Nonlinear Optimization

On the Implementation of an Interior-Point Filter Line-Search Algorithm for Large-Scale Nonlinear Programming

  • Andreas Wächter
  • Lorenz T. Biegler
    • Categories Constrained Nonlinear Optimization

    We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics … Read more

    A filter-trust-region method for unconstrained optimization

  • Nicholas I. M. Gould
  • Caroline Sainvitu
  • Philippe L. Toint
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags ,

    A new filter-trust-region algorithm for solving unconstrained nonlinear optimization problems is introduced. Based on the filter technique introduced by Fletcher and Leyffer, it extends an existing technique of Gould, Leyffer and Toint (SIAM J. Optim., to appear 2004) for nonlinear equations and nonlinear least-squares to the fully general unconstrained optimization problem. The new algorithm is … Read more

    Solving nonconvex SDP problems of structural optimization with stability control

  • Michal Kočvara
  • Michael Stingl
    • Categories Constrained Nonlinear Optimization, Mechanical Engineering, Semi-definite Programming Tags , , ,

    The goal of this paper is to formulate and solve structural optimization problems with constraints on the global stability of the structure. The stability constraint is based on the linear buckling phenomenon. We formulate the problem as a nonconvex semidefinite programming problem and introduce an algorithm based on the Augmented Lagrangian method combined with the … Read more

    On the modeling and control of delamination processes

  • Michal Kočvara
  • Jiri V. Outrata
    • Categories Constrained Nonlinear Optimization, Mechanical Engineering Tags , , ,

    This paper is motivated by problem of optimal shape design of laminated elastic bodies. We use a recently introduced model of delamination, based on minimization of potential energy which includes the free (Gibbs-type) energy and (pseudo)potential of dissipative forces, to introduce and analyze a special mathematical program with equilibrium constraints. The equilibrium is governed by … Read more

    A sufficient optimality criteria for linearly constrained, separable concave minimization problems

  • Tibor Illés
  • Adam Nagy
    • Categories Global Optimization, Global Optimization Theory, Nonlinear Optimization Tags , ,

    Sufficient optimality criteria for linearly constrained, concave minimization problems is given in this paper. Our optimality criteria is based on the sensitivity analysis of the relaxed linear programming problem. Our main result is similar to that of Phillips and Rosen (1993), however our proofs are simpler and constructive. Phillips and Rosen (1993) in their paper … Read more

    On the Global Minimization of the Value-at-Risk

  • Sven Leyffer
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Global Optimization, Nonlinear Optimization

    In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined … Read more

    On an Approximation of the Hessian of the Lagrangian

  • Florian Jarre
    • Categories Constrained Nonlinear Optimization, Optimization Software Design Principles Tags , ,

    In the context of SQP methods or, more recently, of sequential semidefinite programming methods, it is common practice to construct a positive semidefinite approximation of the Hessian of the Lagrangian. The Hessian of the augmented Lagrangian is a suitable approximation as it maintains local superlinear convergence under appropriate assumptions. In this note we give a … Read more

    Some Properties of Regularization and Penalization Schemes for MPECs

  • Daniel Ralph
  • Stephen Wright
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , ,

    Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by … Read more

    Global linear convergence of an augmented Lagrangian algorithm for solving convex quadratic optimization problems

  • Frederic Delbos
  • Jean Charles Gilbert
    • Categories Quadratic Programming Tags , , , , , , ,

    We consider an augmented Lagrangian algorithm for minimizing a convex quadratic function subject to linear inequality constraints. Linear optimization is an important particular instance of this problem. We show that, provided the augmentation parameter is large enough, the constraint value converges {\em globally\/} linearly to zero. This property is viewed as a consequence of the … Read more

    Quadratic interior-point methods in statistical disclosure control

  • Jordi Castro
    • Categories Network Optimization, Quadratic Programming, Statistics

    The safe dissemination of statistical tabular data is one of the main concerns of National Statistical Institutes (NSIs). Although each cell of the tables is made up of the aggregated information of several individuals, the statistical confidentiality can be violated. NSIs must guarantee that no individual information can be derived from the released tables. One … Read more