Constraint Qualifications and Gradient Flows for Block Vanishing Constraint Problems

Abstract: Mathematical Programs with Blocks of Vanishing Constraints (MP-BVCs) are a challenging class of optimization problems due to the loss of standard constraint qualifications. Based on existing theory for classical MPVCs, i.e., MPBVC with block size one, we extend the constraint qualification theory for MPBVCs and show that the known LICQ-type condition for MPBVCs implies the Guignard Constraint Qualification (GCQ). A novel piecewise gradient flow approach converges under GCQ. It directly computes strongly stationary points and is an extension of the piecewise gradient flow approach for classic MPVC problems. Numerical experiments on several truss topology design instances with two and even three applied load forces demonstrate the robustness and effectiveness of the method.