The optimal partition for multiparametric semialgebraic optimization

TL;DR

This paper explores the structure of optimal partitions in multiparametric semialgebraic conic linear optimization, revealing properties like continuity and semialgebraicity, and decomposing feasible sets into linearity and nonlinearity regions.

Contribution

It establishes new properties of set-valued mappings in mpCLOs and uses algebraic geometry to decompose semialgebraic sets into finite linear and nonlinear parts.

Findings

Feasible sets can be decomposed into finite linearity and nonlinearity subsets.

Properties like continuity, monotonicity, and semialgebraicity are characterized for set-valued mappings.

Structural results on the boundary of the feasible set, especially for spectrahedra.

Abstract

In this paper we investigate the optimal partition approach for multiparametric conic linear optimization (mpCLO) problems in which the objective function depends linearly on vectors. We first establish more useful properties of the set-valued mappings early given by us (arXiv:2006.08104) for mpCLOs, including continuity, monotonicity and semialgebraic property. These properties characterize the notions of so-called linearity and nonlinearity subsets of a feasible set, which serve as stability regions of the partition of a conic (linear inequality) representable set. We then use the arguments from algebraic geometry to show that a semialgebraic conic representable set can be decomposed into a union of finite linearity and/or nonlinearity subsets. As an application, we investigate the boundary structure of the feasible set of generic semialgebraic mpCLOs and obtain several nice structural results in this direction, especially for the spectrahedon.