Fundamentals of Structural Optimization (III)

Fundamentals of Structural Optimization (III) – Game-Theory Methods develops a unified, Lie-algebraic and geometric framework for modeling and solving competitive and uncertainty-aware optimization problems as structured games. It shows how static and differential lattice games on semisimple Lie groups can be translated—via mirroring and Lie expansion/retraction—into computable spectral problems. The volume explains when equilibrium selection is governed by eigenvalues (commuting/reference regimes) versus when it is governed by singular values (noncommuting, nonlinear, heterogeneous settings). It further introduces Green–Zerna metrics and Milnor curvature to interpret how Hamiltonian flow sensitivity produces geometric incompatibilities, and provides an asymptotic reduction pathway from differential games to terminal static interaction problems. Finally, it connects the methodology to engineering relevance through heterogeneous games and bilevel structural optimization, where design parameters parameterize lower-level game dynamics and the upper level optimizes the resulting equilibrium value.