Milan Korda

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

    Polynomial argmin for recovery and approximation of multivariate discontinuous functions

  • Didier Henrion
  • Milan Korda
  • Jean-Bernard Lasserre
    • Categories Basic Sciences Applications, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , ,

    We propose to approximate a (possibly discontinuous) multivariate function f(x) on a compact set by the partial minimizer arg min_y p(x,y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when … Read more

    Convex computation of extremal invariant measures of nonlinear dynamical systems and Markov processes

  • Didier Henrion
  • Milan Korda
  • Igor Mezic
    • Categories Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and con- tinuous time. The set of all invariant measures is characterized as the feasible set of an infinite-dimensional linear program (LP). The objective functional of this LP is then used to single-out a specific measure … Read more

    Moments and convex optimization for analysis and control of nonlinear partial differential equations

  • Didier Henrion
  • Jean-Bernard Lasserre
  • Milan Korda
    • Categories Distributed Control, Optimization of Systems modeled by PDEs, Semi-definite Programming Tags , , , ,

    This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a \emph{linear} equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is … Read more

    Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets

  • Didier Henrion
  • Milan Korda
    • Categories Infinite Dimensional Optimization, Semi-definite Programming Tags , , , ,

    Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set $K$. The idea consists of approximating from above the indicator function of $K$ with a sequence of polynomials of increasing degree $d$, so that the integrals of these polynomials generate a convergence sequence of upper bounds … Read more

    Convergence rates of moment-sum-of-squares hierarchies for optimal control problems

  • Didier Henrion
  • Colin N. Jones
  • Milan Korda
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , , ,

    We study the convergence rate of moment-sum-of-squares hierarchies of semidefinite programs for optimal control problems with polynomial data. It is known that these hierarchies generate polynomial under-approximations to the value function of the optimal control problem and that these under-approximations converge in the $L^1$ norm to the value function as their degree $d$ tends to … Read more

    Convex computation of the region of attraction of polynomial control systems

  • Didier Henrion
  • Milan Korda
    • Categories Control Applications, Convex Optimization, Semi-definite Programming Tags , , , , , ,

    We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the … Read more