Mixed-Integer Exponential Cone Programming

• MIECP is a framework that lifts nonlinear exponential and logarithmic constraints into exponential cone form for mixed-integer convex optimization.
• It integrates conic optimization, extended formulations, and branch-and-bound strategies to achieve tighter relaxations and improved solution performance.
• The approach finds practical applications in logistic regression, stochastic matching, and process systems engineering with demonstrable computational speed-ups.

Mixed-Integer Exponential Cone Programming (MIECP) is a mathematical optimization framework that models and solves problems involving both integer and continuous decision variables subject to exponential cone constraints. The exponential cone is a convex, nonpolyhedral set that enables the incorporation of exponential and logarithmic terms commonly found in applications such as logistic regression, maximum likelihood estimation, relative entropy optimization, stochastic matching, and process systems engineering. MIECP generalizes mixed-integer convex programming by “lifting” nonlinear exponential or logarithmic constraints to the exponential cone form and combines strategies from conic optimization, extended formulations, duality theory, and outer-approximation/branch-and-bound algorithms.

1. Mathematical Foundations and Exponential Cone Representation

MIECP is built on the algebraic and geometric properties of the exponential cone, which is defined as

$$K_{\exp} = \text{cl} \left\{ (u,v,w) \in \mathbb{R}^3 : v > 0,\, u \geq v \exp(w/v) \right\} \cup \left\{ (u,0,w) \in \mathbb{R}^3 : u \geq 0,\, w \leq 0 \right\}.$$

Many nonlinear constraints involving $\exp(\cdot)$ or $\log(\cdot)$ can be reformulated as membership in $K_{\exp}$ or in its perspective/hypograph forms. For mixed-integer programs, this embedding is exact and allows efficient conic modeling for problems that would otherwise be challenging under general nonconvex algebraic formulations (Ye et al., 2021, Neira et al., 2021).

The canonical MIECP formulation is

$\begin{align*} \min\ & c^{\top} x \ \text{s.t. } & (A x,\, v,\, w) \in K_{\exp} \ & x_i \in \mathbb{Z} \quad \text{(for some indices $i$)},\ \end{align*}$

where the constraint structure admits arbitrary mixtures of integer and continuous variables with exponential cone membership.

2. Duality, Optimality Conditions, and Extended Formulations

Unlike classical convex programs, MIECP and other mixed-integer convex problems require an extension of standard duality theory. The absence of supporting hyperplanes for the integer-constrained feasible region motivates dual objects built from lattice-free polyhedra, with optimality certified by collections of subgradients and integer points. In particular, for unconstrained minimization,

$$x^\star = \arg\min \{f(x): x \in \mathbb{Z}^n \times \mathbb{R}^d\}$$

is optimal if there exist $k \leq 2^n$ certificate points and subgradients $h_i \in \partial f(x_i)$ such that the polyhedron

$P = \{ x : h_i^{\top}(x - x_i) < 0,\ \forall i \}$

is lattice-free with respect to $\mathbb{Z}^n \times \mathbb{R}^d$ (Baes et al., 2014). When constraints are present, Carathéodory’s theorem ensures sparsity in the multiplier vectors needed to construct these certificates.

Extended formulations “lift” nonlinear profiles in the original variables to higher-dimensional representations, allowing conic constraints and auxiliary variables to encode separable or compositional structure. This yields stronger polyhedral relaxations, reduced OA iterations, and improved solution rates (Lubin et al., 2015). In particular, many MIECP instances (e.g., from the MINLPLIB2 library) admit direct conic reformulation using exponential, power, and second-order cones.

3. Algorithmic Strategies: Outer Approximation, Branch-and-Bound, and Cutting Plane Generation

Solution approaches for MIECP typically use outer-approximation (OA) frameworks, branch-and-bound trees, and conic or polyhedral cutting planes. In OA, the original nonpolyhedral conic constraints are relaxed using finite sets of linear cuts of the form

$z^{\top}(b - Ax) \geq 0 \quad (z\in K_{\exp}^*),$

where $K_{\exp}^*$ is the dual cone (Coey et al., 2018). These “$\mathcal{K}^*$ cuts” are generated via conic certificates returned by continuous conic subproblem solves (for fixed integer assignments) and can be disaggregated into extreme rays for tighter relaxations.

Disjunctive cuts, including lift-and-project and monoidal strengthening procedures, are generalized to the conic setting by formulating cut-generating conic programs (CGCPs). The normalization of dual multipliers is essential for ensuring strong duality and numerical stability (Lodi et al., 2019). Empirical studies show that proper cut selection and normalization yield robust, sparse, and gap-closing polyhedral approximations—competitive with state-of-the-art methods for mixed-integer second-order and exponential cone problems.

Recent solver developments (e.g., Pajarito, MOSEK-IP/OA) exploit external LP and conic solver integration, advanced separation routines, and certificate-based cutting planes to accelerate convergence, especially for instances involving exponential cones (Coey et al., 2018, Ye et al., 2021).

4. Approximation Schemes: SOC and Polyhedral Approximations

For practical efficiency and compatibility with general MILP/MISOCP solvers, several approximation methods for exponential cones are developed:

• Second-Order Cone (SOC) Approximations: The exponential cone is "lifted" to an extended space using generating functions and Gaussian quadrature, producing a collection of SOC constraints that approximate the cone within a controllable error $\epsilon$. With scaling and shifting methods, the number of constraints required scales as $O(\log(1/\epsilon))$ (Ye et al., 2021).
• Polyhedral Outer Approximations: The exponential cone's hypograph can be tightly approximated by tangent planes at selected points, yielding gradient inequalities of the form

$$\log(x_i) + \frac{1}{x_i}(x - x_i) \geq \nu \qquad \forall i.$$

The number of constraints grows as $O(1/\sqrt{\epsilon})$ with proven lower bounds (Ye et al., 2021).

These techniques enable MIECPs to be solved as MILPs or MISOCPs with tractable relaxation gaps and have demonstrated substantial computational speed-ups (often 5–20×) over direct conic interior-point methods in solver MOSEK.

5. Reformulations and Generalized Disjunctive Programming

Convex Generalized Disjunctive Programming (GDP) models naturally admit conic reformulations, either via big-$M$ or hull (convex hull) techniques. Both approaches preserve the conic structure arising from exponential, second-order, or quadratic constraints. The hull reformulation, in particular, introduces perspective variables and avoids the need for approximation, yielding tight continuous relaxations and efficient solver performance (Neira et al., 2021). These frameworks are applicable to a wide range of process systems engineering and machine learning instances, such as logistic regression, network synthesis, and clustering.

The explicit use of conic representations in MIECP thus both improves numerical stability and tightens relaxations, enabling faster and more robust global optimization.

6. Applications and Empirical Performance

MIECP finds application in several domains:

• Stochastic Matching in Logistics: In the recommend-to-match problem with uncertain supply acceptance, MIECP is used to reformulate the expected utility maximization with nonlinear “max” terms via log-sum-exp approximations anchored by exponential cone constraints. This produces a tractable mixed-integer exponential cone model with provable parametric bounds. Experimental results on synthetic and real-world freight data demonstrate near-optimal solution quality and over 90% reduction in computation time compared to MILP or sample average approximation methods (Liu et al., 21 Oct 2025).
• Machine Learning and Statistical Estimation: Logistic regression models with integer constraints are modeled as MIECPs through hypograph representations of the logarithm, with exponential cone constraints capturing the nonlinear terms (Neira et al., 2021). Sparse logistic regression, packing/covering instances, and entropy-based problems are solved using SOC/polyhedral MIECP approximations, achieving optimality gaps below $10^{-4}$ with substantial speed-ups.
• Process Systems Engineering: Configuration problems, process network synthesis, and retrofit synthesis leverage MIECP with conic hull reformulation, improving relaxation tightness and reducing solver run times (Neira et al., 2021).

7. Future Directions and Practical Considerations

Current research in MIECP suggests several directions:

• Further development of valid inequalities and conic cuts (submodular or disjunctive cuts) could reduce relaxation gaps and improve node pruning in branch-and-bound.
• Closing gaps between theoretical lower bounds and practical construction sizes for SOC/polyhedral approximations.
• Robust branch-and-cut and certificate-based OA algorithms that tighten relaxations and reduce the need for expensive conic subproblem solves.
• Expansion to nonsymmetric and nonconvex cones beyond the exponential cone, enabling applications for relative entropy, Kullback–Leibler divergence, and other information-theoretic models.

Practical implementation is increasingly supported by dedicated modeling languages (e.g., JuMP), conic solver APIs (e.g., MathProgBase), and mature solver packages (MOSEK, Pajarito, SCIP), which streamline the transition from high-level MIDCP modeling to efficient mixed-integer exponential cone optimization.

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Key Papers Referenced:

• Duality and optimality framework (Baes et al., 2014).
• Extended formulations and conic representability (Lubin et al., 2015).
• Outer-approximation via conic certificates and solver design (Coey et al., 2018).
• Disjunctive cuts, normalization, and conic lifting (Lodi et al., 2019).
• SOC/polyhedral approximations and computational results (Ye et al., 2021).
• GDP reformulations and conic solvers (Neira et al., 2021).
• Application to stochastic matching with exponential cone reformulation and bounds (Liu et al., 21 Oct 2025).