Martin Kružík

Occupation measure relaxations in variational problems: the role of convexity

  • Didier Henrion
  • Milan Korda
  • Martin Kružík
  • Rodolfo Rios-Zertuche
    • Categories Linear, Cone and Semidefinite Programming Tags

    This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming reformulation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving … Read more

    Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

  • Didier Henrion
  • Martin Kružík
  • Marek Tyburec
  • Jan Zeman
    • Categories Civil and Environmental Engineering, Constrained Nonlinear Optimization, Semi-definite Programming Tags , , , , ,

    The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is … Read more

    Modular-topology optimization with Wang tilings: An application to truss structures

  • Martin Doskar
  • Martin Kružík
  • Matej Lepš
  • Marek Tyburec
  • Jan Zeman
    • Categories Civil and Environmental Engineering, Convex Optimization, Meta Heuristics Tags , , , ,

    Modularity is appealing for solving many problems in optimization. It brings the benefits of manufacturability and reconfigurability to structural optimization, and enables a trade-off between the computational performance of a Periodic Unit Cell (PUC) and the efficacy of non-uniform designs in multi-scale material optimization. Here, we introduce a novel strategy for concurrent minimum-compliance design of … Read more

    Semi-definite relaxations for optimal control problems with oscillation and concentration effects

  • Mathieu Claeys
  • Didier Henrion
  • Martin Kružík
    • Categories Control Applications, Semi-definite Programming

    Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phenomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the control input and typically featuring concentration phenomena, interpreting the control as a measure of time with a … Read more

    On the control of an evolutionary equilibrium in micromagnetics

  • Michal Kočvara
  • Martin Kružík
  • Jiri V. Outrata
    • Categories Basic Sciences Applications, Complementarity and Variational Inequalities, Systems governed by Differential Equations Optimization Tags , , , ,

    We formulate an optimal control problem of magnetization in a ferromagnet as a mathematical program with evolutionary equilibrium constraints. The evolutionary nature of the equilibrium is due to the hysteresis behavior of the respective magnetization process. To solve the problem numerically, we adapted the implicit programming technique. The adjoint equations, needed to compute the subgradients … Read more