Systems governed by Differential Equations Optimization

A Sparsity Preserving Convexification Procedure for Indefinite Quadratic Programs Arising in Direct Optimal Control

  • Moritz Diehl
  • Rien Quirynen
  • Robin Verschueren
  • Mario Zanon
    • Categories Systems governed by Differential Equations Optimization Tags , ,

    Quadratic programs (QP) with an indefinite Hessian matrix arise naturally in some direct optimal control methods, e.g. as subproblems in a sequential quadratic programming (SQP) scheme. Typically, the Hessian is approximated with a positive definite matrix to ensure having a unique solution; such a procedure is called \emph{regularization}. We present a novel regularization method tailored … Read more

    Inexact Newton-Type Optimization with Iterated Sensitivities

  • Moritz Diehl
  • Sebastien Gros
  • Rien Quirynen
    • Categories Systems governed by Differential Equations Optimization Tags , , ,

    This paper presents and analyzes an Inexact Newton-type optimization method based on Iterated Sensitivities (INIS). A particular class of Nonlinear Programming (NLP) problems is considered, where a subset of the variables is defined by nonlinear equality constraints. The proposed algorithm considers an arbitrary approximation for the Jacobian of these constraints. Unlike other inexact Newton methods, … Read more

    Efficient Symmetric Hessian Propagation for Direct Optimal Control

  • Moritz Diehl
  • Boris Houska
  • Rien Quirynen
    • Categories Systems governed by Differential Equations Optimization Tags , , ,

    Direct optimal control algorithms first discretize the continuous-time optimal control problem and then solve the resulting finite dimensional optimization problem. If Newton type optimization algorithms are used for solving the discretized problem, accurate first as well as second order sensitivity information needs to be computed. This article develops a novel approach for computing Hessian matrices … Read more

    Lifted Collocation Integrators for Direct Optimal Control in ACADO Toolkit

  • Moritz Diehl
  • Sebastien Gros
  • Boris Houska
  • Rien Quirynen
    • Categories Systems governed by Differential Equations Optimization Tags , , ,

    This paper presents a class of efficient Newton-type algorithms for solving the nonlinear programs (NLPs) arising from applying a direct collocation approach to continuous time optimal control. The idea is based on an implicit lifting technique including a condensing and expansion step, such that the structure of each subproblem corresponds to that of the multiple … Read more

    pyomo.dae: A Modeling and Automatic Discretization Framework for Optimization with Differential and Algebraic Equations

  • Lorenz T. Biegler
  • Jean-Paul Watson
  • Victor M. Zavala
  • Bethany Nicholson
  • John D. Siirola
    • Categories Modeling Languages and Systems, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , ,

    We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http: //www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, … Read more

    Global optimal control with the direct multiple shooting method

  • Holger Diedam
  • Sebastian Sager
    • Categories Global Optimization Applications, Global Optimization Theory, Systems governed by Differential Equations Optimization Tags , ,

    We propose to solve global optimal control problems with a new algorithm that is based on Bock’s direct multiple shooting method. We provide conditions and numerical evidence for a significant overall runtime reduction compared to the standard single shooting approach. CitationOptimal Control Applications and Methods, Vol. 39 (2), pp. 449–470, 2017 DOI 10.1002/oca.2324 Article online … Read more

    Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

  • Christian Kirches
  • Felix Lenders
  • Paul Manns
    • Categories Approximation Algorithms, Complementarity and Variational Inequalities, Systems governed by Differential Equations Optimization

    We extend recent work on mixed-integer nonlinear optimal control prob- lems (MIOCPs) to the case of integer control functions subject to constraints. Promi- nent examples of such systems include problems with restrictions on the number of switches permitted, or problems that minimize switch cost. We extend a theorem due to [Sager et al., Math. Prog. … Read more

    Constrained Optimization with Low-Rank Tensors and Applications to Parametric Problems with PDEs

  • Sebastian Garreis
  • Michael Ulbrich
    • Categories Optimization of Systems modeled by PDEs, Stochastic Programming, Systems governed by Differential Equations Optimization Tags , , , , , , ,

    Low-rank tensor methods provide efficient representations and computations for high-dimensional problems and are able to break the curse of dimensionality when dealing with systems involving multiple parameters. We present algorithms for constrained nonlinear optimization problems that use low-rank tensors and apply them to optimal control of PDEs with uncertain parameters and to parametrized variational inequalities. … Read more

    A framework for simultaneous aerodynamic design optimization in the presence of chaos

  • Nicolas R. Gauger
  • Stefanie Günther
  • Qiqi Wang
    • Categories Systems governed by Differential Equations Optimization Tags , , , , ,

    Integrating existing solvers for unsteady partial differential equations (PDEs) into a simultaneous optimization method is challenging due to the forward- in-time information propagation of classical time-stepping methods. This paper applies the simultaneous single-step one-shot optimization method to a reformulated unsteady PDE constraint that allows for both forward- and backward-in-time information propagation. Especially in the presence … Read more

    Robust Numerical Calibration for Implied Volatility Expansion Models

  • Radu Baltean-Lugojan
  • Panos Parpas
    • Categories Finance and Economics, Nonlinear Systems and Least-Squares, Systems governed by Differential Equations Optimization Tags , , ,

    Implied volatility expansions allow calibration of sophisticated volatility models. They provide an accurate fit and parametrization of implied volatility surfaces that is consistent with empirical observations. Fine-grained higher order expansions offer a better fit but pose the challenge of finding a robust, stable and computationally tractable calibration procedure due to a large number of market … Read more