Integer Programming

A Parametric Approach for Solving Convex Quadratic Optimization with Indicators Over Trees

  • Aaresh Bhathena
  • Salar Fattahi
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Dynamic Programming, Quadratic Programming Tags , , , , ,

    This paper investigates convex quadratic optimization problems involving $n$ indicator variables, each associated with a continuous variable, particularly focusing on scenarios where the matrix $Q$ defining the quadratic term is positive definite and its sparsity pattern corresponds to the adjacency matrix of a tree graph. We introduce a graph-based dynamic programming algorithm that solves this … Read more

    Tricks from the Trade for Large-Scale Markdown Pricing: Heuristic Cut Generation for Lagrangian Decomposition

  • Robert Streeck
  • Andreas Schmitt
  • Torsten Gellert
  • Asya Dipkaya
  • Vladimir Fux
  • Tim Januschowski
  • Timo Berthold
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences Tags , , , ,

    In automated decision making processes in the online fashion industry, the ‘predict-then-optimize’ paradigm is frequently applied, particularly for markdown pricing strategies. This typically involves a mixed-integer optimization step, which is crucial for maximizing profit and merchandise volume. In practice, the size and complexity of the optimization problem is prohibitive for using off-the-shelf solvers for mixed … Read more

    A Sequential Benders-based Mixed-Integer Quadratic Programming Algorithm

  • Andrea Ghezzi
  • Wim Van Roy
  • Sebastian Sager
  • Moritz Diehl
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Optimization Software Benchmark Tags , , , ,

    For continuous decision spaces, nonlinear programs (NLPs) can be efficiently solved via sequential quadratic programming (SQP) and, more generally, sequential convex programming (SCP). These algorithms linearize only the nonlinear equality constraints and keep the outer convex structure of the problem intact, such as (conic) inequality constraints or convex objective terms. The aim of the presented … Read more

    Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators

  • Peijing Liu
  • Alper Atamturk
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming Tags , , , ,

    In this paper, we consider convex quadratic optimization problems with indicators on the continuous variables. In particular, we assume that the Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in sparse graphical inference problems and others. We describe an explicit convex formulation for the problem by studying the Stieltjes polyhedron arising … Read more

    An MILP-Based Solution Scheme for Factored and Robust Factored Markov Decision Processes

  • Huikang Liu
  • Wolfram Wiesemann
  • Man-Chung Yue
    • Categories (Mixed) Integer Linear Programming, Robust Optimization Tags ,

    Factored Markov decision processes (MDPs) are a prominent paradigm within the artificial intelligence community for modeling and solving large-scale MDPs whose rewards and dynamics decompose into smaller, loosely interacting components. Through the use of dynamic Bayesian networks and context-specific independence, factored MDPs can achieve an exponential reduction in the state space of an MDP and … Read more

    Network Flow Models for Robust Binary Optimization with Selective Adaptability

  • Ian Zhu
    • Categories Combinatorial Optimization, Integer Programming, Robust Optimization

    Adaptive robust optimization problems have received significant attention in recent years, but remain notoriously difficult to solve when recourse decisions are discrete in nature. In this paper, we propose new reformulation techniques for adaptive robust binary optimization (ARBO) problems with objective uncertainty. Without loss of generality, we focus on ARBO problems with “selective adaptability”, a … Read more

    A diving heuristic for mixed-integer problems with unbounded semi-continuous variables

  • Katrin Halbig
  • Alexander Hoen
  • Ambros Gleixner
  • Jakob Witzig
  • Dieter Weninger
    • Categories Integer Programming Tags , , , ,

    Semi-continuous decision variables arise naturally in many real-world applications. They are defined to take either value zero or any value within a specified range, and occur mainly to prevent small nonzero values in the solution. One particular challenge that can come with semi-continuous variables in practical models is that their upper bound may be large … Read more

    Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

  • Johannes Thürauf
  • Julia Grübel
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Energy, Robust Optimization Tags , , , ,

    We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given … Read more

    On the integrality Gap of Small Asymmetric Traveling 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 instances with n nodes, where n is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope ($P^{n}_{ASEP}$) and its vertices. The … Read more

    Representing Integer Program Value Function with Neural Networks

  • Tu Anh-Nguyen
  • Joey Huchette
  • Andrew J. Schaefer
    • Categories (Mixed) Integer Linear Programming, Integer Programming Tags , ,

    We study the value function of an integer program (IP) by characterizing how its optimal value changes as the right-hand side varies. We show that the IP value function can be approximated to any desired degree of accuracy using machine learning (ML) techniques. Since an IP value function is a Chvátal-Gomory (CG) function, we first … Read more