Systems governed by Differential Equations Optimization

Reformulations and Algorithms for the Optimization of Switching Decisions in Nonlinear Optimal Control

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

    In model-based nonlinear optimal control switching decisions that can be optimized often play an important role. Prominent examples of such hybrid systems are gear switches for transport vehicles or valves in chemical engineering. Optimization algorithms need to take the discrete nature of the variables that model these switching decisions into account. Unnecessarily, for many applications … Read more

    Second-order analysis of optimal control problems with control and initial-final state constraints

  • Joseph Frédéric Bonnans
  • Nikolai Osmolovskii
    • Categories Systems governed by Differential Equations Optimization Tags , , , ,

    This paper provides an analysis of Pontryagine mimina satisfying a quadratic growth condition, for optimal control problems of ordinary differential equations with constraints on initial-final state, as well as control constraints satisfying the uniform positive linear independence condition. CitationRapport de Recherche INRIA 6707, Oct. 2008.ArticleDownload View PDF

    On fast integration to steady state and earlier times

  • Uri Ascher
  • Hui Huang
  • Benar Fux Svaiter
  • Kees van den Doel
    • Categories Basic Sciences Applications, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , , ,

    The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best … Read more

    Automatically Assessing the Performance of an Optimization-Based Multigrid Method

  • Robert Michael Lewis
  • Stephen G. Nash
    • Categories Systems governed by Differential Equations Optimization Tags , ,

    Many large nonlinear optimization problems are based upon discretizations of underlying function spaces. Optimization-based multigrid methods—that is, multigrid methods based on solving coarser versions of an optimization problem—are designed to solve such discretized problems efficiently by taking explicit advantage of the family of discretizations. The methods are generalizations of more traditional multigrid methods for solving … Read more

    An Inexact Newton Method for Nonconvex Equality Constrained Optimization

  • Richard H. Byrd
  • Frank E. Curtis
  • Jorge Nocedal
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We present a matrix-free line search algorithm for large-scale equality constrained optimization that allows for inexact step computations. For strictly convex problems, the method reduces to the inexact sequential quadratic programming approach proposed by Byrd et al. [SIAM J. Optim. 19(1) 351–369, 2008]. For nonconvex problems, the methodology developed in this paper allows for the … Read more

    Nonlinear programming without a penalty function or a filter

  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Systems and Least-Squares, Systems governed by Differential Equations Optimization Tags , , ,

    A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a barrier or a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, … Read more

    Modeling and Simulation of Metabolic Networks for Estimation of Biomass Accumulation Parameters

  • Lorenz T. Biegler
  • Bulent Karasozen
  • Metin Turkay
  • Ugur Kaplan
    • Categories Biomedical Applications, Systems governed by Differential Equations Optimization Tags , ,

    Metabolic networks are defined as the collection of biochemical reactions within a cell that define the functions of that cell. Due to the growing need to understand the functions of biological organisms for industrial and medical purposes, modeling and simulation of metabolic networks has attracted a lot of attention recently. Traditionally, metabolic networks are modeled … Read more

    An Inexact SQP Method for Equality Constrained Optimization

  • Richard H. Byrd
  • Frank E. Curtis
  • Jorge Nocedal
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We present an algorithm for large-scale equality constrained optimization. The method is based on a characterization of inexact sequential quadratic programming (SQP) steps that can ensure global convergence. Inexact SQP methods are needed for large-scale applications for which the iteration matrix cannot be explicitly formed or factored and the arising linear systems must be solved … Read more

    The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

  • Andrei V. Dmitruk
  • Alexander Kaganovich
    • Categories Constrained Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , , ,

    We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle. CitationA.V. Dmitruk, A.M. Kaganovich. The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle, Systems & Control Letters, … Read more

    Numerical Experience with a Recursive Trust-Region Method for Multilevel Nonlinear Optimization

  • Serge Gratton
  • Annick Sartenaer
  • Philippe L. Toint
    • Categories Nonlinear Optimization, Systems governed by Differential Equations Optimization, Unconstrained Optimization Tags , , , ,

    We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Sartenaer, Toint (2004), and provide significant numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a very satisfactory compromise between reliability and efficiency. The resulting default algorithm is then compared to alternative … Read more