positive spanning sets

Using orthogonally structured positive bases for constructing positive k-spanning sets with cosine measure guarantees

  • Warren Hare
  • Clément W. Royer
  • Gabriel Jarry-Bolduc
  • Sébastien Kerleau
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    \(\) Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well … Read more

    Direct search based on probabilistic descent

  • Serge Gratton
  • Clément W. Royer
  • Luis Nunes Vicente
  • Z. Zhang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , , ,

    Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets which in turn must have at least n+1 vectors in an n-dimensional … Read more

    Reducing the Number of Function Evaluations in Mesh Adaptive Direct Search Algorithms

  • Charles Audet
  • Sébastien Le Digabel
  • Christophe Tribes
  • Andrea Ianni
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    The mesh adaptive direct search (MADS) class of algorithms is designed for nonsmooth optimization, where the objective function and constraints are typically computed by launching a time-consuming computer simulation. Each iteration of a MADS algorithm attempts to improve the current best-known solution by launching the simulation at a finite number of trial points. Common implementations … Read more

    Direct Multisearch for Multiobjective Optimization

  • Ana Luisa Custódio
  • A. Ismael F. Vaz
  • Luis Nunes Vicente
  • J. F. A. Madeira
    • Categories Multi-Criteria Optimization Tags , , , , , , ,

    In practical applications of optimization it is common to have several conflicting objective functions to optimize. Frequently, these functions are subject to noise or can be of black-box type, preventing the use of derivative-based techniques. We propose a novel multiobjective derivative-free methodology, calling it direct multisearch (DMS), which does not aggregate any of the objective … Read more