Binary Convex Hull Optimization

• Binary convex hull optimization is a method for constructing convex relaxations of mixed-integer problems by integrating binary variables with conic constraints.
• It employs specialized combinatorial and polyhedral techniques, including polymatroid and polar inequalities, to achieve computationally tractable models.
• Efficient algorithmic approaches like rank-based methods and column-generation are used to scale these techniques for applications such as portfolio optimization and scheduling.

Binary convex hull optimization refers to the theory and practice of constructing or characterizing convex hulls of sets in which some or all variables are binary (i.e., take values in $\{0,1\}$), often combined with continuous or conic constraints. This class of problems underlies a wide range of optimization models in operations research, integer programming, nonconvex quadratic optimization, computational geometry, and conic mixed-integer programming. The treatment of such sets necessitates specialized combinatorial and polyhedral analysis, as neither generic convex hull descriptions nor naive linear relaxations suffice for computational tractability or solution tightness.

1. Fundamental Structures and Canonical Problem Statement

Binary convex hull optimization problems typically involve feasible sets defined by both binary variables $z \in \{0,1\}^n$ and (possibly high-dimensional) continuous variables. A general mixed-binary conic set takes the form

$$S\Bigl(\{f_j\}, \{\mathcal K_j\}\Bigr) = \Bigl\{ (x, z) \in \mathbb{R}^{mp} \times \{0,1\}^n : \exists\, y \in \mathbb{R}^p \text{ s.t. } y_j \geq f_j(z),\; A^j x^j + B^j y_j \in \mathcal K_j \; \forall j \Bigr\}$$

where:

• $z$ are the binary or indicator variables;
• $x$ are block-structured continuous variables;
• $y$ are “epigraph” variables, linked to nonlinear functions $f_j:\{0,1\}^n\to\mathbb{R}_+$;
• Each $A^j$, $B^j$, and closed convex pointed cone $\mathcal{K}_j$ (e.g., SOC, PSD cone).

Convexifying such sets—constructing $\operatorname{conv}(S)$—is central to obtaining tight continuous relaxations and effective cutting planes in mixed-integer convex optimization (Kılınç-Karzan et al., 2020).

2. Polyhedral and Conic Hull Characterizations

The convex hull of mixed-binary sets as above is given by

$$\operatorname{conv}\bigl(S(\{f_j\},\{\mathcal K_j\})\bigr) =\Bigl\{(x,z)\in\mathbb{R}^{mp}\times[0,1]^n:\;\exists\,y\in\mathbb{R}_+^p\text{ s.t. } (y,z)\in\operatorname{conv}\bigl(\operatorname{epi}(f_1)\cap\cdots\cap\operatorname{epi}(f_p)\bigr),\; A^j x^j+B^j y_j\in\mathcal K_j\ \forall j\Bigr\}$$

where the only nontrivial inequalities beyond the cones arise from the joint convexification of the epigraphs

$$\operatorname{epi}(f_j) = \bigl\{ (y_j, z) \in \mathbb{R}_+ \times \{0,1\}^n : y_j \geq f_j(z) \bigr\}.$$

For arbitrary nonnegative $f$, valid inequalities—so-called polar inequalities—are of the form

$$y - f(\emptyset) \ge \pi^\top z, \quad \forall \pi \in P_{\tilde f},$$

where $$P_{\tilde f} = \{ \pi \in \mathbb{R}^n : \sum_{i \in S} \pi_i \le \tilde f(S),\ \forall S \subseteq [n] \}$$ and $\tilde f(z) = f(z) - f(0)$. For submodular $f$, the convex hull is exactly described by all extended polymatroid (Edmonds-Lovász) inequalities

$$y - f(0) \ge \sum_{i=1}^n \bigl[ f(V_i) - f(V_{i-1}) \bigr]\,z_{\sigma(i)},\quad\forall\sigma,$$

with $V_i = \{\sigma(1),\dots, \sigma(i)\}$ for permutations $\sigma$ (Kılınç-Karzan et al., 2020).

3. Specialized Constructions: Quadratic and Conic Cases

When $f_j$ have specific structure (e.g., $f(z) = \sqrt{\sigma+\sum c_i z_i}$), the convex hull admits SOC or rotated-SOC representations without recourse to KKT-based analysis. For example, mean–risk portfolio models and best-subset problems can be convexified via this machinery, leading to mixed-integer SOCPs whose relaxations substantially close the integrality gap by virtue of exact polymatroidal inequalities (Kılınç-Karzan et al., 2020).

For quadratic optimization with switching variables (i.e., $x$, $X = xx^T$, and $y \in \{0,1\}^2$), the convex hull for $n=2$ is represented as the projection onto $(x,X,y,Y_{12})$ of a tractably-sized SDP system, involving:

• RLT-style constraints: $\operatorname{diag}(X)\le x \le y$;
• Triangle or “perspective” constraints for cross-product terms $(y_1,y_2,Y_{12})$;
• A single block PSD constraint of size $5 \times 5$;
• Auxiliary variables $\alpha_j$ to effect the lift (2002.04681).

When cross-terms $y_1y_2$ are ignored, a conjectured single-block $5\times 5$ SDP and RLT constraints suffice for exactness (2002.04681).

4. Algorithmic Approaches and Computational Models

Large-scale and high-dimensional binary convex hull problems require computationally efficient algorithms. For dense 2D integer data, the rank-based method defines an $O(n)$ ranking function $f:\mathbb{Z}^2\to\{1,\ldots, m\}$, enabling polygonal chain construction in $O(n + m)$ time and Melkman’s online hull extraction in $O(n)$. For practical data densities ($\rho = n/m \ge 1/p$; $p$ = machine word size, e.g., 32 or 64), this yields end-to-end $O(n)$ complexity, outperforming classical $O(n \log n)$ hull algorithms for binary-image or digital shape applications (Megson et al., 2013).

In parallel computational models, optimal (work, span, cache complexity) hull algorithms are provided in the binary-forking and cache-oblivious models for both presorted ($O(n)$ work, $O(\log n)$ span, $O(n/B)$ cache complexity) and unsorted data ($O(n\log n)$ work, $O(\log n\log\log n)$ span, $O((n/B)\log_M n)$ cache) (Browne et al., 2023).

5. Convex Hull Approximation under Discrete Resource Constraints

The polyhedral convex hull approximation with a hyperplane budget (PCAB) addresses the scenario where a budget $K$ of hyperplanes is imposed, e.g., for separating a positive point set $P$ from negatives $N$ in $\mathbb{R}^d$. The goal is to construct a convex polyhedron $H$ with at most $K$ faces that includes $P$ and (ideally) excludes all points of $N$, or else minimizes the number of negatives in $H$. Mathematical programming models with binary variables assign each negative point to be excluded by some hyperplane or register its “inside” status, with column-generation procedures (COLGEN 2) yielding polytopes that closely approximate exact convex hulls for moderate $K$, scaling to $d=8$ and $n\approx 10^4$; these outperform naive greedy or direct MIP models in both hull tightness and separation accuracy (Barbato et al., 2024).

6. Applications and Theoretical Insights

Binary convex hull characterizations and their computational methods underpin several problem classes:

• Mean-risk and chance-constrained optimization;
• Lot-sizing, scheduling, and sparse SDP relaxations;
• Feature selection and best-subset selection problems;
• Constraint learning via labelled point sets and resource-controlled hull construction.

Critical computational insights include:

• All nontrivial work in convexifying mixed-binary conic sets reduces to the convexification of each epigraph $\operatorname{epi}(f_j)$;
• Submodular $f_j$ admit facet-complete, polyhedral descriptions (polymatroid inequalities), with every extreme point yielding a facet;
• Separability: cone constraints and “$f$–cuts” can be handled in orthogonal subroutines, enabling modular algorithm design;
• Polynomial-time separation of polymatroid inequalities via greedy or Edmonds-Lovász type algorithms;
• Column-generation and Dantzig–Wolfe approaches allow scalable hull approximation with hyperplane budgets and extend classical support vector machine ideas;
• For $n=2$ quadratic indicator problems, the convex hull collapses to a single SDP block, guiding systematic extension to higher dimensions (Kılınç-Karzan et al., 2020, 2002.04681, Barbato et al., 2024).

7. Outlook and Connections

Binary convex hull optimization unifies and generalizes a spectrum of results in mixed-integer conic optimization, combinatorial polyhedral theory, computational geometry, and convexification for nonconvex quadratic optimization. Modern results enable the integration of SOC, PSD, and general conic constraints with arbitrary nonnegative (and in many cases submodular) binary set functions. A promising direction is the further development of tractable lifted SDP or polyhedral descriptions for larger $n$ in quadratic and conic indicator settings, as well as the automated generation of valid cutting planes for black-box conic integer models (Kılınç-Karzan et al., 2020, 2002.04681).