Mixed-Integer Quadratically Constrained Programming

• MIQCP is an optimization framework that integrates discrete and continuous variables with quadratic constraints to solve complex real-world problems.
• Advanced convex relaxations, including SDP and discretization methods, enable strong dual bounds and globally optimal solutions despite nonconvexities.
• Applications in power systems, stochastic programming, and quantum circuit design highlight MIQCP's significant impact on engineering and computational research.

A mixed-integer quadratically constrained program (MIQCP) is an optimization problem in which the objective and constraint functions are quadratic and the decision variables include both continuous and integer or binary components. Such problems integrate combinatorial and nonlinear structure, making them central in operations research, power systems, stochastic programming, and emerging quantum computing applications. MIQCPs are challenging due to the simultaneous presence of discrete decisions and nonconvex quadratic constraints, necessitating advanced relaxation, decomposition, and discretization techniques to yield strong dual bounds and globally optimal solutions, particularly as problem scale increases (Gómez-Casares et al., 23 Sep 2025).

1. Mathematical Formulation and Problem Classes

The canonical form of an MIQCP is: $$\begin{aligned} \min_{x,\,z} \quad & x^\top Q^0 x + c^{0\top} x + d^{0\top} z \ \text{s.t.}\quad & x^\top Q^i x + c^{i\top} x + d^{i\top} z + e^i \le 0, \quad \forall i \in I \ & A x + B z = b \ & z \in \{0,1\}^p,\quad x \in \mathbb{R}^n \end{aligned}$$ where $Q^0$, $Q^i\in\mathbb{R}^{n\times n}$ (not necessarily PSD), $c^0,\,c^i\in\mathbb{R}^n$, $d^0,\,d^i\in\mathbb{R}^p$, $e^i\in\mathbb{R}$, $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{m\times p}$, $b\in\mathbb{R}^m$ (Gómez-Casares et al., 23 Sep 2025). The structure encompasses pure binary/continuous QCQPs, convex/nonconvex instances, and problems with both dense and sparse quadratic interactions.

Many real-world problems can be cast as MIQCPs:

• Energy systems: Unit commitment with AC optimal power flow, featuring binary generator schedules and nonconvex physical constraints (Gómez-Casares et al., 23 Sep 2025), smart inverter placement (Li, 2015)
• Stochastic programming: Two-stage SMIP with quadratic recourse, e.g., scenario-based power dispatch (Belyak et al., 2023)
• Quantum circuit synthesis: Gate selection subject to quadratic constraints enforcing algebraic and physical properties (Rytir et al., 12 Nov 2025)

2. Convex Relaxation Paradigms

Semidefinite programming (SDP) relaxations provide a general and tight convexification pathway for MIQCPs. By lifting $x\in\mathbb{R}^n$ to $W=xx^\top$, quadratic forms become linear in $W$, and the nonconvex restriction $W=xx^\top$ is relaxed to $W\succeq xx^\top$: $$\begin{aligned} \min_{x,W,z}\; & \mathrm{Tr}(Q^0 W) + c^{0\top} x + d^{0\top} z \ \text{s.t.}\; & \mathrm{Tr}(Q^i W) + c^{i\top}x + d^{i\top}z + e^i \le 0,\, \forall i \ & A x + B z = b \ & \begin{pmatrix} 1 & x^\top \ x & W \end{pmatrix}\succeq 0 \ & z\in\{0,1\}^p,\; x\in\mathbb{R}^n \end{aligned}$$ Relaxing $z\in[0,1]^p$ yields the continuous SDP relaxation (Gómez-Casares et al., 23 Sep 2025).

In energy system MIQCPs, the SDP approach is strengthened by incorporating valid linear equalities arising from system physics, and by using disjunctive programming (DP) techniques to encode the effect of integer variables on the rank-1 structure, producing convex hull MISDP relaxations that strictly dominate classical Shor/MIBSDP formulations (Li, 2015).

Diagonal perturbation methods augment the quadratic matrix with a variable diagonal, generating valid convex quadratic relaxations. Multiple adaptive diagonal perturbations yield cutting surfaces in the original variable space, and the associated separation problem reduces to a highly structured SDP solved efficiently by primal-barrier coordinate minimization (Dong, 2014).

Discretization-based MIP relaxations reformulate bilinear terms via piecewise McCormick envelopes or more sophisticated techniques such as hybrid separable (HybS) (Beach et al., 2022) or doubly discretized NMDT (Beach et al., 2023), attaining the convex hull of bilinear forms on uniform grids, with error decaying as $O(2^{-2L})$ for $L$ binary bits. Such formulations are competitive for dense MIQCQPs on moderate dimension.

3. Exploiting Sparsity

Sparsity—both correlative (which pairs of variables couple in quadratic forms) and term sparsity (which monomials appear)—is critical for scaling SDP relaxations to large MIQCPs. Chordal decomposition techniques convert a large semidefinite constraint into many smaller-dimensional PSD constraints on maximal cliques of a chordal extension of the correlative sparsity graph: $$\widehat W \succeq 0 \iff W_{C_k} \succeq 0,\; k=1\dots K$$ where $W_{C_k}$ are principle submatrices on cliques, enforced plus consistency on overlaps (Gómez-Casares et al., 23 Sep 2025). Combined CS and TS (via CS-TSSOS or similar tools) yield block-diagonal forms with hundreds of blocks of dimension $\le 11$ for large power-network instances, reducing root-node times from hours to seconds.

Disjunctive convex hull relaxations exploit explicit sparsity, restricting extended variables to only those bilinear terms involving integer variables, achieving tightest known SDP-based bounds with modest problem growth (Li, 2015).

4. Global Branch-and-Bound and Decomposition Algorithms

MIQCP global solution frameworks are fundamentally branch-and-bound (B&B), embedded with strong convex relaxations:

• SDP-powered B&B: Nodes relax binaries ($z$) to [0,1], solve the sparse SDP, and branch on fractional binaries. Upper bounds are improved by calling local NLP solvers initialized at the SDP solution or via heuristics, with valid inequalities and domain-specific cuts added as needed (Gómez-Casares et al., 23 Sep 2025).
• Discretization-based MIP B&B: Discretized relaxations (e.g., D-NMDT, sawtooth in HybS) are embedded at each node, tightened adaptively by refining discretization along most violated variables (Beach et al., 2023, Beach et al., 2022).
• Dual Decomposition/Progressive Hedging: In stochastic MIQCPs, Lagrangian dualization of nonanticipativity, together with discrete relaxations of bilinear terms (e.g., RNMDT$_p$), allows the use of dual methods (proximal bundle, Frank-Wolfe PH) to yield scenario-separable subproblems, feeding global B&B or best-first-search (Belyak et al., 2023).
• Frank-Wolfe-based Primal Search: Branch-and-bound layered over Frank-Wolfe primal oracles, using heuristics such as rounding, gradient-guided large neighborhood search, and exploitation of active sets, yields high-quality integer-feasible solutions even when lower bounds are weak (Mexi et al., 2 Aug 2025).

5. Reformulation and Cutting-Plane Strategies

MIQCPs often benefit from reformulations that expose structured relaxations and facilitate strong cut generation:

• Second-order-cone (SOC) reformulations for quadratic constraints with positive-definite continuous blocks, replacing quadratic inequalities with SOC constraints with nonlinear right-hand-sides. The convex hull can be characterized using the concave envelope of the right-hand side, and extended formulations with auxiliary $(\eta,\tau)$ are cut by BQP, RLT, and conic-MIR procedures (Du et al., 1 Nov 2025).
• Disjunctive hull and convexification: For integer-influenced matrices, logical disjunctions from $X_{ij}=x_ix_j$ are recast as extended formulations using auxiliary variables and continuous multipliers, yielding the convex hull (CH-MIESDP) with strong theoretical properties (Li, 2015).
• Piecewise or grid-based convexification: Hybrid separable and D-NMDT produce grid-based convex hulls for $z=xy$, sharply controlling relaxation error and binary variable count, proven to outperform previous separable MIP methods on dense MIQCQPs (Beach et al., 2023, Beach et al., 2022).

6. Applications and Computational Performance

MIQCP methodology is established in a spectrum of large-scale domains:

• Power and Energy Systems: The unit commitment problem with ACOPF coupling is routinely solved via sparse SDP relaxations; the 118-bus IEEE system with 24 periods is solvable to optimality gap $<0.1\%$ in minutes using chordal/term sparsity (Gómez-Casares et al., 23 Sep 2025). Distribution system inverter placement attains the tightest known SDP-based bounds via convex hull disjunctive relaxations (Li, 2015).
• Quantum Compilation: Gate synthesis with MIQCQP encodes braid selection, product constraints, and block-structure objectives using quadratic and linearization constraints, reaching optimal CNOT and perfect-entangling circuits in tens of gate layers in the non-semisimple Ising model (Rytir et al., 12 Nov 2025).
• Stochastic Programming: Two-stage stochastic MIQCPs solved by p-branch-and-bound exhibit root-node tightness, exploring very few nodes, with up to $84\%$ time savings over Gurobi in 1% density and zero gap for high-density scenarios (Belyak et al., 2023).
• General MINLP: Global quadratic (paraboloid) underestimators for factorable constraints transform MINLPs into MIQCPs, achieving order-of-magnitude speedups on MINLPLib test sets with modest degradation in dual gap (Göß et al., 8 Jul 2024).

7. Outlook: Scalability, Tightness, and Future Developments

SDP relaxations, particularly with sparsity-exploiting decompositions, produce the strongest dual bounds but at high per-node computational cost; chordal and correlative sparsity recover scalability. Discretization-based MIP relaxations (D-NMDT, HybS, CDA) materially close the duality gap and enable MILP-grade solution strategies with controlled (and often small) model growth in the number of binaries. The ideal convex hull characterization for SOC-structured MIQCPs unlocks next-generation cutting-plane methods (Du et al., 1 Nov 2025).

Anticipated directions include:

• Integration of SDP cuts for warm-starting and branch-and-cut B&B;
• Extension of branch-and-bound to spatial branching on bilinear/continuous variables;
• Adoption of convex hull and MIP-based relaxations in commercial solvers with advanced presolve and cut separation;
• Extension to large-scale stochastic and multi-stage MIQCP via parallelizable decomposition schemes.

These advances collectively bridge the gap between theory and tractable solution of high-dimensional nonconvex multipartite quadratic programs, yielding tools of broad utility across engineering, finance, and quantum computation (Gómez-Casares et al., 23 Sep 2025, Li, 2015, Belyak et al., 2023, Göß et al., 8 Jul 2024, Du et al., 1 Nov 2025).