Topological solution characterization of feasible regions in complex engineering optimization problems with nonlinear constraints

Abstract

In structural design, fluid-structure interaction, and aerodynamic configuration, solving complex engineering optimization problems often involves high-dimensional, highly nonlinear, and non-convex constraints, making it impossible for conventional optimization methods, such as gradient descent and interior-point strategies, to traverse between different regions of the feasible (i.e., feasible) solution set. In most analyses of feasible regions, only local geometry is leveraged, including properties like convexity, star-shapedness, and Karush-Kuhn-Tucker stationarity, therefore neglecting the global topological structure of the feasible region. The present paper proposes a general topological framework for mapping non-linear feasible sets onto CW complexes and stratified manifolds, introduces the precise criteria for contractibility, homotopy equivalence, and path-connectedness of the feasible region, and establishes two theorems connecting specific homotopy-theoretic invariants of the topology with the global convergence of optimization processes. In particular, the presented method employs Betti numbers and persistent homology to estimate the number of connected components and independent loops induced by non-linear equality and inequality constraints, and applies Morse inequalities to derive lower bounds on the total number of stationary points. Sample calculations performed on the truss optimization problem with non-convex characteristics and the synthesis of kinematic mechanisms demonstrate that having four disconnected regions the single-start gradient descent method will, probably, succeed in finding the global optimum 24% of the time, while the interior-point technique will only yield a 29% success rate. At the same time, the topology-aware multi-start algorithm achieving 90% of success under the same condition needed on, average, 51% less iterations than the interior-point strategy. In addition, the persistent homology allows for identifying actual topological voids at the filtration values that are over an order of magnitude larger than the sampling noise error. The above results show that global topological characterization of feasible region is both a mathematically rigorous and practically useful approach to tackle problems related to failure of engineering design optimization processes.