Constrained Nonlinear Optimization

Elastic-Mode Algorithms for Mathematical Programs with Equilibrium Constraints: Global Convergence and Stationarity Properties

  • Mihai Anitescu
  • Paul Tseng
  • Stephen Wright
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study … Read more

    Local Analysis of the Feasible Primal-Dual Interior-Point Method

  • R. Silva
  • Luis Nunes Vicente
  • J. Soares
    • Categories Constrained Nonlinear Optimization Tags , , ,

    In this paper we analyze the rate of local convergence of the Newton primal-dual interior-point method when the iterates are kept strictly feasible with respect to the inequality constraints. It is shown under the classical conditions that the rate is q-quadratic when the functions associated to the inequality constraints define a locally concave feasible region. … Read more

    On Augmented Lagrangian methods with general lower-level constraints

  • Roberto Andreani
  • Ernesto G. Birgin
  • José Mario Martínez
  • María Laura Schuverdt
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lower-level type. Two methods of this class are introduced and analyzed. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved … Read more

    Adaptive Barrier Strategies for Nonlinear Interior Methods

  • Jorge Nocedal
  • Richard A. Waltz
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization

    This paper considers strategies for selecting the barrier parameter at every iteration of an interior-point method for nonlinear programming. Numerical experiments suggest that adaptive choices, such as Mehrotra’s probing procedure, outperform static strategies that hold the barrier parameter fixed until a barrier optimality test is satisfied. A new adaptive strategy is proposed based on the … Read more

    Active Set Identification in Nonlinear Programming

  • Christina Oberlin
  • Stephen Wright
    • Categories Constrained Nonlinear Optimization Tags , ,

    Techniques that identify the active constraints at a solution of a nonlinear programming problem from a point near the solution can be a useful adjunct to nonlinear programming algorithms. They have the potential to improve the local convergence behavior of these algorithms, and in the best case can reduce an inequality constrained problem to an … Read more

    On the Global Convergence of a Trust Region Method for Solving Nonlinear Constraints Infeasibility Problem

  • Choong Ming Chin
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonlinear Systems and Least-Squares Tags , , , ,

    A framework for proving global convergence for a class of nonlinear constraints infeasibility problem is presented without assuming that the Jacobian has full rank everywhere. The underlying method is based on the simple sufficient reduction criteria where trial points are accepted provided there is a sufficient decrease in the constraints violation function. The proposed methods … Read more

    Set Intersection Theorems and Existence of Optimal Solutions

  • Dimitri Bertsekas
  • Paul Tseng
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , ,

    The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We introduce the new notion of … Read more

    Interior Methods for Mathematical Programs with Complementarity Constraints

  • Sven Leyffer
  • Gabriel Lopez-Calva
  • Jorge Nocedal
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    This paper studies theoretical and practical properties of interior-penalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then … Read more

    Convergence Analysis of an Interior-Point Method for Mathematical Programs with Equilibrium Constraints

  • Arun Sen
  • David F. Shanno
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    We prove local and global convergence results for an interior-point method applied to mathematical programs with equilibrium constraints. The global result shows the algorithm minimizes infeasibility regardless of starting point, while one result proves local convergence when penalty functions are exact; another local result proves convergence when the solution is not even a KKT point. … Read more

    Computational experience with an interior point algorithm for large scale contact problems

  • Dominikus Noll
  • Y. Renard
  • G. Tanoh
    • Categories Constrained Nonlinear Optimization, Mechanical Engineering, Quadratic Programming Tags , , ,

    In this paper we present an interior point method for large scale Signorini elastic contact problems. We study the case of an elastic body in frictionless contact with a rigid foundation. Primal and primal-dual algorithms are developed to solve the quadratic optimization problem arising in the variational formulation. Our computational study confirms the efficiency of … Read more