Nonlinear Optimization

Zeroth-order Nonconvex Stochastic Optimization: Handling Constraints, High-Dimensionality, and Saddle-Points

  • Krishnakumar Balasubramanian
  • Saeed Ghadimi
    • Categories Nonlinear Optimization, Stochastic Programming Tags , , , , ,

    In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting, and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To … Read more

    Forecasting conceivable interest rate market scenarios and significant losses on interest rate portfolios using mathematical optimization

  • Katsuhiro Tanaka
    • Categories Constrained Nonlinear Optimization, Finance and Economics Tags , , , , ,

    This study proposes a mathematical optimization programming model that simultaneously forecasts interest rate market scenarios and significant losses on interest rate market portfolios. The model includes three main components. A constraint condition is set using the Mahalanobis distance, which consists of innovation terms in a dynamic conditional correlation-generalized autoregressive conditional heteroscedasticity (DCC-GARCH) model that represent … Read more

    Local minimizers of semi-algebraic functions

  • Tiên-So'n Pham
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , , , ,

    Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so–called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of … Read more

    Intersection disjunctions for reverse convex sets

  • James R. Luedtke
  • Eli Towle
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Global Optimization Theory Tags , , , ,

    We present a framework to obtain valid inequalities for optimization problems constrained by a reverse convex set, which is defined as the set of points in a polyhedron that lie outside a given open convex set. We are particularly interested in cases where the closure of the convex set is either non-polyhedral, or is defined … Read more

    Generalized Conditional Gradient with Augmented Lagrangian for Composite Minimization

  • Jalal Fadili
  • Antonio Silveti-Falls
  • Cesare Molinari
    • Categories Convex and Nonsmooth Optimization, Infinite Dimensional Optimization, Nonlinear Optimization

    In this paper we propose a splitting scheme which hybridizes generalized conditional gradient with a proximal step which we call CGALP algorithm, for minimizing the sum of three proper convex and lower-semicontinuous functions in real Hilbert spaces. The minimization is subject to an affine constraint, that allows in particular to deal with composite problems (sum … Read more

    Escaping local minima with derivative-free methods: a numerical investigation

  • Coralia Cartis
  • Lindon Roberts
  • Oliver Sheridan-Methven
    • Categories Nonlinear Optimization Tags , ,

    We apply a state-of-the-art, local derivative-free solver, Py-BOBYQA, to global optimization problems, and propose an algorithmic improvement that is beneficial in this context. Our numerical findings are illustrated on a commonly-used test set of global optimization problems and associated noisy variants, and on hyperparameter tuning for a machine learning test set. As Py-BOBYQA is a … Read more

    On High-order Model Regularization for Multiobjective Optimization

  • L. Calderón
  • Maria A. Diniz-Ehrhardt
  • José Mario Martínez
    • Categories Constrained Nonlinear Optimization Tags , ,

    A p-order regularization method for finding weak stationary points of multiobjective optimization problems with constraints is introduced. Under Holder conditions on the derivatives of the objective functions, complexity results are obtained that generalize properties recently proved for scalar optimization. Article Download View On High-order Model Regularization for Multiobjective Optimization

    A Single Time-Scale Stochastic Approximation Method for Nested Stochastic Optimization

  • Saeed Ghadimi
  • Andrzej Ruszczyński
  • Mengdi Wang
    • Categories Nonlinear Optimization, Stochastic Programming Tags , ,

    We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single time-scale stochastic approximation algorithm, which we call the Nested Averaged Stochastic Approximation (NASA), to find an approximate stationary point of the problem. The algorithm … Read more

    The convex hull of a quadratic constraint over a polytope

  • Santanu S. Dey
  • Asteroide Santana
    • Categories Global Optimization Theory, Quadratic Programming, Second-Order Cone Programming

    A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global optimality is a well-known NP-hard problem and a traditional approach is to use convex relaxations and branch-and-bound algorithms. This paper makes a … Read more

    An efficient adaptive accelerated inexact proximal point method for solving linearly constrained nonconvex composite problems

  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
  • Weiwei Kong
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP … Read more