Mixed-Integer Second-Order Cone Program

• Mixed-Integer Second-Order Cone Program (MISOCP) is an optimization approach that integrates discrete decisions with second-order conic constraints to manage complex, nonconvex problems.
• Advanced convexification techniques, such as exact convex hull representations and extended formulations, significantly reduce relaxation gaps and improve computational speed.
• Key methodologies including BQP/RLT cuts and perspective techniques tighten relaxations, enabling efficient solution of large-scale problems in robust, stochastic, and network optimization.

A mixed-integer second-order cone program (MISOCP) is an optimization problem that combines linear and second-order cone (SOC) constraints with integrality restrictions on a subset of decision variables. This class arises whenever conic quadratic structure must be enforced in the presence of discrete decisions, typically over binary or general integer sets. MISOCPs capture a wide range of mixed-integer nonconvex quadratic problems via convexification techniques and are fundamental in diverse domains such as robust combinatorial optimization, queueing and facility design, power grid operation, experimental design, and stochastic programming. Recent research has established exact convex hull representations, effective reformulation strategies, and computational practices that yield order-of-magnitude gains in solution quality and speed relative to naive MIQCP models.

1. Defining the Mixed-Integer Second-Order Cone Program

A MISOCP consists of optimizing a linear or convex objective over variables $(x, y)$, where $x \in \mathbb{Z}^n$ (typically binary or discrete) and $y \in \mathbb{R}^m$, subject to a collection of affine, linear, and conic constraints. The standard SOC constraint has the form

$\|A x + B y + d\|_2 \le f(x)$

with $A$, $B$, $d$ data and $f$ possibly nonlinear and nonconvex in $x$. The feasible set often couples the conic structure tightly with the discrete variables, a feature exploited by direct reformulation and convexification techniques. MISOCPs generalize both MILP and integer quadratic programming, providing an expressive, algorithmically tractable framework that admits convex relaxations solvable using interior-point and branch-and-cut algorithms (Du et al., 1 Nov 2025).

2. Convexification of Mixed-Integer Conic Sets

A central theoretical advance is the exact convex hull characterization for sets of the form

$$S = \big\{ (x, y) \in X \times \mathbb{R}^m : \|A x + B y + d\|_2 \le f(x) \big\}$$

where $X$ is a compact mixed-integer set and $f$ is nonconvex, upper semicontinuous. The crucial result is that under mild feasibility and column-space conditions (any $A x + d$ can be offset by $B y$), the convex hull of $S$ is given by

$$\text{conv}(S) = \big\{ (x, y) \in \text{conv}(X) \times \mathbb{R}^m : \|A x + B y + d\|_2 \le \hat{f}(x) \big\}$$

where $\hat{f}$ is the concave envelope of $f$ over $\text{conv}(X)$. This envelope is constructed as the largest concave underestimator, ensuring that the relaxed set is both closed and convex (Du et al., 1 Nov 2025). At any extreme point in this relaxation, the supporting concave envelope is realized by an explicit convex combination of basis points, effectively reducing convex-hull construction to combinatorial decomposition and functional underestimation.

3. MISOCP Reformulation Methods

Given that $\hat{f}$ is typically unknown in closed form, practical MISOCP reformulation introduces auxiliary variables and extended formulations that exploit standard conic and cutting-plane machinery. For $f(x)=\sqrt{q(x)}$, as is common in chance-constrained and robust models, the following variable splitting is employed: $$\begin{array}{ll} (a)\quad&\|A x + B y + d\|_2 \leq \eta \ (b)\quad&\eta^2 \leq \tau \ (c)\quad&\tau \leq q(x) \end{array}$$ The hypograph constraint $\tau \leq q(x)$ is nonconvex, but its convex hull over $X \times \mathbb{R}$ is characterized by the Boolean Quadric Polytope (for binary $x$) and the full family of RLT or McCormick envelopes in the general case. As a result, the extended formulation comprising $(a)$–$(c)$ plus all standard linearizations yields a tight, efficiently solvable MISOCP relaxation (Du et al., 1 Nov 2025).

4. Cutting Plane and Perspective Techniques

To fully describe the convex hull and close the relaxation gap, two classes of cuts are generated:

1. BQP/RLT cuts: For quadratics in the integer domain, all implications required by the Boolean Quadric Polytope or RLT system are enforced, ensuring tight approximation of quadratic (bilinear or squared) terms.
2. Perspective/concave cuts: For fractional solutions $(\bar x, \bar y)$ violating the current envelope, separation routines solve subproblems maximizing $\sum_k \lambda_k f(x^k)$ over convex combinations realizing $\bar x$ as $\sum_k \lambda_k x^k$. This cut tightens the upper bound on $\eta$ at the current node.

Modern solvers rely primarily on the built-in generation of BQP, RLT, and second-order cuts, with general purpose or specialized routines employed to separate envelope cuts only when strict convexification is required (Du et al., 1 Nov 2025).

5. Algorithmic Performance and Computational Results

Comprehensive computational benchmarks on distributionally robust chance-constrained knapsack problems demonstrate the significant strength of the MISOCP convexification paradigm. For $n+m=50,100$:

• The SOC-root relaxation gap is 50–80% smaller than that of CCP (classical MIQCP), directly improving both lower and upper bounds.
• The SOC formulation solves nearly all small/medium-scale instances to optimality; the vanilla MIQCP model frequently times out with optimality gaps exceeding 100%.
• Node counts and CPU times are dramatically reduced: the SOC model explores 1–2 orders of magnitude fewer nodes and achieves 10–1000× faster solution speeds at optimality.
• In large-scale multidimensional knapsack, the SOC model routinely closes 95–99% of the optimality gap, while CCP leaves large gaps (30–140%).

These findings establish that introducing an explicit, convexification-driven MISOCP reformulation, together with automated cut generation, is essential for efficiently solving nonconvex mixed-integer conic and quadratic programs (Du et al., 1 Nov 2025).

6. Broader Methodological Impact and Research Connections

The convexification and reformulation approaches outlined for MISOCP have deep connections to the literature on nonlinear integer programming, robust and chance constraints, and nonconvex quadratic optimization. Extensions of these techniques can be found in:

• Disjunctive conic programming approaches, leveraging duality and aggregation to produce facet-defining SOC inequalities and minimal families of valid cuts (Burer et al., 2014).
• Polyhedral and perspective relaxations in general nonconvex MIQCP and nonconvex quadratic intersection sets (Kilinc-Karzan et al., 2014).
• Applications to queueing system optimization, network design, portfolio selection, and power grid security, where conic and integer structure is central (Ahmadi-Javid et al., 2017, Kayacık et al., 2020, Gholami et al., 2018).
• Algorithmic frameworks for decomposition, outer approximation, and branch-and-cut tailored to the unique structure of MISOCPs (Coey et al., 2018, Luo et al., 2019).

The emergence of these tools has redefined best practice for systems optimization problems characterized by joint conic and discrete structure.

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References:

• "On the Convexification of a Class of Mixed-Integer Conic Sets" (Du et al., 1 Nov 2025)
• "How to Convexify the Intersection of a Second Order Cone and a Nonconvex Quadratic" (Burer et al., 2014)
• "Two-Term Disjunctions on the Second-Order Cone" (Kilinc-Karzan et al., 2014)
• "Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion" (Ahmadi-Javid et al., 2017)
• "A Decomposition Method for Distributionally-Robust Two-stage Stochastic Mixed-integer Cone Programs" (Luo et al., 2019)
• "Outer Approximation With Conic Certificates For Mixed-Integer Convex Problems" (Coey et al., 2018)