On the linear independence constraint qualification in disjunctive programming

TL;DR

This paper introduces a generalized linear independence constraint qualification for mathematical programs with disjunctive constraints, deriving new optimality conditions and applying them to various problem classes, including switching constraints.

Contribution

It proposes an abstract version of LICQ for MPDCs and derives first- and second-order optimality conditions based on strongly stationary points.

Findings

New second-order optimality conditions for switching constraints

Comparison with existing literature on disjunctive programs

Applicability of the constraint qualification to multiple problem classes

Abstract

Mathematical programs with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality-, and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some popular classes of disjunctive programs and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programs with switching constraints are by-products of our approach.