Systems governed by Differential Equations Optimization

Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

  • John W. Pearson
  • Margherita Porcelli
  • Martin Stoll
    • Categories Control Applications, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , , ,

    PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and … Read more

    Improved Regularity Assumptions for Partial Outer Convexification of Mixed-Integer PDE-Constrained Optimization problems

  • Christian Kirches
  • Paul Manns
    • Categories (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags , ,

    Partial outer convexification is a relaxation technique for MIOCPs being constrained by time-dependent differential equations. Sum-Up-Rounding algorithms allow to approximate feasible points of the relaxed, convexified continuous problem with binary ones that are feasible up to an arbitrarily small $\delta > 0$. We show that this approximation property holds for ODEs and semilinear PDEs under … Read more

    Approximation Properties of Sum-Up Rounding in the Presence of Vanishing Constraints

  • Christian Kirches
  • Felix Lenders
  • Paul Manns
    • Categories (Mixed) Integer Nonlinear Programming, Approximation Algorithms, Systems governed by Differential Equations Optimization Tags , ,

    Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak$^*$ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, … Read more

    A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , , ,

    We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally. CitationSubmitted for publicationArticleDownload View PDF

    On Algebraic Proofs of Stability for Homogeneous Vector Fields

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares … Read more

    Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control

  • Sebastian Sager
  • Tobias Weber
  • Clemens Zeile
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags , , , , ,

    Solving mixed-integer nonlinear programs (MINLPs) is hard in theory and practice. Decomposing the nonlinear and the integer part seems promising from a computational point of view. In general, however, no bounds on the objective value gap can be guaranteed and iterative procedures with potentially many subproblems are necessary. The situation is different for mixed-integer optimal … Read more

    Combinatorial Integral Approximation for Mixed-Integer PDE-Constrained Optimization Problems

  • Mirko Hahn
  • Sebastian Sager
    • Categories (Mixed) Integer Nonlinear Programming, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , ,

    We apply the basic principles underlying combinatorial integral approximation methods for mixed-integer optimal control with ordinary differential equations in general, and the sum-up rounding algorithm specifically, to optimization problems with partial differential equation (PDE) constraints. By doing so, we identify two possible generalizations that are applicable to problems involving PDE constraints with mesh-dependent integer variables, … Read more

    A Decision Tool based on a Multi-Objective Methodology for designing High-Pressure Thermal Treatments in Food Industry

  • Miriam R. Ferrández
  • Benjamin Ivorra
  • Pilar M. Ortigosa
  • Ángel M. Ramos
  • Juana L. Redondo
    • Categories Meta Heuristics, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , ,

    In this work, we propose a methodology for designing High-Pressure Thermal processes for food treatment. This approach is based on a multi-objective preference-based evolutionary optimization algorithm, called WASF-GA, combined with a decision strategy which provides the food engineer with the best treatment in accordance with some quality requirements. The resulting method is compared to a … Read more

    CasADi – A software framework for nonlinear optimization and optimal control

  • J A E Andersson
  • Moritz Diehl
  • J B Rawlings
  • J Gillis
  • G Horn
    • Categories Modeling Languages and Systems, Systems governed by Differential Equations Optimization Tags , ,

    We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by … Read more

    Sum of squares certificates for stability of planar, homogeneous, and switched systems

  • Amir Ali Ahmadi
  • Pablo A. Parrilo
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result … Read more