Nonlinear Optimization

Local Convergence of the Method of Multipliers for Variational and Optimization Problems under the Sole Noncriticality Assumption

  • Alexey F. Izmailov
  • A. S. Kurennoy
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    We present local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local superlinear convergence is established under the … Read more

    Some Remarks for a Decomposition of Linear-Quadratic Optimal Control Problems for Two-Steps Systems

  • Shahlar Meherrem
    • Categories Nonlinear Optimization Tags ,

    In this paper we obtained new approach for the problem, which it is described in reference[1,2]. In the references [1], the authors are studied Decomposition of Linear-Quadratic optimal Control problems for Two-Steps Systems. In [1], the authors assumed the switching point is fixed and it is given algorithm for solving Linear-Quadratic optimal Control problem. But … Read more

    Projected subgradient minimization versus superiorization

  • Yair Censor
  • Ran Davidi
  • Gabor T. Herman
  • Reinhard W. Schulte
  • Luba Tetruashvili
    • Categories Biomedical Applications, Constrained Nonlinear Optimization Tags , , , , , , , ,

    The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty and, therefore, the projected subgradient method is applicable only when the feasible region is “simple … Read more

    Convergence Analysis of DC Algorithm for DC programming with subanalytic data

  • Hoai An Le Thi
  • Huynh Van Ngai
  • Tao Pham Dinh
    • Categories Nonlinear Optimization, Transportation Tags , , , ,

    DC Programming and DCA have been introduced by Pham Dinh Tao in 1986 and extensively developed by Le Thi Hoai An and Pham Dinh Tao since 1993. These approaches have been successfully applied to solving real life problems in their large scale setting. In this paper, by using the Lojasiewicz inequality for nonsmooth subanalytic functions, … Read more

    Optimization of running strategies based on anaerobic energy and variations of velocity

  • Amandine Aftalion
  • Joseph Frédéric Bonnans
    • Categories Biomedical Applications, Systems governed by Differential Equations Optimization Tags , , , , , ,

    We present new models, numerical simulations and rigorous analysis for the optimization of the velocity in a race. In a seminal paper, Keller (1973,1974) explained how a runner should determine his speed in order to run a given distance in the shortest time. We extend this analysis, based on the equation of motion and aerobic … Read more

    Algebraic rules for quadratic regularization of Newton’s method

  • Elizabeth Wegner Karas
  • Sandra Augusta Santos
  • Benar Fux Svaiter
    • Categories Unconstrained Optimization Tags , , , ,

    In this work we propose a class of quasi-Newton methods to minimize a twice differentiable function with Lipschitz continuous Hessian. These methods are based on the quadratic regularization of Newton’s method, with algebraic explicit rules for computing the regularizing parameter. The convergence properties of this class of methods are analysed. We show that if the … Read more

    A New Framework for Combining Global and Local Methods in Black Box Optimization

  • Yibo Ji
  • Sujin Kim
  • Wendy Lu Xu
    • Categories Applications - OR and Management Sciences, Bound-constrained Optimization, Global Optimization

    We propose a new framework for the optimization of computationally expensive black box problems, where neither closed-form expressions nor derivatives of the objective functions are available. The proposed framework consists of two procedures. The first constructs a global metamodel to approximate the underlying black box function and explores an unvisited area to search for a … Read more

    Analysis of Copositive Optimization Based Linear Programming Bounds on Standard Quadratic Optimization

  • Gizem Sagol
  • E. Alper Yildirim
    • Categories Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , ,

    The problem of minimizing a quadratic form over the unit simplex, referred to as a standard quadratic optimization problem, admits an exact reformulation as a linear optimization problem over the convex cone of completely positive matrices. This computationally intractable cone can be approximated from the inside and from the outside by two sequences of nested … Read more

    Theoretical aspects of adopting exact penalty elements within sequential methods for nonlinear programming

  • Ademir A. Ribeiro
  • Sandra Augusta Santos
  • Mael Sachine
    • Categories Constrained Nonlinear Optimization

    In the context of sequential methods for solving general nonlinear programming problems, it is usual to work with augmented subproblems instead of the original ones, tackled by the $\ell_1$-penalty function together with the shortcut usage of a convenient penalty parameter. This paper addresses the theoretical reasoning behind handling the original subproblems by such an augmentation … Read more

    Second-order Characterizations of Tilt Stability with Applications to Nonlinear Programming

  • Boris S. Mordukhovich
  • Nghia Tran
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    The paper is devoted to the study of tilt-stable local minimizers of general optimization problems in finite-dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data. The importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization, and this notion has been extensively studied in … Read more