Nonlinear Optimization

A generalized block-iterative projection method for the common fixed point problem induced by cutters

  • Yair Censor
  • Daniel Reem
  • Maroun Zaknoon
    • Categories Convex Optimization, Nonlinear Optimization Tags , , , ,

    The block-iterative projections (BIP) method of Aharoni and Censor [Block-iterative projection methods for parallel computation of solutions to convex feasibility problems, Linear Algebra and its Applications 120, (1989), 165-175] is an iterative process for finding asymptotically a point in the nonempty intersection of a family of closed convex subsets. It employs orthogonal projections onto the … Read more

    Convexity and continuity of specific set-valued maps and their extremal value functions

  • Gabriele Eichfelder
  • Tobias Gerlach
  • Stefan Rocktäschel
    • Categories Convex Optimization, Generalized Convexity/Monoticity, Nonlinear Optimization Tags , , ,

    In this paper, we study several classes of set-valued maps, which can be used in set-valued optimization and its applications, and their respective maximum and minimum value functions. The definitions of these maps are based on scalar-valued, vector-valued, and cone-valued maps. Moreover, we consider those extremal value functions which are obtained when optimizing linear functionals … Read more

    A practical second-order optimality condition for cardinality-constrained problems with application to an augmented Lagrangian method

  • Jean C. A. Medeiros
  • Ademir A. Ribeiro
  • Mael Sachine
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , ,

    This paper addresses the mathematical programs with cardinality constraints (MPCaC). We first define two new tailored (strong and weak) second-order necessary conditions, MPCaC-SSONC and MPCaC-WSONC. We then propose a constraint qualification (CQ), namely, MPCaC-relaxed constant rank constraint qualification (MPCaC-RCRCQ), and establish the validity of MPCaC-SSONC at minimizers under this new CQ. All the concepts proposed … Read more

    Riemannian Interior Point Methods for Constrained Optimization on Manifolds

  • Zhijian Lai
  • Akiko Yoshise
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , , ,

    We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method (RIPM), is for solving Riemannian  constrained optimization problems. We establish its local superlinear and quadratic convergence  under the standard assumptions. Moreover, we show its global convergence when it is combined with … Read more

    A decomposition method for lasso problems with zero-sum constraint

  • Andrea Cristofari
    • Categories Nonlinear Optimization Tags , , , ,

    In this paper, we consider lasso problems with zero-sum constraint, commonly required for the analysis of compositional data in high-dimensional spaces. A novel algorithm is proposed to solve these problems, combining a tailored active-set technique, to identify the zero variables in the optimal solution, with a 2-coordinate descent scheme. At every iteration, the algorithm chooses … Read more

    Advancements in the computation of enclosures for multi-objective optimization problems

  • Gabriele Eichfelder
  • Leo Warnow
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    A central goal for multi-objective optimization problems is to compute their nondominated sets. In most cases these sets consist of infinitely many points and it is not a practical approach to compute them exactly. One solution to overcome this problem is to compute an enclosure, a special kind of coverage, of the nondominated set. One … Read more

    Accelerating Stochastic Sequential Quadratic Programming for Equality Constrained Optimization using Predictive Variance Reduction

  • Albert S. Berahas
  • Jiahao Shi
  • Zihong Yi
  • Baoyu Zhou
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In this paper, we propose a stochastic variance reduction method for solving equality constrained optimization problems. Specifically, we develop a method based on the sequential quadratic programming paradigm that utilizes gradient approximations via predictive variance reduction techniques. Under reasonable assumptions, we prove that a measure of first-order stationarity evaluated at the iterates generated by our … Read more

    An Adaptive Riemannian Gradient Method Without Function Evaluations

  • Geovani Nunes Grapiglia
  • Gabriel Stella
    • Categories Nonlinear Optimization Tags , ,

    In this paper we propose an adaptive gradient method for optimization on Riemannian manifolds. The update rule for the stepsizes relies only on gradient evaluations. Assuming that the objective function is bounded from below and that its gradient field is Lipschitz continuous, we establish worst-case complexity bounds for the number of gradient evaluations that the … Read more

    Direct search based on probabilistic descent in reduced spaces

  • Lindon Roberts
  • Clément W. Royer
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , ,

    Derivative-free algorithms seek the minimum value of a given objective function without using any derivative information. The performance of these methods often worsen as the dimension increases, a phenomenon predicted by their worst-case complexity guarantees. Nevertheless, recent algorithmic proposals have shown that incorporating randomization into otherwise deterministic frameworks could alleviate this effect for direct-search methods. … Read more

    Learning for Spatial Branching: An Algorithm Selection Approach

  • Bissan Ghaddar
  • Ignacio Gómez-Casares
  • Julio González-Díaz
  • Brais González-Rodríguez
  • Beatriz Pateiro-López
  • Sofía Rodríguez-Ballesteros
    • Categories Nonlinear Optimization Tags , , , ,

    The use of machine learning techniques to improve the performance of branch-and-bound optimization algorithms is a very active area in the context of mixed integer linear problems, but little has been done for non-linear optimization. To bridge this gap, we develop a learning framework for spatial branching and show its efficacy in the context of … Read more