Systems governed by Differential Equations Optimization

A Collection of 1,300 Dynamical Systems for Testing Data Fitting, Optimal Control, Experimental Design, Identification, Simulation or Similar Software – User’s Guide

  • Klaus Schittkowski
    • Categories Modeling Languages and Systems, Optimization Software Benchmark, Systems governed by Differential Equations Optimization

    We describe a collection of test problems which have been used to develop and test data fitting software for identifying parameters in explicit model functions, dynamical systems of equations, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time-dependent partial differential equations with or without algebraic equations. The test cases … Read more

    Data Fitting and Experimental Design in Dynamical Systems with EASY-FIT ModelDesign

  • Klaus Schittkowski
    • Categories Nonlinear Systems and Least-Squares, Problem Solving Environments, Systems governed by Differential Equations Optimization

    EASY-FIT is an interactive software system to identify parameters and compute optimal designs in explicit model functions, steady-state systems, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time dependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e. observation times and measurements, the minimum … Read more

    A Factorization with Update Procedures for a KKT Matrix Arising in Direct Optimal Control

  • Hans Georg Bock
  • Christian Kirches
  • Sebastian Sager
  • Johannes P. Schlöder
    • Categories Quadratic Programming, Systems governed by Differential Equations Optimization Tags , ,

    Quadratic programs obtained for optimal control problems of dynamic or discrete–time processes usually involve highly block structured Hessian and constraints matrices. Efficient numerical methods for the solution of such QPs have to respect and exploit this block structure. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite … Read more

    Block Structured Quadratic Programming for the Direct Multiple Shooting Method for Optimal Control

  • Hans Georg Bock
  • Christian Kirches
  • Sebastian Sager
  • Johannes P. Schlöder
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming, Systems governed by Differential Equations Optimization Tags , ,

    In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved … Read more

    The Integer Approximation Error in Mixed-Integer Optimal Control

  • Hans Georg Bock
  • Moritz Diehl
  • Sebastian Sager
    • Categories (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP) in two ways. We improve a theorem that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike … Read more

    A Fast Moving Horizon Estimation Algorithm Based on Nonlinear Programming Sensitivity

  • Lorenz T. Biegler
  • Carl D. Laird
  • Victor M. Zavala
    • Categories Control Applications, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    Moving Horizon Estimation (MHE) is an efficient optimization-based strategy for state estimation. Despite the attractiveness of this method, its application in industrial settings has been rather limited. This has been mainly due to the difficulty to solve, in real-time, the associated dynamic optimization problems. In this work, a fast MHE algorithm able to overcome this … Read more

    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. Citation Rapport de Recherche INRIA 6707, Oct. 2008. Article Download View Second-order analysis of optimal control problems … Read more

    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