Systems governed by Differential Equations Optimization

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. Citation Optimal Control Applications and Methods, Vol. 39 (2), pp. 449–470, 2017 DOI 10.1002/oca.2324 Article … 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

    Combinatorial Optimal Control of Semilinear Elliptic PDEs

  • Christoph Buchheim
  • Renke Kuhlmann
  • Christian Meyer
    • Categories Cutting Plane Approaches, Semi-infinite Programming, Systems governed by Differential Equations Optimization Tags , , , ,

    Optimal control problems (OCP) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, it results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we … Read more

    A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function

  • Radu Ioan Bot
  • Ernö Robert Csetnek
    • Categories Nonsmooth Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We address the minimization of the sum of a proper, convex and lower semicontinuous with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The … Read more

    Nonlinear Programming Strategies on High-Performance Computers

  • Naiyuan Chiang
  • Carl D. Laird
  • Victor M. Zavala
  • Jia Kang
    • Categories Constrained Nonlinear Optimization, Parallel Algorithms, Systems governed by Differential Equations Optimization

    We discuss structured nonlinear programming problems arising in control applications, and we review software and hardware capabilities that enable the efficient exploitation of such structures. We focus on linear algebra parallelization strategies and discuss how these interact and influence high-level algorithmic design elements required to enforce global convergence and deal with negative curvature in a … Read more

    Bridging the Gap Between Multigrid, Hierarchical, and Receding-Horizon Control

  • Victor M. Zavala
    • Categories Distributed Control, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We analyze the structure of the Euler-Lagrange conditions of a lifted long-horizon optimal control problem. The analysis reveals that the conditions can be solved by using block Gauss-Seidel schemes and we prove that such schemes can be implemented by solving sequences of short-horizon problems. The analysis also reveals that a receding-horizon control scheme is equivalent … Read more