On robust mathematical programs with vanishing constraints under data uncertainty

The main objective of this paper is to study mathematical programs with vanishing constraints involving data uncertainty (UMPVC) and to address them using a robust optimization framework that accounts for the worst-case scenario. We begin by formulating the model and presenting an illustrative example from truss topology optimization under uncertain loading conditions. Robust Fritz-John optimality conditions are derived for UMPVC, and an extended no nonzero abnormal multiplier constraint qualification (ENNAMCQ) is introduced to obtain robust Karush-Kuhn-Tucker (KKT) necessary optimality conditions for UMPVC. Additionally, we identify the robust strong stationary points of UMPVC and establish robust sufficient optimality conditions under generalized convexity assumptions. We also determine robust weak stationary points of UMPVC using a tightened nonlinear programming approach to seek robust necessary and sufficient optimality conditions. The robust versions of several constraint qualifications (CQs), like Abadie CQ, Mangasarian-Fromovitz CQ, and linearly independent CQ, are developed to handle the uncertainties associated with the special structure of the vanishing constraints. Algorithms are proposed to implement the theoretical results, and several illustrative examples are provided to demonstrate their effectiveness.