Constraint qualification analysis and optimality conditions for nonlinear engineering programming problems

Abstract

Constraint qualifications provide an analytical basis for the validity of both first- and second-order optimality principles in nonlinear programming models that have applications in engineering design areas such as structural, mechanical, and network engineering. In many cases of non-convex equality and inequality constraints, feasible regions do not fully satisfy the regularity conditions of such concepts as linear independence constraint qualification, making it impossible to prove the need of the Karush-Kuhn-Tucker theory at degenerate points. The framework of qualification theory is proposed that offers a connection between generalized Mangasarian-Fromovitz and Abadie-type conditions, and conditions of the metric and sub-regularity qualification theory for the problems of coupled, possibly non-smooth systems of engineering constraints. The concepts of tangent and linearized feasibility cones are introduced in order to describe the geometry of regularity of constraint qualifications, and to derive the sharp descriptions of the boundedness and compactness of sets of Lagrange multipliers. Based on the proposed theory, it is proved that generalized conditions of first-order stationarity and conditions of second-order sufficiency are valid for the entire critical cone of the Lagrangian function. The practical relevance of the theoretical results is illustrated in a benchmark optimization problem in structural engineering which includes various constraints of stress and deformation that are non-convex in nature.