Constrained Nonlinear Optimization

Interior-point method for nonlinear programming with complementarity constraints

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , , ,

    In this report, we propose an algorithm for solving nonlinear programming problems with com-plementarity constraints, which is based on the interior-point approach. Main theoretical results concern direction determination and step-length selection. We use an exact penalty function to remove complementarity constraints. Thus a new indefinite linear system is defined with a tridiagonal low-right submatrix. Inexact … Read more

    An adaptive cubic regularisation algorithm for nonconvex optimization with convex constraints and its function-evaluation complexity

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Constrained Nonlinear Optimization Tags , , , , ,

    The adaptive cubic overestimation algorithm described in Cartis, Gould and Toint (2007) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, but without any Lipschitz continuity requirement on the objective’s Hessian. A worst-case complexity analysis in … Read more

    An Active Set Strategy for Solving Optimization Problems with up to 200,000,000 Nonlinear Constraints

  • Klaus Schittkowski
    • Categories Constrained Nonlinear Optimization

    We propose a numerical algorithm for solving smooth nonlinear programming problems with a large number of constraints, but a moderate number of variables. The active set method proceeds from a given bound mw for the maximum number of expected violated constraints, where mw is a user-provided parameter less than the total number of constraints. A … 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

    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. Citation Dept. … 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

    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

    Optimal structure of gas transmission trunklines

  • Jean André
  • Joseph Frédéric Bonnans
    • Categories Constrained Nonlinear Optimization, Facility Planning and Design Tags , , , ,

    In this paper, we consider the optimal design of a straight pipeline system. Suppose a gas pipeline is to be designed to transport a specified flowrate from the entry point to the gas demand point. Physical and contractual requirements at supply and delivery nodes are known as well as the costs to buy and lay … Read more

    A second derivative SQP method: local convergence

  • Nicholas I. M. Gould
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization Tags , , ,

    Gould and Robinson (NAR 08/18, Oxford University Computing Laboratory, 2008) gave global convergence results for a second-derivative SQP method for minimizing the exact $\ell_1$-merit function for a \emph{fixed} value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. … Read more

    A Sequential Quadratic Programming Algorithm with an Additional Equality Constrained Phase

  • Jose Luis Morales
  • Jorge Nocedal
  • Yuchen Wu
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags ,

    A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods. It avoids the difficulties associated with indefinite quadratic programming subproblems by defining this subproblem to be always convex. The novel feature of the approach is the addition of an equality constrained phase that promotes fast … Read more