Integer Programming

A Generalization of Benders’ Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse

  • Anahita Hassanzadeh
  • Ted K. Ralphs
    • Categories (Mixed) Integer Linear Programming, Stochastic Programming Tags ,

    We describe a generalization of Benders’ method for solving two-stage stochastic linear optimization problems in which there are both continuous and integer variables in the first and second stages. Benders’ method relies on finding effective lower approximations for the value function of the second-stage problem. In this setting, the value function is a discontinuous, non-convex, … Read more

    On the Value Function of a Mixed Integer Linear Optimization Problem and an Algorithm for its Construction

  • Anahita Hassanzadeh
  • Ted K. Ralphs
    • Categories (Mixed) Integer Linear Programming, Stochastic Programming Tags , , ,

    This paper addresses the value function of a general mixed integer linear optimization problem (MILP). The value function describes the change in optimal objective value as the right-hand side is varied and understanding its structure is central to solving a variety of important classes of optimization problems. We propose a discrete representation of the MILP … Read more

    New symmetries in mixed-integer linear optimization

  • Philipp M. Christophel
  • Menal Guzelsoy
  • Imre Pólik
    • Categories (Mixed) Integer Linear Programming Tags , ,

    We present two novel applications of symmetries for mixed-integer linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new … Read more

    Cut Generation for Optimization Problems with Multivariate Risk Constraints

  • Simge Küçükyavuz
  • Nilay Noyan
    • Categories Integer Programming, Multi-Criteria Optimization, Stochastic Programming

    We consider a class of multicriteria stochastic optimization problems that features benchmarking constraints based on conditional value-at-risk and second-order stochastic dominance. We develop alternative mixed-integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints. We give the complete linear description of two non-convex substructures appearing in these … Read more

    A Feasible Active Set Method with Reoptimization for Convex Quadratic Mixed-Integer Programming

  • Christoph Buchheim
  • Marianna De Santis
  • Stefano Lucidi
  • Francesco Rinaldi
  • Long Trieu
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming Tags , ,

    We propose a feasible active set method for convex quadratic programming problems with non-negativity constraints. This method is specifically designed to be embedded into a branch-and-bound algorithm for convex quadratic mixed integer programming problems. The branch-and-bound algorithm generalizes the approach for unconstrained convex quadratic integer programming proposed by Buchheim, Caprara and Lodi to the presence … Read more

    Solving Bilevel Mixed Integer Program by Reformulations and Decomposition

  • Yu An
  • Bo Zeng
    • Categories (Mixed) Integer Linear Programming, Complementarity and Variational Inequalities Tags , , ,

    In this paper, we study bilevel mixed integer programming (MIP) problem and present a novel computing scheme based on reformulations and decomposition strategy. By converting bilevel MIP into a constrained mathematical program, we present its single-level reformulations that are friendly to perform analysis and build insights. Then, we develop a decomposition algorithm based on column-and-constraint … Read more

    Constraint Qualification Failure in Action

  • Hassan L. Hijazi
  • Leo Liberti
    • Categories (Mixed) Integer Nonlinear Programming, Second-Order Cone Programming Tags , , , ,

    This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers’ failure. CitationH. L. Hijazi and L.Liberti “Constraint … Read more

    Mixed-integer Quadratic Programming is in NP

  • Alberto Del Pia
  • Santanu S. Dey
  • Marco Molinaro
    • Categories Integer Programming Tags , ,

    Mixed-integer quadratic programming (MIQP) is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing … Read more

    A scalable bounding method for multi-stage stochastic integer programs

  • Osman Y. Ozaltin
  • Burhaneddin Sandikçi
    • Categories Integer Programming, Parallel Algorithms, Stochastic Programming Tags , , , ,

    Many dynamic decision problems involving uncertainty can be appropriately modeled as multi-stage stochastic programs. However, most practical instances are so large and/or complex that it is impossible to solve them on a single computer, especially due to memory limitations. Extending the work of Sandikci et al. (2013) on two-stage stochastic mixed-integer-programs (SMIPs), this paper develops … Read more

    Branch-and-bound for bi-objective integer programming

  • Sophie N. Parragh
  • Fabien Tricoire
    • Categories Combinatorial Optimization, Integer Programming, Multi-Criteria Optimization Tags , , , ,

    In Pareto bi-objective integer optimization the optimal result corresponds to a set of non- dominated solutions. We propose a generic bi-objective branch-and-bound algorithm that uses a problem-independent branching rule exploiting available integer solutions, and cutting plane generation taking advantage of integer objective values. The developed algorithm is applied to the bi-objective team orienteering problem with … Read more