Systems governed by Differential Equations Optimization

Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

  • Alfio Borzì
  • Valentina De Simone
  • Daniela di Serafino
    • Categories Systems governed by Differential Equations Optimization Tags , , , ,

    We present algebraic multilevel preconditioners for linear systems arising from the discretization of systems of coupled elliptic partial differential equations (PDEs). These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary … Read more

    On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous Vector Fields

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

    It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this … Read more

    TACO – A Toolkit for AMPL Control Optimization

  • Christian Kirches
  • Sven Leyffer
    • Categories Modeling Languages and Systems, Problem Solving Environments, Systems governed by Differential Equations Optimization

    We describe a set of extensions to the AMPL modeling language to conveniently model mixed-integer optimal control problems for ODE or DAE dynamic processes. These extensions are realized as AMPL user functions and suffixes and do not require intrusive changes to the AMPL language standard or implementation itself. We describe and provide TACO, a Toolkit … Read more

    Efficient Direct Multiple Shooting for Nonlinear Model Predictive Control on Long Horizons

  • Hans Georg Bock
  • Christian Kirches
  • Johannes P. Schlöder
  • Leonard Wirsching
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , ,

    We address direct multiple shooting based algorithms for nonlinear model predictive control, with a focus on problems with long prediction horizons. We describe different efficient multiple shooting variants with a computational effort that is only linear in the horizon length. Proposed techniques comprise structure exploiting linear algebra on the one hand, and approximation of derivative … Read more

    Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin’s Maximum Principle

  • Sebastian Sager
    • Categories (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags , ,

    Optimum Experimental Design (OED) problems are optimization problems in which an experimental setting and decisions on when to measure – the so-called sampling design – are to be determined such that a follow-up parameter estimation yields accurate results for model parameters. In this paper we use the interpretation of OED as optimal control problems with … Read more

    A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

  • Hédy Attouch
  • Benar Fux Svaiter
    • Categories Complementarity and Variational Inequalities, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , , , , , , , ,

    We introduce non-autonomous continuous dynamical systems which are linked to Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant … Read more

    NONSMOOTH OPTIMIZATION OVER THE (WEAKLY OR PROPERLY) PARETO SET OF A LINEAR-QUADRATIC MULTI-OBJECTIVE CONTROL PROBLEM : EXPLICIT OPTIMALITY CONDITIONS

  • Henri Bonnel
  • Ngoc Sang Pham
    • Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization, Systems governed by Differential Equations Optimization Tags , ,

    We present explicit optimality conditions for a nonsmooth functional defined over the (properly or weakly) Pareto set associated to a multiobjective linear-quadratic control problem. This problem is very difficult even in a finite dimensional setting, i.e. when, instead of a control problem, we deal with a mathematical programming problem. Amongst different applications, our problem may … Read more

    Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions

  • Samir Adly
  • Abderrahim Hantoute
  • Michel Thera
    • Categories Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , , , , , ,

    The main objective of this paper is to provide new explicit criteria to characterize weak lower semi-continuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by the means of the proximal … Read more

    Convergence and Descent Properties for a Class of Multilevel Optimization Algorithms

  • Stephen G. Nash
    • Categories Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags ,

    I present a multilevel optimization approach (termed MG/Opt) for the solution of constrained optimization problems. The approach assumes that one has a hierarchy of models, ordered from fine to coarse, of an underlying optimization problem, and that one is interested in finding solutions at the finest level of detail. In this hierarchy of models calculations … Read more

    Lipschitz solutions of optimal control problems with state constraints of arbitrary order

  • Joseph Frédéric Bonnans
    • Categories Control Applications, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , ,

    In this paper we generalize to an arbitrary order, under minimal hypotheses, some sufficient conditions for Lipschitz continuity of the solution of a state constrained optimal control problems. The proof combines the approach by Hager in 1979 for dealing with first-order state constraints, and the high-order alternative formulation of the optimality conditions. Citation Published as … Read more