Nonlinear Optimization

Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems

  • Everton Silva
  • Elizabeth Wegner Karas
  • Lucelina Santos
    • Categories Multi-Criteria Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness … Read more

    Small polygons with large area

  • Christian Bingane
  • Michael J. Mossinghoff
    • Categories Constrained Nonlinear Optimization, Global Optimization Applications Tags , ,

    A polygon is {\em small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ … Read more

    Accelerating nuclear-norm regularized low-rank matrix optimization through Burer-Monteiro decomposition

  • Ching-pei Lee
  • Ling Liang
  • Tianyun Tang
  • Kim-Chuan Toh
    • Categories Global Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    This work proposes a rapid algorithm, BM-Global, for nuclear-norm-regularized convex and low-rank matrix optimization problems. BM-Global efficiently decreases the objective value via low-cost steps leveraging the nonconvex but smooth Burer-Monteiro (BM) decomposition, while effectively escapes saddle points and spurious local minima ubiquitous in the BM form to obtain guarantees of fast convergence rates to the … Read more

    An implicit function formulation for optimization of discretized index-1 differential algebraic systems

  • Robert Parker
  • Bethany Nicholson
  • John D. Siirola
  • Carl D. Laird
  • Lorenz T. Biegler
    • Categories Systems governed by Differential Equations Optimization Tags , , ,

    A formulation for the optimization of index-1 differential algebraic equation systems (DAEs) that uses implicit functions to remove algebraic variables and equations from the optimization problem is described. The formulation uses the implicit function theorem to calculate derivatives of functions that remain in the optimization problem in terms of a reduced space of variables, allowing … Read more

    An Exact Approach for Solving Pickup-and-Delivery Traveling Salesman Problems with Neighborhoods

  • Cai Gao
  • Ningji Wei
  • Jose L. Walteros
    • Categories Nonlinear Optimization, Production and Logistics, Transportation Tags , , , ,

    This paper studies a variant of the traveling salesman problem called the pickup-and-delivery traveling salesman problem with neighborhoods that combines traditional pickup and delivery requirements with the flexibility of visiting the customers at locations within compact neighborhoods of arbitrary shape. We derive two optimality conditions for the problem, a local condition that verifies whether a … Read more

    Duality assertions in vector optimization w.r.t. relatively solid convex cones in real linear spaces

  • Christian Günther
  • Bahareh Khazayel
  • Christiane Tammer
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization Tags , , , , , , ,

    We derive duality assertions for vector optimization problems in real linear spaces based on a scalarization using recent results concerning the concept of relative solidness for convex cones (i.e., convex cones with nonempty intrinsic cores). In our paper, we consider an abstract vector optimization problem with generalized inequality constraints and investigate Lagrangian type duality assertions … Read more

    A Gauss-Newton-based Decomposition Algorithm for Nonlinear Mixed-Integer Optimal Control Problems

  • Adrian Bürger
  • Clemens Zeile
  • Angelika Altmann-Dieses
  • Sebastian Sager
  • Moritz Diehl
    • Categories (Mixed) Integer Nonlinear Programming, Control Applications, Systems governed by Differential Equations Optimization Tags , ,

    For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is … Read more

    Worst-Case Complexity of TRACE with Inexact Subproblem Solutions for Nonconvex Smooth Optimization

  • Frank E. Curtis
  • Qi Wang
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , ,

    An algorithm for solving nonconvex smooth optimization problems is proposed, analyzed, and tested. The algorithm is an extension of the Trust Region Algorithm with Contractions and Expansions (TRACE) [Math. Prog. 162(1):132, 2017]. In particular, the extension allows the algorithm to use inexact solutions of the arising subproblems, which is an important feature for solving large-scale … Read more

    Improved RIP-Based Bounds for Guaranteed Performance of Two Compressed Sensing Algorithms

  • Yun-Bin Zhao
  • Zhi-Quan Luo
    • Categories Constrained Nonlinear Optimization, Global Optimization Applications, Nonsmooth Optimization Tags , , , ,

    Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two mainstream compressed sensing algorithms using the hard thresholding operator. The guaranteed performance of the two algorithms for signal recovery was mainly analyzed in terms of the restricted isometry property (RIP) of sensing matrices. At present, the best known bound using RIP of order … Read more

    A Generalized Formulation for Group Selection via ADMM

  • Chengyu Ke
  • Sunyoung Shin
  • Yifei Lou
  • Miju Ahn
    • Categories Nonlinear Optimization Tags , , ,

    This paper studies a statistical learning model where the model coefficients have a pre-determined non-overlapping group sparsity structure. We consider a combination of a loss function and a regularizer to recover the desired group sparsity patterns, which can embrace many existing works. We analyze the stationary solution of the proposed formulation, obtaining a sufficient condition … Read more