Mixed-Integer Optimization Overview

Updated 3 February 2026

Mixed-Integer Optimization (MIO) is an optimization problem that combines integer and continuous variables to capture both combinatorial and logical structures.
Modern MIO methods employ branch-and-bound, cutting-plane, and decomposition techniques to achieve global optimality and address nonconvex constraints.
MIO has widespread applications in machine learning, scheduling, and network design, with recent innovations leveraging ML techniques to enhance solver efficiency.

A mixed-integer optimization (MIO) problem is an optimization problem in which some decision variables are restricted to take integer values while others remain continuous. MIO arises in a wide spectrum of applications, encoding combinatorial and logical structure alongside continuous modeling. The mathematical and algorithmic framework of MIO enables the global solution of problems with discrete, algebraic, logical, or combinatorial constraints, under linear, convex, or nonconvex objectives and constraints. The rapidly improving capabilities of modern MIO solvers have driven both methodological advances and practical impact in operations research, engineering, machine learning, computational sciences, and beyond.

1. Mathematical Formulation and Core Models

A generic MIO problem can be written as

$\min_{x\in\mathbb{R}^n,\;z\in\mathbb{Z}^m}\; f(x,z)\ \text{s.t.}\; g_i(x,z)\le0,\; i=1,\dots,p; \; h_j(x,z)=0,\; j=1,\dots,q,$

where $x$ are continuous variables, $z$ are integer/binary variables, $f$ is the objective and $g_i, h_j$ are constraint functions. The presence of integer restrictions makes the feasible set nonconvex and endows MIO with expressive power for modeling decisions such as selection, assignment, sequencing, switching, and all forms of logical structure.

Prominent subclasses include:

Mixed-Integer Linear Optimization (MILO): $f$ , $g_i$ , $h_j$ affine.
Mixed-Integer Quadratic Optimization (MIQO/MIQP): $f$ quadratic, $g_i$ , $h_j$ linear.
Mixed-Integer Conic Optimization (MICO): conic constraints (e.g., second-order, PSD).
Mixed-Integer Nonlinear Optimization (MINLO): general nonlinear $f$ , $g_i$ , $h_j$ .

Common modeling constructs involve logical implications (Big-M or indicator-restriction constraints), group or sparsity constraints (e.g., $\ell_0$ -cardinality restrictions), general network flows, and combinatorial substructure (e.g., change-point detection, subset selection, tree/graph models) (Justin et al., 9 May 2025, Bertsimas et al., 2019, Hendrych et al., 2022, Prokhorov et al., 2024).

2. Algorithms and Theoretical Guarantees

The pillar algorithmic strategies for exact MIO solve are:

Branch-and-Bound (B&B): Systematic enumeration of feasible integer variable assignments, where each node relaxes integrality to continuous values to provide lower bounds. Pruning occurs when continuous relaxation is infeasible or its bound cannot improve the incumbent integer solution. B&B is finite for bounded MILO/MIQP/MICO under typical assumptions, and it produces global optimality certificates (Justin et al., 9 May 2025, Hendrych et al., 2022).
Cutting-Plane Methods: Strengthen continuous relaxations by adding valid inequalities ("cuts") violated by current fractional solutions. These can be problem-generic (Gomory, lift-and-project) or model-specific (cover inequalities in knapsack, shattering/packing cuts in trees) (Alston et al., 2024).
Decomposition Schemes: Benders (single-tree or nested), Dantzig-Wolfe, Lagrangian, and outer-approximation techniques, especially for structured problems with loosely coupled subsystems or complicating variables (Bertsimas et al., 2019, Hendrych et al., 2022, Mexi et al., 2 Aug 2025).
Metaheuristics and Hybridization: Including evolutionary strategies, Frank-Wolfe projection-free and nonconvex penalty-based methods, and recent hybrid convex-integer frameworks for situations such as MIQCQP with unbounded variables (Zepko et al., 2024, Mexi et al., 2 Aug 2025, Hendrych et al., 2022).

Well-separated continuous relaxations supply lower (for minimization) or upper (for maximization) bounds, while cutting planes and branching drive search. Modern solvers use sophisticated presolve, bound tightening, warm-starts, intelligent branching, and cut management to accelerate progress (Quirynen et al., 2022, Hendrych et al., 2022, Mexi et al., 2 Aug 2025).

Theoretical properties include finite convergence of B&B (for finite variable domains or under compactness), and for convex MIO, convergence to globally optimal solutions with certified MIP-gap criteria. For nonconvex MINLO, convergence guarantees are generally local unless the global solution space is sampled or explored exhaustively.

3. Key Methodological Innovations

Several major methodological advances in MIO broaden its modeling, tractability, and scalability:

Outer-approximation and convex indicator constraints: By formulating logical conditions (e.g., "if $z_i=0$ then $x_i=0$ ") via convex constraint sets and perspective or regularized reformulation, one can bypass weak Big-M relaxations and enable scalable, cut-based, single-tree algorithms for large-scale sparse regression, portfolio selection, network design, and unit commitment (Bertsimas et al., 2019).
Projection-Free Methods and Frank-Wolfe for Mixed-Integer Convex Optimization: For convex MINLO with challenging polytopal or combinatorial structure, Frank-Wolfe or conditional-gradient methods use MILOs as linear oracles within continuous relaxations, either in B&B frameworks or as primal heuristics for nonconvex problems (Hendrych et al., 2022, Mexi et al., 2 Aug 2025). This approach enables efficient warm-starting, active set reuse, and gradient-based pruning.
Exact Cardinality and Support Constraints: $\ell_0$ -sparsity, group and structured selection (e.g., for subsets of features, clusters, or groups) are modeled directly, enabling exact variable or group selection with global optimality certificates (Bertsimas et al., 2022, Sankaranarayanan et al., 2023, Shi et al., 2022, Prokhorov et al., 2024). Recent work illustrates the statistical and computational advantages of such formulations over $\ell_1$ or relaxed heuristics.
Learning-Accelerated and ML-Enhanced Techniques: Deep learning or machine learning methods predict variable fixings or solution structures to reduce problem dimension or warm-start optimization, dramatically accelerating solve times for large-scale and repetitive MIOs in facility-location, scheduling, and machine learning applications (Triantafyllou et al., 2024, Bertsimas et al., 2019, Jr et al., 2023).
Primal Heuristics and Metaheuristics: Specialized primal heuristics exploit solutions of relaxed subproblems (e.g., collected integer points in power-penalized Frank–Wolfe, discrete exploratory searches combining gradient and primitive directions), yielding high-quality feasible solutions quickly even in large nonconvex MIQCQP regimes (Mexi et al., 2 Aug 2025, Lapucci et al., 2024).

4. Applications Across Domains

MIO is a foundational tool across operations research, engineering, and scientific computing. Select application areas include:

Responsible Machine Learning: Explicitly encoding interpretability (e.g., sparse regression, decision trees (Alston et al., 2024)), statistical fairness (subgroup fairness, intersectional constraints (Němeček et al., 27 Jan 2026)), robustness, and privacy as explicit constraints in MIO-based supervised learning (Justin et al., 9 May 2025). MIO provides mechanisms for globally optimal, interpretable, and certifiably fair machine learning models.
Sparse Dynamics Discovery and Model Selection: Learning governing nonlinear ODE/PDEs, cluster-aware mixed effects models, or kernel SVM feature-selection with provable support recovery and high noise robustness (Bertsimas et al., 2022, Sankaranarayanan et al., 2023, Shi et al., 2022, Tamura et al., 2022). Exact $\ell_0$ selection and physics-based constraints are key advantages.
Change-Point Detection and Structural Breaks: Simultaneous estimation of the number and location of breaks and segment-specific coefficients with global optimality and statistical consistency (Prokhorov et al., 2024).
Energy Systems and Scheduling: Unit commitment, multi-energy optimization, and network flows with complex technical, logical, and multi-objective constraints—often via large MILO models, which can be modeled in flexible node-port or minimal arc-centric forms for scalability (Riedmüller et al., 20 May 2025).
Embedded and Real-Time Control: Model predictive control problems (MI-MPC) and motion planning under hybrid constraints, solved in real-time through presolve acceleration, embedded branch-and-bound, and custom active-set interior point solvers (Quirynen et al., 2022, Bertsimas et al., 2019).
Disjunctive and Logical Modeling: Unified frameworks for handling logical relationships (e.g., "if-then," network design, combinatorial assignments) using strong convex and outer-approximation methods, which enable scalability to $10^4$ – $10^5$ variables and constraints (Bertsimas et al., 2019, Jr et al., 2023).

5. Computational Performance, Scalability, and Software

State-of-the-art commercial and open-source solvers (Gurobi, CPLEX, SCIP, CBC, Bonmin) routinely solve MILP/MIQP instances with millions of continuous variables and $10^5$ binary/integer variables under strong relaxations (Justin et al., 9 May 2025, Zepko et al., 2024). For MICO/MINLO and highly structured or convex MIOs, scalable algorithms for problem-specific relaxations, cut-generation, presolve, active set and warm-starts, parallel computation, and ML-accelerated variable fixing are essential (Hendrych et al., 2022, Jr et al., 2023, Triantafyllou et al., 2024).

For practical computational performance:

Strong convex relaxations (e.g., via perspective reformulation, indicator constraints, problem-specific cut generation) and reduction in model dimension through learning are paramount (Bertsimas et al., 2019, Triantafyllou et al., 2024).
Algorithmic enhancements, such as sampling constraints (for large combinatorial margins), lazy constraint separation, and cross-validation embedded within MIO (for hyperparameter tuning or model selection), contribute to scalability and flexibility (Shiina et al., 9 Jan 2026, Shi et al., 2022).
Pareto-efficient tradeoffs between interpretability/fairness/responsibility and accuracy can be explored through explicit multiobjective MILO (Alston et al., 2024).
For problem families with repeated solve patterns (e.g., MPC, parametric MIO), neural shortcutting and strategy prediction yield two to three orders of magnitude speedups (Bertsimas et al., 2019).

6. Research Directions and Open Problems

Key challenges and frontiers in MIO research include:

Pushing scalability for classes where existing solvers time out (e.g., large unbounded integer MIQCQP, high-dimensional MINLO with nonconvex constraints), especially through metaheuristics, ML-enhanced presolve, and projection-free methods (Zepko et al., 2024, Mexi et al., 2 Aug 2025).
Statistical theory for exact MIO estimators, including generalization and out-of-sample guarantees for learning models obtained via combinatorial optimization (Justin et al., 9 May 2025).
Embedding privacy, robustness, and multiple responsible ML criteria within a unified MIO paradigm at large scale (Justin et al., 9 May 2025).
Automated learning of problem structure (complicating variables, symmetry breaking) to accelerate solution (Triantafyllou et al., 2024).
Nonconvex logic-dominated settings (e.g., complex disjunctive programs, deep constraint learning) where current approaches rely on relaxations or metaheuristics (Maragno et al., 2021).

As solver technology, modeling formalisms, and integration with statistical learning advance, MIO continues to provide a rigorous, versatile, and powerful foundation for computationally challenging problems at the intersection of discrete algorithms, convex analysis, and data-driven modeling.