Nonlinear Optimization

An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations

  • Frank E. Curtis
  • Olaf Schenk
    • Categories Nonlinear Optimization, Optimization of Systems modeled by PDEs Tags , , , , , ,

    We present a line-search algorithm for large-scale continuous optimization. The algorithm is matrix-free in that it does not require the factorization of derivative matrices. Instead, it uses iterative linear system solvers. Inexact step computations are supported in order to save computational expense during each iteration. The algorithm is an interior-point approach derived from an inexact … Read more

    A New Relaxation Scheme for Mathematical Programs with Equilibrium Constraints

  • Michael Ulbrich
  • Sonja Veelken
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently non-degenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The … 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

    Self-correcting geometry in model-based algorithms for derivative-free unconstrained optimization

  • Katya Scheinberg
  • Philippe L. Toint
    • Categories Optimization of Simulated Systems, Unconstrained Optimization Tags , ,

    Several efficient methods for derivative-free optimization (DFO) are based on the construction and maintenance of an interpolation model for the objective function. Most of these algorithms use special “geometry-improving” iterations, where the geometry (poisedness) of the underlying interpolation set is made better at the cost of one or more function evaluations. We show that such … Read more

    The Advanced Step NMPC Controller: Optimality, Stability and Robustness

  • Lorenz T. Biegler
  • Victor M. Zavala
    • Categories Control Applications, Nonlinear Optimization, Optimization of Systems modeled by PDEs

    Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear … 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

    An interior point algorithm for nonlinear minimax problems

  • Enitan Obasanjo
  • Berc Rustem
  • George Tzallas-Regas
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We present a primal-dual interior point method for constrained nonlinear, discrete minimax problems where the objective functions and constraints are not necessarily convex. The algorithm uses two merit functions to ensure progress towards the points satisfying the first order optimality conditions of the original problem. Convergence properties are described and numerical results provided. CitationDept. of … Read more

    Identifying Activity

  • Adrian Lewis
  • Stephen Wright
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Identification of active constraints in constrained optimization is of interest from both practical and theoretical viewpoints, as it holds the promise of reducing an inequality-constrained problem to an equality-constrained problem, in a neighborhood of a solution. We study this issue in the more general setting of composite nonsmooth minimization, in which the objective is a … Read more

    Hybrid MPI/OpenMP parallel support vector machine training

  • Jacek Gondzio
  • Kristian Woodsend
    • Categories Data-Mining, Parallel Algorithms, Quadratic Programming Tags , , ,

    Support Vector Machines are a powerful machine learning technology, but the training process involves a dense quadratic optimization problem and is computationally challenging. A parallel implementation of Support Vector Machine training has been developed, using a combination of MPI and OpenMP. Using an interior point method for the optimization and a reformulation that avoids the … Read more

    A Line Search Exact Penalty Method Using Steering Rules

  • Richard H. Byrd
  • Gabriel Lopez-Calva
  • Jorge Nocedal
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    Line search algorithms for nonlinear programming must include safeguards to enjoy global convergence properties. This paper describes an exact penalization approach that extends the class of problems that can be solved with line search SQP methods. In the new algorithm, the penalty parameter is adjusted at every iteration to ensure sufficient progress in linear feasibility … Read more