Constrained Nonlinear Optimization

On QPCCs, QCQPs and Completely Positive Programs

  • Lijie Bai
  • John E. Mitchell
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming

    This paper studies several classes of nonconvex optimization problems defined over convex cones, establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs. The problems being studied include: a quadratically constrained quadratic program (QCQP), a quadratic program with complementarity constraints (QPCC), and rank constrained semidefinite programs. Our results … Read more

    Characterization of properly optimal elements with variable ordering structures

  • Gabriele Eichfelder
  • Tobias Gerlach
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization Tags , , ,

    In vector optimization with a variable ordering structure the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications it was started to develop a comprehensive theory for these vector optimization problems. Thereby also notions of proper efficiency … Read more

    New active set identification for general constrained optimization and minimax problems

  • Jinbao Jian
  • Yi Liu
    • Categories Constrained Nonlinear Optimization

    The purpose of this paper is to discuss the problem of identifying the active constraints for general constrained nonlinear programming and constrained minimax problems at an isolated local solution. Facchinei et al. [F. Facchinei, A. Fischer, and C. Kanzow, On the accurate identification of active constraints, SIAM J. Optim., 9(1998), 14-32] proposed an effective technique … Read more

    Iterative Reweighted Linear Least Squares for Exact Penalty Subproblems on Product Sets

  • James V. Burke
  • Frank E. Curtis
  • Hao Wang
  • Jiashan Wang
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonlinear Optimization Tags , , , , ,

    We present two matrix-free methods for solving exact penalty subproblems on product sets that arise when solving large-scale optimization problems. The first approach is a novel iterative reweighting algorithm (IRWA), which iteratively minimizes quadratic models of relaxed subproblems while automatically updating a relaxation vector. The second approach is based on alternating direction augmented Lagrangian (ADAL) … Read more

    An Interior-Point Trust-Funnel Algorithm for Nonlinear Optimization

  • Frank E. Curtis
  • Nicholas I. M. Gould
  • Daniel P. Robinson
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , ,

    We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, … Read more

    Lagrangean Decomposition for Mean-Variance Combinatorial Optimization

  • Frank Baumann
  • Christoph Buchheim
  • Anna Ilyina
    • Categories Combinatorial Optimization, Constrained Nonlinear Optimization, Robust Optimization Tags , ,

    We address robust versions of combinatorial optimization problems, focusing on the uncorrelated ellipsoidal uncertainty case, which corresponds to so-called mean-variance optimization. We present a branch and bound-algorithm for such problems that uses lower bounds obtained from Lagrangean decomposition. This approach allows to separate the uncertainty aspect in the objective function from the combinatorial structure of … Read more

    A Primal Heuristic for MINLP based on Dual Information

  • Armin Fügenschuh
  • Jesco Humpola
  • Thomas Lehmann
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Meta Heuristics

    We present a novel heuristic algorithm to identify feasible solutions of a mixed-integer nonlinear programming problem arising in natural gas transportation: the selection of new pipelines to enhance the network’s capacity to a desired level in a cost-efficient way. We solve this problem in a linear programming based branch-and-cut approach, where we deal with the … Read more

    Narrowing the difficulty gap for the Celis-Dennis-Tapia problem

  • Immanuel M. Bomze
  • Michael L. Overton
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Quadratic Programming Tags , , , ,

    We study the {\em Celis-Dennis-Tapia (CDT) problem}: minimize a non-convex quadratic function over the intersection of two ellipsoids. In contrast to the well-studied trust region problem where the feasible set is just one ellipsoid, the CDT problem is not yet fully understood. Our main objective in this paper is to narrow the difficulty gap that … Read more

    A Regularized SQP Method with Convergence to Second-Order Optimal Points

  • Philip E. Gill
  • Vyacheslav Kungurtsev
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , ,

    Regularized and stabilized sequential quadratic programming methods are two classes of sequential quadratic programming (SQP) methods designed to resolve the numerical and theoretical difficulties associated with ill-posed or degenerate nonlinear optimization problems. Recently, a regularized SQP method has been proposed that provides a strong connection between augmented Lagrangian methods and stabilized SQP methods. The method … Read more