Piecewise Polyhedral Formulations for a Multilinear Term

Polyhedral Analysis of Symmetric Multilinear Polynomials over Box Constraints

It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification question for multilinear polynomials that are symmetric with respect to permutations of variables. Such a permutation-invariant structure naturally implies a quadratic-sized extended formulation for the envelopes through the use of disjunctive programming. The optimization and separation problems are answered directly without using this extension. The probl…

Convex Hull Formulations for Mixed-Integer Multilinear Functions

In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables. We develop two equivalent convex relaxations of an MIMF and study their polyhedral properties in their corresponding higher-dimensional spaces. We numerically observe that the proposed formulations consistently perform better than state-of-the-art relaxation approaches.

分段多面体映射的极大单调性

Maximal monotonicity of piecewise polyhedral mappings

研究分段多面体映射的极大单调性问题，核心方法是利用分段多面体结构绕过通常涉及相对内部交集非空的约束条件，主要贡献是解决了相关构造是否保持极大单调性的关键问题。

Solution Uniqueness of Convex Piecewise Affine Functions Based Optimization with Applications to Constrained $\ell_1$ Minimization

In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems incorporates many important polyhedral constrained $\ell_1$ recovery problems arising from sparse optimization, such as basis pursuit, LASSO, and basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging the max-formulation of convex piecewise affine functions and convex analysis tools, we develop dual variables based necessary and sufficien…

Discrete-to-Continuous Extensions: piecewise multilinear extension, min-max theory and spectral theory

We introduce the homogeneous and piecewise multilinear extensions and the eigenvalue problem for locally Lipschitz function pairs, in order to develop a systematic framework for relating discrete and continuous min-max problems. This also enables us to investigate spectral properties for pairs of $p$-homogeneous functions and to propose a critical point theory for zero-homogeneous functions. The main contributions are: (1) We provide several min-max relations between an original discrete formulation and its piecewise multilinear extension. We introduce the concept of perfect domain pairs to vi…

坐标方式多面体方法用于引出多元线性效用和单变量非线性效用函数

Coordinate-wise Polyhedral Method for Eliciting Multivariate Linear Utility and Univariate Nonlinear Utility Functions

提出坐标方式多面体方法（CPM），通过预定义坐标切割并求解线性方程组设计查询，实现线性收敛率，并扩展至单变量分段线性及一般Lipschitz连续效用函数。