Month: November 2016

On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

  • Matthias Köppe
  • Yuan Zhou
    • Categories (Mixed) Integer Linear Programming, Cutting Plane Approaches Tags , ,

    We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory–Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing … Read more

    On the Existence of Pareto Solutions for Polynomial Vector Optimization Problems

  • Do Sang Kim
  • Tiên-So'n Pham
  • Nguyen Van Tuyen
    • Categories Multi-Criteria Optimization, Unconstrained Optimization Tags , , , , ,

    We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{\rm Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m – 1$ containing the set of Pareto values of … Read more

    Preconditioning PDE-constrained optimization with L^1-sparsity and control constraints

  • Margherita Porcelli
  • Valeria Simoncini
  • Martin Stoll
    • Categories Control Applications, Systems governed by Differential Equations Optimization Tags , , , , ,

    PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L^1-terms as well as additional inclusion of box … Read more

    Characterizations of Mixed Binary Convex Quadratic Representable Sets

  • Alberto Del Pia
  • Jeffrey Poskin
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory

    Representability results play a fundamental role in optimization since they provide characterizations of the feasible sets that arise from optimization problems. In this paper we study the sets that appear in the feasibility version of mixed binary convex quadratic optimization problems. We provide a complete characterization of the sets that can be obtained as the … Read more

    A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

  • Natashia Boland
  • Brian Dandurand
  • Andrew C. Eberhard
  • Fabricio Oliveira
  • Jeffrey Christiansen
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Parallel Algorithms Tags , , , ,

    We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. The dual solution approach needs … Read more

    Efficient solution of quadratically constrained quadratic subproblems within the MADS algorithm

  • Nadir Amaioua
  • Charles Audet
  • Andrew R. Conn
  • Sébastien Le Digabel
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization

    The Mesh Adaptive Direct Search algorithm (MADS) is an iterative method for constrained blackbox optimization problems. One of the optional MADS features is a versatile search step in which quadratic models are built leading to a series of quadratically constrained quadratic subproblems. This work explores different algorithms that exploit the structure of the quadratic models: … Read more

    Rigorous results in electronic structure calculations

  • Denis Chaykin
  • Christian Jansson
  • Frerich Keil
  • Marko Lange
  • Kai Torben Ohlhus
  • Siegfried M. Rump
    • Categories Chemical Engineering, Semi-definite Programming Tags , , , ,

    Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced … Read more

    Locally weighted regression models for surrogate-assisted design optimization

  • Charles Audet
  • Michael Kokkolaras
  • Sébastien Le Digabel
  • Bastien Talgorn
    • Categories Mechanical Engineering, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Locally weighted regression combines the advantages of polynomial regression and kernel smoothing. We present three ideas for appropriate and effective use of LOcally WEighted Scatterplot Smoothing (LOWESS) models for surrogate optimization. First, a method is proposed to reduce the computational cost of LOWESS models. Second, a local scaling coefficient is introduced to adapt LOWESS models … Read more

    Statistical inference and hypotheses testing of risk averse stochastic programs

  • Vincent Guigues
  • Alexander Shapiro
  • Volker Krätschmer
    • Categories Stochastic Programming Tags , , , , ,

    We study statistical properties of the optimal value and optimal solutions of the Sample Average Approximation of risk averse stochastic problems. Central Limit Theorem type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete Kusuoka representation. The obtained results … Read more

    The Rate of Convergence of Augmented Lagrange Method for a Composite Optimization Problem

  • Jihong Zhang
  • Liwei Zhang
  • Yule Zhang
    • Categories Applications - OR and Management Sciences, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper we analyze the rate of local convergence of the augmented Lagrange method for solving optimization problems with equality constraints and the objective function expressed as the sum of a convex function and a twice continuously differentiable function. The presence of the non-smoothness of the convex function in the objective requires extensive tools … Read more