Month: January 2025

Improved Approximation Algorithms for Low-Rank Problems Using Semidefinite Optimization

  • Ryan Cory-Wright
  • Jean Pauphilet
    • Categories (Mixed) Integer Linear Programming, Approximation Algorithms, Semi-definite Programming

    Inspired by the impact of the Goemans-Williamson algorithm on combinatorial optimization, we construct an analogous relax-then-sample strategy for low-rank optimization problems. First, for orthogonally constrained quadratic optimization problems, we derive a semidefinite relaxation and a randomized rounding scheme, which obtains provably near-optimal solutions, mimicking the blueprint from Goemans and Williamson for the Max-Cut problem. We … Read more

    Bi-Parameterized Two-Stage Stochastic Min-Max and Min-Min Mixed Integer Programs

  • Sumin Kang
  • Manish Bansal
    • Categories Integer Programming, Network Optimization, Stochastic Programming Tags , , , ,

    We introduce two-stage stochastic min-max and min-min integer programs with bi-parameterized recourse (BTSPs), where the first-stage decisions affect both the objective function and the feasible region of the second-stage problem. To solve these programs efficiently, we introduce Lagrangian-integrated L-shaped (\(L^2\)) methods, which guarantee exact solutions when the first-stage decisions are pure binary. For mixed-binary first-stage … Read more

    High-Probability Polynomial-Time Complexity of Restarted PDHG for Linear Programming

  • Zikai Xiong
    • Categories Linear Programming Tags , , ,

    The restarted primal-dual hybrid gradient method (rPDHG) is a first-order method that has recently received significant attention for its computational effectiveness in solving linear program (LP) problems. Despite its impressive practical performance, the theoretical iteration bounds for rPDHG can be exponentially poor. To shrink this gap between theory and practice, we show that rPDHG achieves … Read more