Christian Kirches

Improved Regularity Assumptions for Partial Outer Convexification of Mixed-Integer PDE-Constrained Optimization problems

  • Christian Kirches
  • Paul Manns
    • Categories (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags , ,

    Partial outer convexification is a relaxation technique for MIOCPs being constrained by time-dependent differential equations. Sum-Up-Rounding algorithms allow to approximate feasible points of the relaxed, convexified continuous problem with binary ones that are feasible up to an arbitrarily small $\delta > 0$. We show that this approximation property holds for ODEs and semilinear PDEs under … Read more

    Approximation Properties of Sum-Up Rounding in the Presence of Vanishing Constraints

  • Christian Kirches
  • Felix Lenders
  • Paul Manns
    • Categories (Mixed) Integer Nonlinear Programming, Approximation Algorithms, Systems governed by Differential Equations Optimization Tags , ,

    Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak$^*$ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, … Read more

    Combining Multi-Level Real-time Iterations of Nonlinear Model Predictive Control to Realize Squatting Motions on Leo

  • Christian Kirches
  • Ivan Koryakovskiy
  • Manuel Kudruss
  • Katja Mombaur
  • Heike Vallery
    • Categories Constrained Nonlinear Optimization, Control Applications, Nonlinear Systems and Least-Squares Tags , , , , ,

    Today’s humanoid robots are complex mechanical systems with many degrees of freedom that are built to achieve locomotion skills comparable to humans. In order to synthesize whole-body motions, real-tme capable direct methods of optimal control are a subject of contemporary research. To this end, Nonlinear Model Predictive Control is the method of choice to realize … Read more

    High-Level Interfaces for the Multiple Shooting Code for Optimal Control MUSCOD

  • Christian Kirches
  • Manuel Kudruss
  • Felix Lenders
    • Categories Modeling Languages and Systems, Optimization Software Design Principles, Problem Solving Environments Tags , , , ,

    The demand for model-based simulation and optimization solutions requires the availability of software frameworks that not only provide computational capabilities, but also help to ease the formulation and implementation of the respective optimal control problems. In this article, we present and discuss recent development efforts and applicable work flows using the example of MUSCOD, the … Read more

    Numerical solution of optimal control problems with explicit and implicit switches

  • Hans Georg Bock
  • Christian Kirches
  • Andreas Meyer
  • Andreas Potschka
    • Categories Constrained Nonlinear Optimization, Systems governed by Differential Equations Optimization

    In this article, we present a unified framework for the numerical solution of optimal control problems constrained by ordinary differential equations with both implicit and explicit switches. We present the problem class and qualify different types of implicitly switched systems. This classification significantly affects opportunities for solving such problems numerically. By using techniques from generalized … Read more

    trlib: A vector-free implementation of the GLTR method for iterative solution of the trust region problem

  • Christian Kirches
  • Andreas Potschka
  • Felix Lenders
    • Categories Optimization Software and Modeling Systems, Quadratic Programming Tags , , ,

    We describe trlib, a library that implements a Variant of Gould’s Generalized Lanczos method (Gould et al. in SIAM J. Opt. 9(2), 504–525, 1999) for solving the trust region problem. Our implementation has several distinct features that set it apart from preexisting ones. We implement both conjugate gradient (CG) and Lanczos iterations for assembly of … 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

    An Active-Set Quadratic Programming Method Based On Sequential Hot-Starts

  • Travis C. Johnson
  • Christian Kirches
  • Andreas Wächter
    • Categories Quadratic Programming Tags , , , , , , ,

    A new method for solving sequences of quadratic programs (QPs) is presented. For each new QP in the sequence, the method utilizes hot-starts that employ information computed by an active-set QP solver during the solution of the first QP. This avoids the computation and factorization of the full matrices for all but the first problem … Read more

    Mixed-Integer Nonlinear Optimization

  • Pietro Belotti
  • Christian Kirches
  • Sven Leyffer
  • Jeff Linderoth
  • James R. Luedtke
  • Ashutosh Mahajan
    • Categories (Mixed) Integer Nonlinear Programming Tags , , ,

    Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling … Read more

    On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

  • Michael N. Jung
  • Christian Kirches
  • Sebastian Sager
    • Categories (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags ,

    Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, … Read more