Nonlinear Optimization

A primal-infeasible interior point algorithm for linearly constrained convex programming

  • Yan-jin Wang
    • Categories Constrained Nonlinear Optimization

    In the paper a primal-infeasible interior point algorithm is proposed for linearly constrained convex programming. The starting point is any positive primal-infeasible dual-feasible point in a large region. The method maintains positivity of the iterates which point satisfies primal-infeasible dual-feasible point. At each iterates it requires to solve approximately a nonlinear system. It is shown … Read more

    Efficient and cheap bounds for (standard) quadratic optimization

  • Immanuel M. Bomze
  • Marco Locatelli
  • F. Tardella
    • Categories Global Optimization Applications, Quadratic Programming Tags , , , ,

    A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. A number of problems can be transformed into a StQP, including the general quadratic problem over a polytope and the maximum clique problem in a graph. In this paper we present several polynomial-time bounds for StQP ranging from very simple … Read more

    Density-based Globally Convergent Trust-Region Methods for Self-Consistent Field Electronic Structure Calculations

  • Juliano B. Francisco
  • Leandro Martínez
  • José Mario Martínez
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    A theory of globally convergent trust-region methods for self-consistent field electronic structure calculations that use the density matrices as variables is developed. The optimization is performed by means of sequential global minimizations of a quadratic model of the true energy. The global minimization of this quadratic model, subject to the idempotency of the density matrix … Read more

    Knitro: An Integrated Package for Nonlinear Optimization

  • Richard H. Byrd
  • Jorge Nocedal
  • Richard A. Waltz
    • Categories Nonlinear Optimization, Optimization Software and Modeling Systems Tags , ,

    This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, … Read more

    A primal-dual interior point method for nonlinear optimization over second order cones

  • Hiroshi Yabe
  • Hiroshi Yamashita
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming Tags , ,

    In this paper, we are concerned with nonlinear minimization problems with second order cone constraints. A primal-dual interior point method is proposed for solving the problems. We also propose a new primal-dual merit function by combining the barrier penalty function and the potential function within the framework of the line search strategy, and show the … Read more

    A General Robust-Optimization Formulation for Nonlinear Programming

  • Yin Zhang
    • Categories Constrained Nonlinear Optimization, Robust Optimization Tags ,

    Most research in robust optimization has so far been focused on inequality-only, convex conic programming with simple linear models for uncertain parameters. Many practical optimization problems, however, are nonlinear and non-convex. Even in linear programming, coefficients may still be nonlinear functions of uncertain parameters. In this paper, we propose robust formulations that extend the robust-optimization … Read more

    Semidefinite-Based Branch-and-Bound for Nonconvex Quadratic Programming

  • Samuel Burer
  • Dieter Vandenbussche
    • Categories Global Optimization, Quadratic Programming, Semi-definite Programming Tags , , , ,

    This paper presents a branch-and-bound algorithm for nonconvex quadratic programming, which is based on solving semidefinite relaxations at each node of the enumeration tree. The method is motivated by a recent branch-and-cut approach for the box-constrained case that employs linear relaxations of the first-order KKT conditions. We discuss certain limitations of linear relaxations when handling … Read more

    Constructing Generalized Mean Functions Using Convex Functions with Regularity Conditions

  • S.C. Fang
  • Duan Li
  • Yun-Bin Zhao
    • Categories Convex Optimization, Generalized Convexity/Monoticity, Nonlinear Optimization Tags , , , ,

    The generalized mean function has been widely used in convex analysis and mathematical programming. This paper studies a further generalization of such a function. A necessary and sufficient condition is obtained for the convexity of a generalized function. Additional sufficient conditions that can be easily checked are derived for the purpose of identifying some classes … Read more

    The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications

  • Defeng Sun
    • Categories Nonlinear Optimization, Semi-definite Programming Tags , ,

    For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others. CitationTechnical Report, Department of Mathematics, … Read more