Nonlinear Optimization

A Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties.

  • Sasan Bakhtiari
  • Craig T. Lawrence
  • André L. Tits
  • Thomas J. Urban
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization Tags , ,

    An exact-penalty-function-based scheme—inspired from an old idea due to Mayne and Polak (Math. Prog., vol.~11, 1976, pp.~67–80)—is proposed for extending to general smooth constrained optimization problems any given feasible interior-point method for inequality constrained problems. It is shown that the primal-dual interior-point framework allows for a simpler penalty parameter update rule than that discussed and … Read more

    On graphs with stability number equal to the optimal value of a convex quadratic program

  • Domingos M. Cardoso
    • Categories Graphs and Matroids, Quadratic Programming

    Since the Motzkin-Straus result on the clique number of graphs, published in 1965, where they show that the size of the largest clique in a graph can be obtained by solving a quadratic programming problem, several results on the continuous approach to the determination of the clique number of a graph or, equivalently, to the … Read more

    A new exact penalty function

  • Waltraud Huyer
  • Arnold Neumaier
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    For constrained smooth or nonsmooth optimization problems, new continuously differentiable penalty functions are derived. They are proved exact in the sense that under some nondegeneracy assumption, local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function. This is achieved by augmenting the dimension of the program by a variable that … Read more

    Model Problems for the Multigrid Optimization of Systems Governed by Differential Equations

  • Robert Michael Lewis
  • Stephen G. Nash
    • Categories Infinite Dimensional Optimization, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags ,

    We present a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems have many applications, and are a broader class of problems than systems of equations. Using several model problems we give evidence (both theoretical and numerical) that a multigrid approach can often be successful in the setting of optimization. … Read more

    Pattern Search Methods for User-Provided Points:Application to Molecular Geometry Problems

  • Pedro Alberto
  • Luis Nunes Vicente
  • Fernando Nogueira
  • Humberto Rocha
    • Categories Basic Sciences Applications, Unconstrained Optimization Tags , , , , ,

    This paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry. The goal is to find a configuration of a cluster or a molecule with minimum total energy. The minimization problems in this class of geometry molecular … Read more

    Combinatorial Structures in Nonlinear Programming

  • Stefan Scholtes
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization, Robust Optimization Tags , , ,

    Non-smoothness and non-convexity in optimization problems often arise because a combinatorial structure is imposed on smooth or convex data. The combinatorial aspect can be explicit, e.g. through the use of ”max”, ”min”, or ”if” statements in a model, or implicit as in the case of bilevel optimization where the combinatorial structure arises from the possible … Read more

    The Penalty Interior Point Method fails to converge for Mathematical Programs with Equilibrium Constraints

  • Sven Leyffer
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , ,

    This paper presents a small example for which the Penalty Interior Point Method converges to a non-stationary point. The reasons for this adverse behaviour are discussed. CitationNumerical Analysis Report NA/208, Department of Mathematics, University of Dundee, February 2002.ArticleDownload View PDF

    Local convergence of SQP methods for Mathematical Programs with Equilibrium Constraints

  • Roger Fletcher
  • Sven Leyffer
  • Daniel Ralph
  • Stefan Scholtes
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , ,

    Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs). In particular, Sequential Quadratic Programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. It is shown that SQP converges superlinearly under reasonable assumptions … Read more

    Relations between divergence of multipliers and convergence to infeasible points in primal-dual interior methods for nonconvex nonlinear programming

  • Anders Forsgren
  • Göran Sporre
    • Categories Constrained Nonlinear Optimization Tags , , ,

    Recently, infeasibility issues in interior methods for nonconvex nonlinear programming have been studied. In particular, it has been shown how many line-search interior methods may converge to an infeasible point which is on the boundary of the feasible region with respect to the inequality constraints. The convergence is such that the search direction does not … Read more

    The least-intensity feasible solution for aperture-based inverse planning in radiation therapy.

  • Yair Censor
  • James M. Galvin
  • Darek Michalski
  • Ying Xiao
    • Categories Biomedical Applications, Convex Optimization, Nonlinear Systems and Least-Squares Tags , , , , ,

    Aperture-based inverse planning (ABIP) for intensity modulated radiation therapy (IMRT) treatment planning starts with external radiation fields (beams) that fully conform to the target(s) and then superimposes sub-fields called segments to achieve complex shaping of 3D dose distributions. The segments’ intensities are determined by solving a feasibility problem. The least-intensity feasible (LIF) solution, proposed and … Read more