Mixed-Integer Convex Formulation

• Mixed-integer convex formulation is an optimization approach that integrates discrete variables with convex continuous relaxations to enable tractable global optimization.
• It employs extended formulations and outer approximation algorithms to tighten relaxations and reduce computational iterations.
• The framework has diverse applications in engineering, energy systems, machine learning, and operations research, supporting scalable and exact solutions.

A mixed-integer convex formulation is an optimization framework in which some variables are constrained to take discrete (typically integer or binary) values while the constraints and objective exhibit convexity in the continuous variables. This paradigm unifies the modeling power of convex analysis—allowing efficient global optimization of continuous variables in the presence of convex functions and sets—with the structural expressiveness of integer programming, which encodes combinatorial or logical decisions. The interplay between integer constraints and convex structure gives rise to a rich mathematical theory, powerful algorithmic techniques (notably outer approximation algorithms), a spectrum of extended formulations (particularly conic and disjunctive), and diverse applications spanning engineering, network design, statistics, machine learning, and operations research.

1. Foundations and Formalism of Mixed-Integer Convex Formulation

Let $x \in \mathbb{Z}^n$ and $y \in \mathbb{R}^p$ represent the integer and continuous variables, respectively. The canonical mixed-integer convex programming (MICP) problem is stated as

$$\begin{aligned} \text{minimize}_{x, y} \quad & f(x, y) \ \text{subject to} \quad & g_j(x, y) \le 0 \,\,\, \forall j \in J, \ & L \le x \le U, \quad x \in \mathbb{Z}^{n}, \quad y \in \mathbb{R}_{+}^{p}, \end{aligned}$$

where $f$ and $g_j$ are convex functions in the continuous variables for all fixed integer values. The feasible set is hence the union of convex sets over all admissible discrete assignments.

Convexity of the objective and constraint functions combined with bounded integer variables guarantees, in principle, tractability: the continuous relaxations are efficiently solvable, but the resulting feasible set is generally nonconvex due to integrality conditions, necessitating sophisticated solution approaches.

A central generalization is the mixed-integer conic form:

$$\begin{aligned} \min_{x, z}\quad & c^T z \ \text{subject to}\quad & A_x x + A_z z = b, \ & L \le x \le U, \quad x \in \mathbb{Z}^n, \quad z \in \mathcal{K}, \end{aligned}$$

where $\mathcal{K}$ is a closed convex cone—e.g., the nonnegative orthant, second-order (SOC), exponential (EXP), or power (POW) cone. Virtually all practical instances arising in benchmark libraries, such as MINLPLIB2, can be recast in such a conic form using standard cones (Lubin et al., 2015, Lubin et al., 2016).

2. Extended Formulations: Structure, Lifting, and Conic Representability

A defining advancement in mixed-integer convex optimization is the systematic use of extended formulations—reformulating the original problem in a higher-dimensional space by introducing auxiliary variables and constraints. The principal motivation is that direct (original space) polyhedral approximations of complicated convex nonlinear constraints are often extremely weak; for instance, approximating the unit $\ell_2$ ball in $\mathbb{R}^n$ through tangent hyperplanes requires $2^n$ inequalities. By exposing separability or other structure via auxiliary variables (i.e., lifting), one can obtain polyhedral or conic outer approximations with dramatically fewer constraints and tighter relaxations (Lubin et al., 2015).

A representative extended formulation for separable convex constraints is:

$$\sum_k t_k \le 0, \qquad g_k(x_k) \le t_k \forall k.$$

Conic representability offers a unifying extended formulation mechanism, as many convex sets and functions (including nonsmooth examples) can be encoded via epigraphs of conic representable forms. The epigraph of the geometric mean $-\sqrt{xy}$ for $x, y \geq 0$ is representable by an affine transformation into the SOC (rotated second-order cone). Similarly, power productions, exponential functions, and norms map naturally into power and exponential cones (Lubin et al., 2015, Lubin et al., 2016).

This approach allows almost all practical MICP problems to be reformulated with a small number of cone types. Empirical evidence from MINLPLIB2 demonstrates conic representability is universal for tested instances, with the SOC and EXP cones accounting for the vast majority of model queries (Lubin et al., 2015, Lubin et al., 2016).

3. Outer Approximation Algorithms and Polyhedral Approximation

Solving MICPs efficiently relies on outer approximation (OA) algorithms, which alternate between solving mixed-integer linear (polyhedral or conic) master problems and continuous convex subproblems. OA constructs a relaxation of the feasible region by successively adding valid inequalities (cuts) derived from convexity—typically tangent planes or dual certificates—over integer feasible candidates.

Mathematically, for smooth convex $g_j$, the tangent inequality

$g_j(x') + \nabla g_j(x')^T (x - x') \leq 0$

is a global valid cut for all $x \in X$, tight at $x = x'$. The iterative OA method builds a polyhedral or conic set $P$ as the intersection of supporting hyperplanes, solves the mixed-integer linear/conic relaxation (MIOA), fixes the discrete variables, and solves the resulting continuous subproblem. If the subproblem is feasible and optimal, global optimality is achieved; otherwise, new cuts are generated (Lubin et al., 2016).

In the conic context, the OA master (MIOA) takes the form:

$$\begin{aligned} \min_{x, z} \quad & c^T z \ \text{subject to} \quad & A_x x + A_z z = b, \ & L \le x \le U, \quad x \in \mathbb{Z}^n, \ & \beta^T z \geq 0 \quad \forall \beta \in T, \end{aligned}$$

where $T \subset \mathcal{K}^*$ accumulates dual vectors from continuous subproblem solutions. Finite convergence of OA in the conic setting is guaranteed due to the finiteness of integer assignments (Lubin et al., 2015).

The key computational insight is that extended (lifted) OA formulations—via conic or polyhedral embeddings—require fewer master iterations and yield tighter relaxations, directly improving solution times and tractability on challenging instances (Lubin et al., 2016).

4. Automated Modeling: Disciplined Convex Programming and the DCP–Formulation Correspondence

A principal challenge in constructing effective extended formulations is extracting and exploiting algebraic structure in the model. To overcome the intractability of arbitrary algebraic convexity detection (which is NP-hard), disciplined convex programming (DCP) prescribes that models be built from a catalogue of convex atomic functions (atoms), assembled according to rules guaranteeing the convexity of composite expressions (Lubin et al., 2015, Lubin et al., 2016).

The crucial insight is that every DCP-compliant composition corresponds to an extended formulation. For example, if $h(x) = g(f(x))$ with $g$ convex, nondecreasing, and $f$ convex, then the epigraph

$E_h = \{ (x, t): h(x) \le t \}$

is represented by introducing an auxiliary variable $s$ and writing:

$(x, s) \in E_f, \qquad (s, t) \in E_g.$

This is a lifted, stronger formulation equivalent to the algebraic structure exposed by DCP. Notably, this mechanism works regardless of whether the resulting set is conic representable; thus, DCP supplies both a convexity certificate and the basis for automatable extended formulation construction (Lubin et al., 2015, Lubin et al., 2016).

DCP-oriented modeling languages (CVX, CVXPY, Convex.jl) leverage these principles to automatically produce extended formulations that are both theoretically strong and computationally tractable.

5. Advanced Polyhedral Descriptions and Separation Algorithms

Beyond separable or standard conic-epigraphic forms, certain mixed-integer convex sets admit polyhedral convex hull descriptions with strong facet-defining inequalities, crucial for addressing bilinear and multilinear structures that arise (for example) in covering, assignment, or multilinear function constraints.

Convex hull descriptions can be exponentially large but admit compact extended formulations through lifting and auxiliary variables. For bounded mixed-integer bilinear covering sets,

$$S^U = \{ (x, y) \in \mathbb{Z}_+^n \times \mathbb{R}_+^n : \sum_{i=1}^n x_i y_i \ge r, x \le u \},$$

the convex hull can be described via an exponentially-sized but systematic family of inequalities or, alternatively, through an extended formulation with a pseudopolynomial number of constraints and auxiliary variables $w \in \mathbb{R}^n$ (serving as proxies for the tightest linearization in each coordinate) (Rahman et al., 2017).

Efficient separation algorithms are pivotal: the paper (Rahman et al., 2017) devises a linear-time procedure for checking violation of the convex hull inequalities and generating cuts, thereby facilitating integration into cutting-plane or branch-and-cut algorithms without exponential computational overhead.

6. Numerical Performance, Benchmarking, and Solver Advances

Empirical studies on benchmark libraries (such as MINLPLIB2) confirm that extended formulations and conic representations, either derived algebraically or via DCP translation, lead to significant algorithmic gains. Concrete results include:

• Substantial reduction in the number of OA iterations (e.g., 994 vs. 2685 for Bonmin on benchmark sets) and overall solution times.
• Resolution of previously unsolved instances (e.g., “gams01” solved in 6 iterations using conic/DCP-based extended formulation (Lubin et al., 2016)).
• Superior or competitive performance relative to established solvers such as Bonmin, SCIP, Artelys Knitro, and even commercial conic solvers (for problems representable in the SOC/EXP form) (Lubin et al., 2016).
• Compactness of the new formulations: many benchmark instances only need a small collection of standard cones, establishing general representability and scalability.

These computational gains are directly linked to the strength of the underlying extended formulation and the exploitation of lifted algebraic structure.

7. Applications and Impact

Mixed-integer convex formulations have broad applicability, enabling globally optimal solutions in domains such as:

• Process systems engineering (design and operation scheduling with discrete switches and convex economics).
• Energy systems modeling (unit commitment, optimal power flow).
• Supply chain and logistics (facility location with economies of scale).
• Portfolio optimization and selection (cardinality constraints and convex risk objectives).
• Machine learning and signal processing (sparse regression, compressed sensing).
• Non-rigid shape matching and image analysis (via convex MIP models for correspondence and transformation).

Their capacity to encode combinatorial decision-making within a provably tractable convex relaxation—strengthened by extended formulations and advanced OA algorithms—constitutes a central methodology enabling scalable and exact global optimization across scientific and industrial domains.

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In summary, mixed-integer convex formulation is distinguished by its exploitation of convexity under integer constraints, advanced lifting and conic/epigraphic representations, algorithmic paradigms such as OA with finite convergence, and automated structure extraction via DCP. The resulting theory and algorithms yield strong computational outcomes and support a wide range of modern optimization applications (Lubin et al., 2015, Lubin et al., 2016, Rahman et al., 2017).