dual gradient projection (GPAD) provides an efficient, memory-light method for embedded applications, delivering high-quality node relaxations and allowing for dual and binary warm-start strategies (Naik et al., 2021).
• Early fixing of binary variables via feasibility heuristics and warm-started trees can drastically reduce tree size.

Low-Rank Augmented Lagrangian for DNN Relaxations

Riemannian ALM methods such as RNNAL and RiNNAL+ apply a Burer–Monteiro factorization to the SDP/DNN relaxation, solving over low-rank matrix manifolds, which:

• Dramatically reduce computational cost and memory (projection and retraction scale as $O(mr^2)$), with observed rank typically $r \ll n$ (Hou et al., 18 Jul 2025, Hou et al., 19 Feb 2025).
• Outer AL iteration interleaves with fast Riemannian GD; metric projection onto the constraint manifold is formulated as a convex geometric median problem.
• The hybrid two-phase solution (low-rank phase and lifting-projection) ensures global convergence and escape from saddle points in nonconvex penalty landscapes.
• Global ALM convergence holds under mild constraint qualifications and regularity conditions.

Numerical benchmarks consistently show 10–1000× acceleration over classical SDP solvers and tightness of bounds at scale $n>5000$.

Quantum-Classical Hybrid Solvers

Multi-block ADMM decompositions separate MBQPs with quadratic and convex structure into binary unconstrained subproblems (QUBOs) and convex continuous subproblems (Gambella et al., 2020):

• The binary QUBO is mapped to Ising models and solved by quantum variational eigensolvers (VQE) or QAOA on present-day quantum hardware, while the convex part is solved classically.
• Convergence to stationary points is established under semi-algebraic path assumptions, especially in the three-block ADMM setting.
• Empirical results suggest hybrid quantum–classical methods can produce nontrivial feasible solutions for small- to moderate-sized MBQPs, with noise in quantum subproblem solvers not necessarily degrading empirical solution quality.

4. Relaxation, Approximation, and Bounding Theory

Semidefinite Relaxations and Rounding

SDP relaxations for MBQPs/MIPQCQPs are effective for both feasibility guarantees and bound tightness.

• In MBQCQPs with cardinality and on/off constraints (e.g., selection problems), specialized randomized rounding procedures can convert continuous SDP solutions to feasible binary-continuous pairs, with explicit worst-case approximation guarantees (Xu et al., 2013, Xu et al., 2014).
• For minimization, the approximation ratio is $O(Q^2(M-Q+1)+M^2)$ (real case) and $O(M(M-Q+1))$ (complex), independent of dimension.
• For maximization, the best achievable (and tight) guarantee is $O(1/\ln M)$.

Sensitivity Analysis and Duality

Sensitivity analysis in MBQP is inherently hard: even given optimal solutions, approximating changes under perturbation is NP-hard (Cifuentes et al., 2023). Dual bounding approaches leverage the completely positive reformulation and its co-positive dual:

• Strong duality (exactness of bound) holds if the feasible set is bounded or if the quadratic form is convex (PSD), but can fail otherwise, resulting in positive duality gaps or infeasible duals.
• Multiplicity in the co-positive dual induces ambiguity in sensitivity predictions; extra care must be taken to select the most informative dual solution.

DNN Relaxation Equivalence

Recent results establish equivalence between DNN and SDP-RLT relaxations for MBQP; the latter matches DNN in tightness but with reduced matrix dimension, and the best current incremental algorithms leverage this connection for efficiency (Hou et al., 18 Jul 2025).

5. Implementation, Scalability, and Computational Results

Convex MBQP Reformulation and CPLEX

For box-constrained MBQPs, mixed-binary convex reformulation (MBQPₑ*) enables large-scale solution by off-the-shelf solvers:

• On ternary and extended domain instances up to $n=50$, MBQPₑ* is orders of magnitude faster than SDP-based B&B and ellipsoidal relaxation methods, consistently solving nearly all instances in seconds (Xia et al., 2014).
• The formulation exploits both the diagonal shift (convexifying QP relaxations) and the parsimonious binary encoding.

Low-Rank Riemannian ALM

RNNAL and RiNNAL+ solve DNN and SDP-RLT relaxations up to $n=10,000$ variables, eclipsing previous SDP solvers' practical limits. For all tested classes (BIQ, QAP, stable set, QKP), RNNAL converges within one hour, with solution ranks well below problem dimension and computational times one to three orders of magnitude faster than SDPNAL+ (Hou et al., 19 Feb 2025, Hou et al., 18 Jul 2025).

Discrete Approximations in Dense MIQCQP

Hybrid separable (HybS) MIP relaxation reduces binary count and duality gap for full MIQCQP, showing 2–5× computational speedup over classical separable models, with sharpest relaxations for moderate-to-coarse discretizations (e.g., $L\leq 4$) (Beach et al., 2022).

6. Applications and Future Directions

MBQP methodologies support an expansive array of application domains:

• Energy systems and AC-OPF, wireless resource allocation, portfolio optimization, combinatorial machine learning (clustering, unsupervised learning as MBQP feasibility (Berrones-Santos, 2020)), and quantum-inspired optimization for scheduling and knapsack-type problems (Xu et al., 2014, Gambella et al., 2020, Berrones-Santos, 2020).

Advances in low-rank optimization, hybrid quantum-classical algorithms, and stronger mixed-integer relaxations continue to improve both the scale and quality of MBQP solution, with ongoing research into:

• Structure-exploiting preconditioners, parallel/embedded implementations,
• Robustification and adaptive constraint management for challenging constraint qualifications,
• Quantum algorithm architectures that handle scalar or block constraints more efficiently as quantum hardware scales,
• Tightening relaxations for MBQCQP and MIQCQP with network or complementarity structure.

7. References and Key Literature

• Mixed-binary convex reformulation: Xia and Han (Xia et al., 2014)
• Low-rank Riemannian ALM: Hou, Tang, Toh (Hou et al., 18 Jul 2025); Dong et al. (Hou et al., 19 Feb 2025)
• Quantum-classical ADMM: Barkoutsos et al. (Gambella et al., 2020)
• Semidefinite relaxations and approximation bounds: Xu, Hong, Luo (Xu et al., 2013); Luo, Zhang (Xu et al., 2014)
• Discretization-based MIP relaxations: Bonami et al. (Beach et al., 2022)
• Sensitivity and duality in MBQP: Cifuentes et al. (Cifuentes et al., 2023)
• Embedded and heuristic solvers: Stellato et al. (Naik et al., 2021)
• Learning as MBQP: Berrones-Santos (Berrones-Santos, 2020)