This monograph studies optimization problems for rigid punches in elastic media and for high-speed penetration of rigid strikers into deformed elastoplastic, concrete, and composite media using variational calculations, tools from functional analysis, and stochastic and min-max (guaranteed) optimization approaches with incomplete data. The book presents analytical and numerical results developed by the authors during the last ten years. Contents Part I: Quasi-statics of contact interaction Optimal contact pressure and optimal punch shape Influence of forces applied outside contact region Shape optimization of punch moving with friction Optimization in contact problems under incomplete data Application of probabilistic data Minimization of material wear Multiobjective optimization of contact pressure, wear of material, and energy dissipation Part II: High-speed penetration into deformable medium Optimization of impactor shape Strikers of nonaxisymmetric optimal shape Multiobjective optimization of rigid shell Optimization of truncated rotating strikers Penetration into solid medium under strength constraints Some problems of global multipurpose structural optimization Analytical and numerical estimations of optimal parameters of layered slab Striker shape optimization using condition of minimal ballistic limit velocity for layered slabs Part III: Appendices Nonadditive functionals and extremum conditions Multivariant estimation of functionals and their sensitivity Basic ideas of multicriteria optimization Evolutionary algorithm of global optimization