Constrained Nonlinear Optimization

An Algorithm for Degenerate Nonlinear Programming with Rapid Local Convergence

  • Stephen Wright
    • Categories Constrained Nonlinear Optimization Tags , ,

    The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. The first is any conventional method for nonlinear programming that produces estimates of the Lagrange multipliers at each … Read more

    Convex- and Monotone- Transformable Mathematical Programming Problems and a Proximal-Like Point Method

  • João Xavier da Cruz Neto
  • Orizon Pereira Ferreira
  • L. R. Lucambio Pérez
  • S. Z. Nemeth
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity, Global Optimization Theory Tags , ,

    The problem of finding singularities of monotone vectors fields on Hadamard manifolds will be considered and solved by extending the well-known proximal point algorithm. For monotone vector fields the algorithm will generate a well defined sequence, and for monotone vector fields with singularities it will converge to a singularity. It will be also shown how … Read more

    KNITRO-Direct: A Hybrid Interior Algorithm for Nonlinear Optimization

  • Jose Luis Morales
  • Jorge Nocedal
  • Dominique Orban
  • Richard A. Waltz
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags ,

    A hybrid interior-point method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search based method which computes steps by factoring the primal-dual equations and an iterative method using a conjugate gradient algorithm and globalized by means of trust regions. Steps computed by a direct factorization are always tried first, … Read more

    Sharpening the Karush-John optimality conditions

  • Arnold Neumaier
  • Hermann Schichl
    • Categories Constrained Nonlinear Optimization Tags , ,

    A refined version of the Karush-John first order optimality conditions is presented which reduces the number of constraints for which a constraint qualification is needed. This version is a generalization both of the Karush-John conditions and of the first order optimality conditions for concave constraints. ArticleDownload View PDF

    Finding the projection of a point onto the intersection of convex sets via projections onto halfspaces

  • L.M. Bregman
  • Yair Censor
  • Simeon Reich
  • Y. Zepkowitz-Malachi
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Convex Optimization Tags , , , ,

    We present a modification of Dykstra’s algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a halfspace or onto the intersection of two halfspaces. Convergence of the algorithm is established and special choices of the halfspaces are proposed. The option to project onto halfspaces instead of general … Read more

    An Interior Point Method for Mathematical Programs with Complementarity Constraints (MPCCs)

  • Lorenz T. Biegler
  • Arvind U. Raghunathan
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    Interior point methods for nonlinear programs (NLPs) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primal-dual algorithm has been adapted with a modified step calculation. The algorithm is shown to … Read more

    A Starting-Point Strategy for Nonlinear Interior Methods

  • Michael Gertz
  • Jorge Nocedal
  • Annick Sartenaer
    • Categories Constrained Nonlinear Optimization Tags ,

    This paper presents a strategy for choosing the initial point, slacks and multipliers in interior methods for nonlinear programming. It consists of first computing a Newton-like step to estimate the magnitude of these three variables and then shifting the slacks and multipliers so that they are sufficiently positive. The new strategy has the option of … Read more

    On the Convergence of Successive Linear Programming Algorithms

  • Richard H. Byrd
  • Nicholas I. M. Gould
  • Jorge Nocedal
  • Richard A. Waltz
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    We analyze the global convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more specifically the SLP-EQP approach presented in \cite{ByrdGoulNoceWalt02}. Every iteration requires the solution of two trust region subproblems involving linear and quadratic models, respectively. The interaction between the trust regions of these … Read more

    A hybrid algorithm for nonlinear equality constrained optimization problems: global and local convergence theory

  • Mohammedi El Hallabi
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Robust Optimization Tags , , , , , , ,

    In this paper we combine both trust-region and linesearch globalization strategies in a globally convergent hybrid algorithm to solve a continuously differentiable nonlinear equality constrained minimization problem. First, the trust-region approach is used to determine a descent direction of the augmented Lagrangian chosen as the merit function, and second, linesearch techniques are used to obtain … Read more

    A Local Convergence Theory of a Filter Line Search Method for Nonlinear Programming

  • Choong Ming Chin
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    In this paper the theory of local convergence for a class of line search filter type methods for nonlinear programming is presented. The algorithm presented here is globally convergent (see Chin [4]) and the rate of convergence is two-step superlinear. The proposed algorithm solves a sequence of quadratic progrmming subproblems to obtain search directions and … Read more