Global solution characterization and landscape analysis of nonlinear engineering optimization problems

Abstract

The existence of non-convexity is prevalent in modern engineering design such as aircraft structure design, multi-modal construction mechanics, and huge power grid scheduling which induces local optimization traps and prevents successful convergence to the globally optimal design. In contrast to the standard engineering design methods where the design space is considered as a numerical domain of search, in this paper the design space is considered as a manifold which enables to identify the critical points, basins of attractions, and barrier heights of optimization. In this way, some important connections have been found between classical nonlinear programming techniques — generalized Mangasarian-Fromovitz Constraint Qualification (GMFCQ) and metric sub-regularity on the one hand and global Morse-theoretic landscape explanations on the other hand. This results in the definition of a rigorous characterization of the global solutions able to overcome difficulties of classical Linear Independence Constraint Qualification (LICQ) methods. Hence, new conditions generalized Karush-Kuhn-Tucker (G-KKT) have been introduced which allowed to demonstrate the bounded, compact and convex multiplier properties under GMFCQ assumptions and to prove the conditions of uniqueness of the local minima. The effectiveness of the proposed approach has been confirmed by results of numerical experiments on benchmark case of multi-member truss design which resulted in the success of GMFCQ which provided bounded multipliers while classical approaches failed.