Nonlinear Optimization

Alternating direction methods for non convex optimization with applications to second-order least-squares and risk parity portfolio selection

  • Xi Bai
  • Katya Scheinberg
    • Categories Finance and Economics, Nonlinear Optimization, Nonlinear Systems and Least-Squares Tags , , , , ,

    In this paper we mainly focus on optimization of sums of squares of quadratic functions, which we refer to as second-order least-squares problems, subject to convex constraints. Our motivation arises from applications in risk parity portfolio selection. We generalize the setting further by considering a class of nonlinear, non convex functions which admit a (non … Read more

    A Fast Branch-and-Bound Algorithm for Non-convex Quadratic Integer Optimization Subject To Linear Constraints Using Ellipsoidal Relaxations

  • Christoph Buchheim
  • Marianna De Santis
  • Laura Palagi
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Quadratic Programming

    We propose two exact approaches for non-convex quadratic integer minimization subject to linear constraints where lower bounds are computed by considering ellipsoidal relaxations of the feasible set. In the first approach, we intersect the ellipsoids with the feasible linear subspace. In the second approach we penalize exactly the linear constraints. We investigate the connection between … Read more

    On the Quadratic Shortest Path Problem

  • Christoph Buchheim
  • Federico Malucelli
  • Borzou Rostami
  • Davide Frey
    • Categories Combinatorial Optimization, Quadratic Programming Tags , ,

    Finding the shortest path in a directed graph is one of the most important combinatorial optimization problems, having applications in a wide range of fields. In its basic version, however, the problem fails to represent situations in which the value of the objective function is determined not only by the choice of each single arc, … Read more

    Lower bounding procedure for the Asymmetric Quadratic Traveling Salesman Problem

  • Pietro Belotti
  • Stefano Gualandi
  • Federico Malucelli
  • Borzou Rostami
    • Categories (Mixed) Integer Linear Programming, Combinatorial Optimization, Quadratic Programming Tags , , , ,

    In this paper we consider the Asymmetric Quadratic Traveling Salesman Problem. Given a directed graph and a function that maps every pair of consecutive arcs to a cost, the problem consists in finding a cycle that visits every vertex exactly once and such that the sum of the costs is minimum. We propose an extended … Read more

    An external penalty-type method for multicriteria

  • Ellen H. Fukuda
  • L. M. Graña Drummond
  • Fernanda M. P. Raupp
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization Tags , , ,

    We propose an extension of the classical real-valued external penalty method to the multicriteria optimization setting. As its single objective counterpart, it also requires an external penalty function for the constraint set, as well as an exogenous divergent sequence of nonnegative real numbers, the so-called penalty parameters, but, differently from the scalar procedure, the vector-valued … Read more

    On the Performance of SQP Methods for Nonlinear Optimization

  • Philip E. Gill
  • Michael A. Saunders
  • Elizabeth Wong
    • Categories Constrained Nonlinear Optimization, Optimization Software Benchmark Tags , , ,

    This paper concerns some practical issues associated with the formulation of sequential quadratic programming (SQP) methods for large-scale nonlinear optimization. SQP methods find an approximate solution of a sequence of quadratic programming (QP) subproblems in which a quadratic model of the objective function is minimized subject to the linearized constraints. Extensive numerical results are given … Read more

    Regret Analysis of Block Coordinate Gradient Methods for Online Convex Programming

  • Hua Xiaoqin
  • Nobuo Yamashita
    • Categories Applications - OR and Management Sciences, Nonlinear Optimization, Stochastic Programming Tags , , ,

    In this paper, we propose two block coordinate gradient (BCG) methods for the online convex programming: the BCG method with the cyclic rule and the BCG method with the random rule. The proposed methods solve a low dimensional problem at each iteration, and hence they are efficient for large scale problems. For the proposed methods, … Read more

    Trust-region methods without using derivatives: Worst case complexity and the non-smooth case

  • R. Garmanjani
  • Luis Nunes Vicente
  • Diogo Júdice
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , , ,

    Trust-region methods are a broad class of methods for continuous optimization that found application in a variety of problems and contexts. In particular, they have been studied and applied for problems without using derivatives. The analysis of trust-region derivative-free methods has focused on global convergence, and they have been proved to generate a sequence of … Read more

    A trust-funnel method for nonlinear optimization problems with general nonlinear constraints and its application to derivative-free optimization

  • Phillipe Sampaio
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization Tags , , ,

    A trust-funnel method is proposed for solving nonlinear optimization problems with general nonlinear constraints. It extends the one presented by Gould and Toint (Math. Prog., 122(1):155-196, 2010), originally proposed for equality-constrained optimization problems only, to problems with both equality and inequality constraints and where simple bounds are also considered. As the original one, our method … Read more

    Object-Parallel Infrastructure for Implementing First-Order Methods, with an Example Application to LASSO

  • Jonathan Eckstein
  • Gyorgy Matyasfalvi
    • Categories Bound-constrained Optimization, Parallel Algorithms, Unconstrained Optimization Tags , ,

    We describe the design of a C++ vector-manipulation substrate that allows first-order optimization algorithms to be expressed in a concise and readable manner, yet still achieve high performance in parallel computing environments. We use standard object-oriented techniques of encapsulation and operator overloading, combined with a novel “symbolic temporaries” delayed-evaluation system that greatly reduces the overhead … Read more