Nonlinear Optimization

On the use of the energy norm in trust-region and adaptive cubic regularization subproblems

  • Elhoucine Bergou
  • Youssef Diouane
  • Serge Gratton
    • Categories Unconstrained Optimization Tags , , , , , , ,

    We consider solving unconstrained optimization problems by means of two popular globalization techniques: trust-region (TR) algorithms and adaptive regularized framework using cubics (ARC). Both techniques require the solution of a so-called “subproblem” in which a trial step is computed by solving an optimization problem involving an approximation of the objective function, called “the model”. The … Read more

    A decoupled first/second-order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds

  • Serge Gratton
  • Clément W. Royer
  • Luis Nunes Vicente
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    In order to be provably convergent towards a second-order stationary point, optimization methods applied to nonconvex problems must necessarily exploit both first and second-order information. However, as revealed by recent complexity analyzes of some of these methods, the overall effort to reach second-order points is significantly larger when compared to the one of approaching first-order … Read more

    Bilevel optimization with a multiobjective problem in the lower level

  • Roberto Andreani
  • Sandra Augusta Santos
  • Viviana A. Ramirez
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    Bilevel problems model instances with a hierarchical structure. Aiming at an efficient solution of a constrained multiobjective problem according with some pre-defined criterion, we reformulate this optimization but non standard problem as a classic bilevel one. This reformulation intents to encompass all the objectives, so that the properly efficient solution set is recovered by means … Read more

    Iteration-Complexity of a Linearized Proximal Multiblock ADMM Class for Linearly Constrained Nonconvex Optimization Problems

  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    This paper analyzes the iteration-complexity of a class of linearized proximal multiblock alternating direction method of multipliers (ADMM) for solving linearly constrained nonconvex optimization problems. The subproblems of the linearized ADMM are obtained by partially or fully linearizing the augmented Lagrangian with respect to the corresponding minimizing block variable. The derived complexity bounds do not … Read more

    MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems

  • Martin Gugat
  • Günter Leugering
  • Martin Schmidt
  • Mathias Sirvent
  • David Wintergerst
  • Alexander Martin
    • Categories (Mixed) Integer Nonlinear Programming, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , ,

    We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations … Read more

    A Bregman alternating direction method of multipliers for sparse probabilistic Boolean network problem

  • Deng Kangkang
  • Peng Zheng
    • Categories Applications - Science and Engineering, Biomedical Applications, Constrained Nonlinear Optimization Tags , , , ,

    The main task of genetic regulatory networks is to construct a sparse probabilistic Boolean network (PBN) based on a given transition-probability matrix and a set of Boolean networks (BNs). In this paper, a Bregman alternating direction method of multipliers (BADMM) is proposed to solve the minimization problem raised in PBN. All the customized subproblem-solvers of … Read more

    On generalized-convex constrained multi-objective optimization

  • Christian Günther
  • Christiane Tammer
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity, Multi-Criteria Optimization Tags , , , , ,

    In this paper, we consider multi-objective optimization problems involving not necessarily convex constraints and componentwise generalized-convex (e.g., semi-strictly quasi-convex, quasi-convex, or explicitly quasi-convex) vector-valued objective functions that are acting between a real linear topological pre-image space and a finite dimensional image space. For these multi-objective optimization problems, we show that the set of (strictly, weakly) … Read more

    Automatic Differentiation of the Open CASCADE Technology CAD System and its coupling with an Adjoint CFD Solver

  • Salvatore Auriemma
  • Herve Legrand
  • Mladen Banovic
  • Jens-Dominik Müller
  • Orest Mykhaskiv
  • Andrea Walther
    • Categories Optimization of Systems modeled by PDEs, Unconstrained Optimization Tags , , ,

    Automatic Differentiation (AD) is applied to the open-source CAD system Open CASCADE Technology using the AD software tool ADOL-C (Automatic Differentiation by OverLoading in C++). The differentiated CAD system is coupled with a discrete adjoint CFD solver, thus providing the first example of a complete differentiated design chain built from generic, multi-purpose tools. The design … Read more

    Linear Convergence of Proximal Incremental Aggregated Gradient Methods under Quadratic Growth Condition

  • Hui Zhang
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    Under the strongly convex assumption, several recent works studied the global linear convergence rate of the proximal incremental aggregated gradient (PIAG) method for minimizing the sum of a large number of smooth component functions and a non-smooth convex function. In this paper, under the quadratic growth condition{a strictly weaker condition than the strongly convex assumption, … Read more

    Dantzig Wolfe decomposition and objective function convexification for binary quadratic problems: the cardinality constrained quadratic knapsack case

  • Alberto Ceselli
  • Lucas Létocart
  • Emiliano Traversi
    • Categories (Mixed) Integer Nonlinear Programming, Combinatorial Optimization, Quadratic Programming Tags , , ,

    The purpose of this paper is to provide strong reformulations for binary quadratic problems. We propose a first methodological analysis on a family of reformulations combining Dantzig-Wolfe decomposition and Quadratic Convex Reformulation principles. As a representative case study, we apply them to a cardinality constrained quadratic knapsack problem, providing extensive experimental insights. We show that … Read more