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Convexification of classes of mixed-integer sets with L$^\natural$-convexity

Qimeng Yu, Simge Küçükyavuz

TL;DR

This work develops a comprehensive polyhedral framework for convexifying mixed-integer sets governed by $L^\natural$-convex functions. By introducing shifted extremal polymatroid inequalities (SEPIs) and a polynomial-time separation algorithm, the authors provide complete convex hull descriptions of epigraphs for single and multiple LC functions, as well as a novel mixed-integer extension $H$ whose epigraph convex hull is described by mixed-integer SEPIs (MISEPIs). They show hidden $L^\natural$-convexity in classic MIP structures such as the mixing set and continuous mixing set, unifying existing polyhedral results and deriving new ones for multi-capacity variants. The results enable tractable minimization over mixed-integer domains with LC-like structure and illuminate the relationship between LC, submodularity, and polyhedral descriptions. Overall, the paper advances exact polyhedral descriptions and efficient separation for a broad class of nonconvex problems with practical relevance in inventory, production, and related optimization domains.

Abstract

L$^\natural$ (natural)-convex functions encompass a large class of nonlinear functions over general integer domains and arise in a wide range of real-world applications. We explore the minimization of L$^\natural$-convex functions, of multiple L$^\natural$-convex functions with common variables, and of a mixed-integer extension of L$^\natural$-convex functions -- functions defined over a mixed-integer domain with properties that resemble L$^\natural$-convexity. For each of these families of minimization problems, we propose valid linear inequalities and provide convex hull descriptions for the corresponding epigraphs. For all classes of proposed inequalities, we discuss their facet conditions, develop exact separation methods, and analyze the complexity of the separation problem. We discover hidden L$^\natural$-convexity in well-known mixed-integer structures in the integer programming literature, namely the (general integer) mixing set and the continuous mixing set. We show that our findings subsume the existing polyhedral results for these sets and establish new results for the multi-capacity variant of the continuous mixing set.

Convexification of classes of mixed-integer sets with L$^\natural$-convexity

TL;DR

This work develops a comprehensive polyhedral framework for convexifying mixed-integer sets governed by -convex functions. By introducing shifted extremal polymatroid inequalities (SEPIs) and a polynomial-time separation algorithm, the authors provide complete convex hull descriptions of epigraphs for single and multiple LC functions, as well as a novel mixed-integer extension whose epigraph convex hull is described by mixed-integer SEPIs (MISEPIs). They show hidden -convexity in classic MIP structures such as the mixing set and continuous mixing set, unifying existing polyhedral results and deriving new ones for multi-capacity variants. The results enable tractable minimization over mixed-integer domains with LC-like structure and illuminate the relationship between LC, submodularity, and polyhedral descriptions. Overall, the paper advances exact polyhedral descriptions and efficient separation for a broad class of nonconvex problems with practical relevance in inventory, production, and related optimization domains.

Abstract

L (natural)-convex functions encompass a large class of nonlinear functions over general integer domains and arise in a wide range of real-world applications. We explore the minimization of L-convex functions, of multiple L-convex functions with common variables, and of a mixed-integer extension of L-convex functions -- functions defined over a mixed-integer domain with properties that resemble L-convexity. For each of these families of minimization problems, we propose valid linear inequalities and provide convex hull descriptions for the corresponding epigraphs. For all classes of proposed inequalities, we discuss their facet conditions, develop exact separation methods, and analyze the complexity of the separation problem. We discover hidden L-convexity in well-known mixed-integer structures in the integer programming literature, namely the (general integer) mixing set and the continuous mixing set. We show that our findings subsume the existing polyhedral results for these sets and establish new results for the multi-capacity variant of the continuous mixing set.

Paper Structure

This paper contains 18 sections, 52 theorems, 229 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Outline
  4. Preliminaries
  5. Examples and property preserving operations
  6. New results on L$^\natural$-convex and submodular functions
  7. Epigraphs of L$^\natural$-Convex Functions
  8. Mixed-Integer Sets Described by Multiple L$^\natural$-Convex Functions with Common Variables
  9. A Mixed-Integer Extension of L$^\natural$-Convexity
  10. Properties of $\mathcal{P}^H$ for general $H$
  11. Convexify $\mathcal{P}^H$ with L$^\natural$-convexity
  12. Properties of $F_{u_0, \mathbf{u}}$
  13. Separation of MISEPIs
  14. Facet-defining MISEPIs
  15. Examples of structured $H$ and $\text{conv}\left({\mathcal{P}^H}\right)$
  16. ...and 3 more sections

Key Result

Theorem 2.5

fujishige2000notes A function $f:\mathcal{X}\rightarrow\mathbb{R}$ is L$^\natural$-convex if and only if it is integrally convex and lattice submodular.

Figures (4)

  • Figure 1: A visual illustration of \ref{['eq:LC_def_mpc']}, where $f$ represents an L$^\natural$-convex function, the solid dots represent the function values at arbitrary pairs of elements in the domain, and the hollow dots represent the function values at the discrete midpoint(s).
  • Figure 2: A non-convex function $f(\mathbf{x}) = 10x_1^2 - x_2^2$ (left) and an L$^\natural$-convex function (right) obtained by restricting $f$ to $\mathcal{X} = \{\mathbf{x}\in\mathbb{Z}^2: \mathbf{0} \leq \mathbf{x} \leq [2\,\, 1]^\top \}$.
  • Figure 3: A Venn diagram displaying the relation between L$^\natural$-convexity and submodularity.
  • Figure 4: A visual illustration of the cycle $C=\{(1,4), (4,3), (3,1)\}$, where $N(C) = \{1,3,4\}$ and $L(C) = \{3,4\}$ because $(4,3), (3,1)$ are backward arcs.

Theorems & Definitions (115)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 105 more