Global Optimization

A Branch-and-Bound based Algorithm for Nonconvex Multiobjective Optimization

  • Gabriele Eichfelder
  • Julia Niebling
    • Categories Global Optimization, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    A new branch-and-bound based algorithm for smooth nonconvex multiobjective optimization problems with convex constraints is presented. The algorithm computes an $(\varepsilon,\delta)$-approximation of all globally optimal solutions. We introduce the algorithm which uses selection rules, discarding and termination tests. The discarding tests are the most important aspect, as they examine in different ways whether a box … Read more

    An algorithm for computing Frechet means on the sphere

  • Gabriele Eichfelder
  • Thomas Hotz
  • Johannes Wieditz
    • Categories Global Optimization, Statistics Tags , ,

    For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special global problem over the sphere, namely the determination of Frechet means, which are points minimising the mean distance … Read more

    Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning

  • Thomas Kleinert
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Applications - OR and Management Sciences, Global Optimization Tags , , , ,

    We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in [25] that does not include a network design problem. The two classical discrete optimization … Read more

    Exact Semidefinite Formulations for a Class of (Random and Non-Random) Nonconvex Quadratic Programs

  • Samuel Burer
  • Yinyu Ye
    • Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming Tags , ,

    We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature … Read more

    Extended formulations for convex hulls of some bilinear functions

  • Akshay Gupte
  • Thomas Kalinowski
  • Fabian Rigterink
  • Hamish Waterer
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Polyhedra Tags , , ,

    We consider the problem of characterizing the convex hull of the graph of a bilinear function $f$ on the $n$-dimensional unit cube $[0,1]^n$. Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose the systematic study of properties of $f$ … Read more

    Deterministic Global Optimization with Artificial Neural Networks Embedded

  • Alexander Mitsos
  • Artur M Schweidtmann
    • Categories Chemical Engineering, Global Optimization, Nonlinear Optimization Tags , , , , , , ,

    Artificial neural networks (ANNs) are used in various applications for data-driven black-box modeling and subsequent optimization. Herein, we present an efficient method for deterministic global optimization of ANN embedded optimization problems. The proposed method is based on relaxations of algorithms using McCormick relaxations in a reduced-space [\textit{SIOPT}, 20 (2009), pp. 573-601] including the convex and … Read more

    Optimization of Stochastic Problems with Probability Functions via Differential Evolution

  • Bayrammyrat Myradov
    • Categories Basic Sciences Applications, Stochastic Approaches, Stochastic Programming

    Chance constrained programming, quantile/Value-at-Risk (VaR) optimization and integral quantile / Conditional Value-at-Risk (CVaR) optimization problems as Stochastic Programming Problems with Probability Functions (SPP-PF) are one of the most widely studied optimization problems in recent years. As a rule real-life SPP-PF is nonsmooth nonconvex optimization problem with complex geometry of objective function. Moreover, often it cannot … Read more

    Global Optimisation of Multi-Plant Manganese Alloy Production

  • Henrik Andersson
  • Stein Wasbø
  • Martin Digernes
  • Brage Rugstad Knudsen
  • Lars Rudi
  • Magnus Stålhane
    • Categories Global Optimization Applications, Production and Logistics, Supply Chain Management

    This paper studies the problem of multi-plant manganese alloy production. The problem consists of finding the optimal furnace feed of ores, fluxes, coke, and slag that yields output products which meet customer specifications, and to optimally decide the volume, composition, and allocation of the slag. To solve the problem, a nonlinear pooling problem formulation is … Read more

    Run-and-Inspect Method for Nonconvex Optimization and Global Optimality Bounds for R-Local Minimizers

  • Yifan Chen
  • Wotao Yin
  • Yuejiao Sun
    • Categories Global Optimization Theory, Unconstrained Optimization Tags , , , ,

    Many optimization algorithms converge to stationary points. When the underlying problem is nonconvex, they may get trapped at local minimizers and occasionally stagnate near saddle points. We propose the Run-and-Inspect Method, which adds an “inspect” phase to existing algorithms that helps escape from non-global stationary points. The inspection samples a set of points in a … Read more

    Computational Aspects of Bayesian Solution Estimators in Stochastic Optimization

  • Gérard Cornuéjols
  • Danial Davarnia
  • Burak Kocuk
    • Categories Statistics, Stochastic Approaches, Stochastic Programming Tags , , , ,

    We study a class of stochastic programs where some of the elements in the objective function are random, and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two common … Read more