Linear, Cone and Semidefinite Programming

Strengthening Lasserre’s Hierarchy in Real and Complex Polynomial Optimization

  • Jie Wang
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Semi-definite Programming Tags , , , , ,

    We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre’s hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve … Read more

    On the integrality Gap of Small Asymmetric Travelling Salesman Problems: A Polyhedral and Computational Approach

  • Eleonora Vercesi
  • Luca Maria Gambardella
  • Stefano Gualandi
  • Monaldo Mastrolilli
  • Janos Barta
    • Categories Combinatorial Optimization, Integer Programming, Linear, Cone and Semidefinite Programming Tags ,

    \(\) In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for small-sized instances. In particular, we focus on the geometric properties and symmetries of the ASEP polytope ( \(P_{ASEP}^n\) ) and its vertices. The polytope’s … Read more

    An inexact infeasible arc-search interior-point method for linear programming problems

  • Einosuke Iida
  • Makoto Yamashita
    • Categories Linear Programming, Linear, Cone and Semidefinite Programming

    Inexact interior-point methods (IPMs) are a type of interior-point methods that inexactly solve the linear equation system for obtaining the search direction. On the other hand, arc-search IPMs approximate the central path with an ellipsoidal arc obtained by solving two linear equation systems in each iteration, while conventional line-search IPMs solve one linear system. Therefore, … Read more

    Data Collaboration Analysis Over Matrix Manifolds

  • Keiyu Nosaka
  • Akiko Yoshise
    • Categories Linear, Cone and Semidefinite Programming, Optimization in Data Science, Other Topics

    The effectiveness of machine learning (ML) algorithms is deeply intertwined with the quality and diversity of their training datasets. Improved datasets, marked by superior quality, enhance the predictive accuracy and broaden the applicability of models across varied scenarios. Researchers often integrate data from multiple sources to mitigate biases and limitations of single-source datasets. However, this … Read more

    A Primal-Dual Frank-Wolfe Algorithm for Linear Programming

  • Matthew Hough
  • Stephen A. Vavasis
    • Categories Linear Programming Tags , , ,

    \(\) We present two first-order primal-dual algorithms for solving saddle point formulations of linear programs, namely FWLP (Frank-Wolfe Linear Programming) and FWLP-P. The former iteratively applies the Frank-Wolfe algorithm to both the primal and dual of the saddle point formulation of a standard-form LP. The latter is a modification of FWLP in which regularizing perturbations … Read more

    The SCIP Optimization Suite 9.0

  • Suresh Bolusani
  • Mathieu Besançon
  • Ksenia Bestuzheva
  • Antonia Chmiela
  • Joao Dionisio
  • Tim Donkiewicz
  • Jasper van Doornmalen
  • Leon Eifler
  • Mohammed Ghannam
  • Ambros Gleixner
  • Christoph Graczyk
  • Katrin Halbig
  • Ivo Hedtke
  • Alexander Hoen
  • Christopher Hojny
  • Rolf van der Hulst
  • Dominik Kamp
  • Thorsten Koch
  • Kevin Kofler
  • Jurgen Lentz
  • Julian Manns
  • Gioni Mexi
  • Erik Mühmer
  • Marc E. Pfetsch
  • Franziska Schlösser
  • Felipe Serrano
  • Yuji Shinano
  • Mark Turner
  • Stefan Vigerske
  • Dieter Weninger
  • Liding Xu
    • Categories Integer Programming, Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems Tags , , , , , , , , ,

    The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in the SCIP Optimization Suite 9.0. The updates in SCIP 9.0 include improved symmetry handling, additions and improvements of nonlinear handlers and primal heuristics, a … Read more

    On Averaging and Extrapolation for Gradient Descent

  • Alan Luner
  • Benjamin Grimmer
    • Categories Convex Optimization, Semi-definite Programming Tags , , , ,

    \(\) This work considers the effect of averaging, and more generally extrapolation, of the iterates of gradient descent in smooth convex optimization. After running the method, rather than reporting the final iterate, one can report either a convex combination of the iterates (averaging) or a generic combination of the iterates (extrapolation). For several common stepsize … Read more

    T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization

  • Hiroki Marumo
  • Sunyoung Kim
  • Makoto Yamashita
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , , ,

    We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into … Read more

    Accurate and Warm-Startable Linear Cutting-Plane Relaxations for ACOPF

  • Daniel Bienstock
  • Matias Villagra
    • Categories Convex Optimization, Energy, Linear, Cone and Semidefinite Programming Tags , ,

    We present a linear cutting-plane relaxation approach that rapidly proves tight lower bounds for the Alternating Current Optimal Power Flow Problem (ACOPF). Our method leverages outer-envelope linear cuts for well-known second-order cone relaxations for ACOPF along with modern cut management techniques. These techniques prove effective on a broad family of ACOPF instances, including the largest … Read more