Nonlinear Optimization

Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

  • David Ek
  • Anders Forsgren
    • Categories Bound-constrained Optimization Tags , , ,

    The focus in this paper is interior-point methods for bound-constrained nonlinear optimization where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. … Read more

    Survey of Sequential Convex Programming and Generalized Gauss-Newton Methods

  • Katrin Baumgärtner
  • Moritz Diehl
  • Florian Messerer
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions … Read more

    Conditional gradient method for multiobjective optimization

  • Pedro Bonfim Assunção
  • Orizon Pereira Ferreira
  • Leandro Fonseca Prudente
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    We analyze the conditional gradient method, also known as Frank-Wolfe method, for constrained multiobjective optimization. The constraint set is assumed to be convex and compact, and the objectives functions are assumed to be continuously differentiable. The method is considered with different strategies for obtaining the step sizes. Asymptotic convergence properties and iteration-complexity bounds with and … Read more

    On monotonicity and search traversal in copositivity detection algorithms

  • Leocadio G Casado
  • Eligius M.T. Hendrix
  • Boglárka Tóth
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Quadratic Programming Tags , , ,

    Matrix copositivity has an important theoretical background. Over the last decades, the use of algorithms to check copositivity has made a big progress. Methods are based on spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the … Read more

    Using gradient directions to get global convergence of Newton-type methods

  • Daniela di Serafino
  • Gerardo Toraldo
  • Marco Viola
    • Categories Unconstrained Optimization

    The renewed interest in Steepest Descent (SD) methods following the work of Barzilai and Borwein [IMA Journal of Numerical Analysis, 8 (1988)] has driven us to consider a globalization strategy based on SD, which is applicable to any line-search method. In particular, we combine Newton-type directions with scaled SD steps to have suitable descent directions. … Read more

    Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

  • Stefano Coniglio
  • Fabio Furini
  • Ivana Ljubić
    • Categories Cutting Plane Approaches, Integer Programming, Nonlinear Optimization Tags , ,

    We study a family of discrete optimization problems asking for the maximization of the expected value of a concave, strictly increasing, and differentiable function composed with a set-union operator. The expected value is computed with respect to a set of coefficients taking values from a discrete set of scenarios. The function models the utility function … Read more

    An inexact scalarized proximal algorithm with quasi- distance for convex and quasiconvex multi-objective minimization

  • Rogério Azevedo Rocha
  • Ronaldo Malheiros Gregório
  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Multi-Criteria Optimization, Nonlinear Optimization Tags

    In the paper of Rocha et al., J Optim Theory Appl (2016) 171:964979, the authors introduced a proximal point algorithm with quasi-distances to solve unconstrained convex multi-objective minimization problems. They proved that all accumulation points are ecient solutions of the problem. In this pa- per we analyze an inexact proximal point algorithm to solve convex … Read more

    On the convergence of augmented Lagrangian strategies for nonlinear programming

  • Roberto Andreani
  • A. R. V. Cárdenas
  • Alberto Ramos
  • Ademir A. Ribeiro
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , , , ,

    Augmented Lagrangian algorithms are very popular and successful methods for solving constrained optimization problems. Recently, the global convergence analysis of these methods have been dramatically improved by using the notion of the sequential optimality conditions. Such conditions are optimality conditions independently of the fulfilment of any constraint qualifications and provide theoretical tools to justify stopping … Read more

    A numerical study of transformed mixed-integer optimal control problems

  • Sebastian Sager
  • Manuel Tetschke
  • Clemens Zeile
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Systems governed by Differential Equations Optimization Tags , , , ,

    Time transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal con trol problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. … Read more

    A proximal bundle variant with optimal iteration-complexity for a large range of prox stepsizes

  • Jiaming Liang
  • Renato D.C. Monteiro
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    This paper presents a proximal bundle variant, namely, the relaxed proximal bundle (RPB) method, for solving convex nonsmooth composite optimization problems. Like other proximal bundle variants, RPB solves a sequence of prox bundle subproblems whose objective functions are regularized composite cutting-plane models. Moreover, RPB uses a novel condition to decide whether to perform a serious … Read more