Nonlinear Optimization

Adaptive cubic regularization methods with dynamic inexact Hessian information and applications to finite-sum minimization

  • Stefania Bellavia
  • Benedetta Morini
  • Gianmarco Gurioli
    • Categories Unconstrained Optimization Tags

    Abstract. We consider the Adaptive Regularization with Cubics approach for solving nonconvex optimization problems and propose a new variant based on inexact Hessian information chosen dynamically. The theoretical analysis of the proposed procedure is given. The key property of ARC framework, constituted by optimal worst-case function/derivative evaluation bounds for first- and second-order critical point, is … Read more

    Alternative DC Algorithm for Partial DC programming

  • Le Thi Hoai An
  • Huynh Van Ngai
  • Tao Pham Dinh
    • Categories Nonlinear Optimization Tags , , , ,

    In this paper, we introduce an alternative DC algorithm for solving partial DC programs. This proposed algorithm is an natural extension of the standard DC algorithm. Furthermore, we also consider an inexact version of this alternative DC algorithm. The convergence of these proposed algorithms (both the exact and inexact versions) are investigated. The applications to … Read more

    Spectral properties of Barzilai-Borwein rules in solving singly linearly constrained optimization problems subject to lower and upper bounds

  • Serena Crisci
  • Federica Porta
  • Luca Zanni
  • Valeria Ruggiero
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In 1988, Barzilai and Borwein published a pioneering paper which opened the way to inexpensively accelerate first-order methods. More in detail, in the framework of unconstrained optimization, Barzilai and Borwein developed two strategies to select the steplength in gradient descent methods with the aim of encoding some second-order information of the problem without computing and/or … Read more

    On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization

  • Geovani Nunes Grapiglia
  • Ya-xiang Yuan
    • Categories Nonlinear Optimization Tags , , ,

    In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of $\mathcal{O}(|\log(\epsilon)|)$ outer iterations for the referred algorithm to generate an $\epsilon$-approximate KKT point, for $\epsilon\in (0,1)$. When the penalty parameters are unbounded, we prove … Read more

    An accelerated inexact proximal point method for solving nonconvex-concave min-max problems

  • Renato D.C. Monteiro
  • Weiwei Kong
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    Abstract This paper presents a quadratic-penalty type method for solving linearly-constrained composite nonconvex-concave min-max problems. The method consists of solving a sequence of penalty subproblems which, due to the min-max structure of the problem, are potentially nonsmooth but can be approximated by smooth composite nonconvex minimization problems. Each of these penalty subproblems is then solved … Read more

    Hybrid Stochastic Gradient Descent Algorithms forStochastic Nonconvex Optimization

  • M. Lam Nguyen
  • Quoc Tran-Dinh
  • H. Nhan Pham
  • T. Dzung Phan
    • Categories Nonlinear Optimization, Stochastic Programming

    We introduce a hybrid stochastic estimator to design stochastic gradient algorithms for solving stochastic optimization problems. Such a hybrid estimator is a convex combination of two existing biased and unbiased estimators and leads to some useful property on its variance. We limit our consideration to a hybrid SARAH-SGD for nonconvex expectation problems. However, our idea … Read more

    Multiphase Mixed-Integer Nonlinear Optimal Control of Hybrid Electric Vehicles

  • Francesco Braghin
  • Nicolò Robuschi
  • Sebastian Sager
  • Federico Cheli
  • Clemens Zeile
    • Categories (Mixed) Integer Nonlinear Programming, Control Applications, Systems governed by Differential Equations Optimization

    This paper considers the problem of computing the non-causal minimum-fuel energy management strategy of a hybrid electric vehicle on a given driving cycle. Specifically, we address the multiphase mixed-integer nonlinear optimal control problem arising when optimal gear choice, torque split and engine on/off controls are sought in off-line evaluations. We propose an efficient model by … Read more

    Using interior point solvers for optimizing progressive lens models with spherical coordinates

  • Glòria Casanellas
  • Jordi Castro
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , , ,

    Designing progressive lenses is a complex problem that has been previously solved by formulating an optimization model based on Cartesian coordinates. In this work a new progressive lens model using spherical coordinates is presented, and interior point solvers are used to solve this new optimization model. Although this results in a highly nonlinear, nonconvex, continuous … Read more

    Hybrid methods for nonlinear least squares problems

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , , , , , ,

    This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function $F(x) = (1/2) f^T(x) f(x)$, where $x \in R^n$ and $f \in R^m$, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved … Read more

    Numerical solution of generalized minimax problems

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , , , , ,

    This contribution contains the description and investigation of four numerical methods for solving generalized minimax problems, which consists in the minimization of functions which are compositions of special smooth convex functions with maxima of smooth functions (the most important problem of this type is the sum of maxima of smooth functions). Section~1 is introductory. In … Read more