Mixed-Integer Quadratic Programming Overview

• Mixed-Integer Quadratic Programming (MIQP) is an optimization class where a quadratic objective is minimized over polyhedral sets with some variables constrained to be integers.
• It employs techniques like branch-and-bound, relaxations, and heuristics to tackle computationally challenging NP-complete problems.
• Applications span power systems, model predictive control, portfolio management, and emerging quantum-classical hybrid methods for scalable optimization.

Mixed-Integer Quadratic Programming (MIQP) is the problem class in which a quadratic objective is optimized over a polyhedral set, with some decision variables required to take integer values, often subject to linear or quadratic constraints. MIQP encompasses a fundamental intersection of continuous and combinatorial optimization and is foundational in fields ranging from power systems, embedded control, portfolio management, to statistical learning. The general MIQP problem can be NP-complete, may be convex or nonconvex, and, depending on problem structure, demands state-of-the-art methods in convexification, duality theory, decomposition, relaxation, and algorithmic design.

1. Formal Structure, Complexity, and Theoretical Foundations

A generic MIQP is formulated as: $$\begin{align*} \text{minimize}\quad & \frac{1}{2} x^T H x + c^T x \ \text{subject to} \quad & Ax \leq b \ & x_i \in \mathbb{Z}, \quad i \in \mathcal{I} \subseteq \{1,\ldots,n\} \end{align*}$$ where $H$ may be (positive semi)definite (convex case) or indefinite (nonconvex case), and $\mathcal{I}$ denotes the integer-constrained indices.

The decision version of MIQP is in NP; that is, given any feasible instance, there exists a certificate (a feasible solution) of polynomial bit-size in the size of the input. Thus, MIQP is NP-complete (Pia et al., 2014). The proof framework unifies previous results for quadratic programming (QP) and integer linear programming, constructing size-bounded certificates by decomposing the feasible set into polytopes and integer cones and applying quadratic form analysis on the recession cone. This means that while MIQP is intractable in the worst case, its feasible solutions are always accessible to polynomial verification, which informs practical solution strategies and benchmarking.

2. Algorithmic Methods: Branch-and-Bound, Relaxation, and Heuristic Schemes

The backbone for solving MIQP in practice is the branch-and-bound (B&B) framework, wherein nodes correspond to relaxations (typically as quadratic programs or semidefinite programs at each branch) and pruning is achieved by bounding techniques or feasibility/dominance cuts (Arnström et al., 2022, Shoja et al., 20 Mar 2025). Contemporary MIQP solvers may embed the following elements:

• Active-Set QP Solvers: Frequently used within B&B to solve QP relaxations efficiently. The complexity (e.g., number of active-set iterations) can be analyzed and certified a priori in multi-parametric settings by tracking regions of constant active-set structure (Shoja et al., 2022, Shoja et al., 20 Mar 2025).
• Dual Active-Set and Warm-Starting: Dual active-set methods operate directly in the dual space, support hot starting from parent nodes, and are particularly suited for embedded and memory-limited settings (Arnström et al., 2022). Warm-starting is also powerful in receding-horizon MPC where consecutive MIQP instances change incrementally (Marcucci et al., 2019).
• Heuristics and ADMM-type Routines: Fast, suboptimal solutions are obtained by extending ADMM to the nonconvex (mixed-integer) domain (Takapoui et al., 2015), or by using accelerated dual projection (GPAD) for fast QP relaxations and then greedily fixing integer variables (Naik et al., 2021). While convergence to the global optimum is not guaranteed, these methods can deliver near-optimal or even optimal solutions efficiently for problems of moderate size.

3. Convexification, Duality, and Outer Approximation

For convex MIQPs, Lagrangian and augmented Lagrangian duality play a critical role. The augmented Lagrangian dual (ALD), constructed by adding a penalty function for equality violations, can achieve an exact (zero) duality gap for convex MIQP with a finite penalty parameter when the penalty is any norm; moreover, the paper establishes that the necessary weight can be polynomially bounded with respect to problem data (Gu et al., 2019). This enables effective primal-dual algorithms and guides relaxation-based approaches.

Outer approximation (OA) methods have seen broad adoption for MIQP, particularly when indicator variables activate quadratic terms (e.g., sparse regression, portfolio selection). OA leverages perspective reformulations and cuts derived from convex relaxations (typically subgradients of marginal functions) to iteratively strengthen master MILP relaxations, showing remarkable large-scale performance and notable speedups over earlier methods (Wei et al., 2023).

4. Specialized Relaxations for Nonconvex MIQCQPs

Nonconvex MIQCQPs—those with indefinite quadratic forms or bilinear constraints—arise in energy systems and other engineering applications. For these, convex relaxations are constructed by:

• Semidefinite Programming (SDP) Liftings: Lifting quadratic products into a symmetric matrix W (with $W = xx^T$), then relaxing to semidefiniteness ($W \succeq xx^T \rightarrow W \succeq 0$). Chordal decomposition exploits problem sparsity by dividing the global SDP constraint into small PSD constraints on variable cliques, drastically improving tractability for large systems such as unit commitment with AC optimal power flow constraints (Gómez-Casares et al., 23 Sep 2025).
• Discretization and Convex Envelope Approaches: Enhanced separable reformulations, sawtooth and D-NMDT relaxations, and hybrid separable (HybS) constructions produce hereditary sharpness for univariate or bilinear terms, providing provably tight polyhedral or MIP relaxations (Beach et al., 2022, Beach et al., 2023).

5. Embedded, Real-Time, and Distributed MIQP

MIQP has crucial impact in embedded optimization, especially in control and decision-making for systems requiring millisecond bounded solutions:

• Embedded Solvers: Customized B&B solvers paired with highly efficient QP routines (e.g., dual active-set with compact node encoding) allow real-time closed-loop MPC with limited computational resources (e.g., MCU with <30kB RAM) (Arnström et al., 2022, Quirynen et al., 2023).
• Heuristic and ADMM-based Distributed Methods: Proximal ADMM with constraint tightening supports efficient decomposition of large MIQP problems in networked environments, such as distributed traffic control with connected vehicles, by exploiting problem decomposability and sequential tightening heuristics (for big-M constraints) to steer relaxations towards integer feasibility (Le et al., 6 Apr 2025).
• Warm Start and Parametric Certification: Hybrid MPC applications benefit from warm-starting both branch-and-bound trees and dual bounds across time steps, with complexity-certification frameworks partitioning parameter regions for a priori real-time computational guarantees (Marcucci et al., 2019, Shoja et al., 2022, Shoja et al., 20 Mar 2025).

6. Approximation Algorithms and Quantum-Classical Hybrids

For certain classes of indefinite MIQP, ε-approximation algorithms exist with polynomial-time guarantees when the rank of the quadratic form and the number of integer variables is fixed. Such algorithms use strongly polynomial symmetric decomposition, mesh partition, and flatness certificates to systematically reduce the ‘combinatorial explosion’ of the solution space (Pia, 2022).

Quantum-classical hybrid algorithms integrate quantum annealing solvers (e.g., D-Wave’s CQM) within classical decomposition frameworks (such as extended Benders decomposition) to solve MIQP master problems—particularly the hard binary quadratic models. Empirical results show exponential speedups on instances where the quadratic coupling is especially strong, positioning hybrid computation as a frontier for large-scale MIQP (Yoshihara et al., 4 Oct 2025).

7. Applications: Energy, Control, Decision Making, and Verification

Principal applications of MIQP include:

• Power Systems: Economic load dispatch (ELDP) with valve-point effects is solved to global optimality by MIQP formulations using surrogate piecewise-quadratic approximations (Azzam et al., 2014). Unit commitment with ACOPF constraints is addressed using sparse SDP relaxations and strong feasible solution heuristics (Gómez-Casares et al., 23 Sep 2025).
• Model Predictive Control and Robotics: Real-time vehicle maneuvering and intersection control exploit MIQP for integrating logic (lane changes, collision constraints) and continuous control in receding-horizon frameworks (Quirynen et al., 2023, Le et al., 6 Apr 2025).
• Statistical Learning and Finance: Portfolio selection and sparse regression exploit outer approximation and perspective reformulation for tractable MIQP formulations (Wei et al., 2023).
• Verification of Neural Network Controllers: MIQP is employed for Lyapunov stability certification where candidate NN policies are compared to robust MPC baselines, with region of attraction guarantees established by MIQP-based verifications (Schwan et al., 2022).

The theory and methodologies for MIQP thus address both exact and approximate problem solving, undergirded by complexity theory, duality and relaxation frameworks, decomposition, heuristics, and emerging quantum-classical hybridization. The field continues to expand with new applications, theoretical insights, and computational advances.