On the linear independence constraint qualification in disjunctive programming

Mathematical programmes 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 programmes and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programmes with switching constraints are by-products of our approach.