Integer Programming

On two relaxations of quadratically-constrained cardinality minimization

  • Dennis Wei
    • Categories (Mixed) Integer Nonlinear Programming, Approximation Algorithms, Branch and Cut Algorithms Tags , , , ,

    This paper considers a quadratically-constrained cardinality minimization problem with applications to digital filter design, subset selection for linear regression, and portfolio selection. Two relaxations are investigated: the continuous relaxation of a mixed integer formulation, and an optimized diagonal relaxation that exploits a simple special case of the problem. For the continuous relaxation, an absolute upper … Read more

    Chance-constrained binary packing problems

  • Simge Küçükyavuz
  • James R. Luedtke
  • Yongjia Song
    • Categories Integer Programming, Stochastic Programming Tags ,

    We consider a class of packing problems with uncertain data, which we refer to as the chance-constrained binary packing problem. In this problem, a subset of items is selected that maximizes the total profit so that a generic packing constraint is satisfied with high probability. Interesting special cases of our problem include chance-constrained knapsack and … Read more

    On Defining Design Patterns to Generalize and Leverage Automated Constraint Solving

  • Thiago Serra
    • Categories Combinatorial Optimization, Integer Programming, Optimization Software Design Principles Tags , ,

    This position paper reflects on the generalization of adaptive methods for Constraint Programming (CP) solving mechanisms, and suggests the use of application-oriented descriptions as a means to broaden CP adoption in the industry. We regard as an adaptive method any procedure that modifies the behavior of the solving process according to previous experience gathered from … Read more

    On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

  • Michael N. Jung
  • Christian Kirches
  • Sebastian Sager
    • Categories (Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization Tags ,

    Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, … Read more

    Automatic Dantzig-Wolfe Reformulation of Mixed Integer Programs

  • Martin Bergner
  • Alberto Caprara
  • Alberto Ceselli
  • Fabio Furini
  • Marco E. Lübbecke
  • Enrico Malaguti
  • Emiliano Traversi
    • Categories (Mixed) Integer Linear Programming Tags , , , , ,

    Dantzig-Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That … Read more

    Threshold Boolean Form for Joint Probabilistic Constraints with Random Technology Matrix

  • Alexander Kogan
  • Miguel A. Lejeune
    • Categories (Mixed) Integer Nonlinear Programming, Data-Mining, Stochastic Programming Tags , , , , ,

    We develop a new modeling and exact solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the … Read more

    An Outer-Inner Approximation for separable MINLPs

  • Pierre Bonami
  • Hassan L. Hijazi
  • Adam Ouorou
    • Categories (Mixed) Integer Nonlinear Programming, Optimization Software and Modeling Systems Tags , ,

    A common structure in convex mixed-integer nonlinear programs is separable nonlinear functions. In the presence of such structures, we propose three improvements to the outer approximation algorithms. The first improvement is a simple extended formulation, the second is a refined outer approximation, and the third is a heuristic inner approximation of the feasible region. These … Read more

    Approximating the solution for the multiparametric 0-1-mixed integer linear programming problem with interval data

  • Alejandro Crema
  • Fernando Crema
  • Edgar Hugo Peraza
    • Categories (Mixed) Integer Linear Programming Tags , ,

    In this paper we present algorithms to approximate the solution for the multiparametric 0-1-mixed integer linear programming problem relative to the objective function. We consider the uncertainty for the parameters that de fine the cost vector corresponding to a subset of 0-1-variables by assuming that each parameter belongs to a known interval. We suppose that we … Read more

    On valid inequalities for quadratic programming with continuous variables and binary indicators

  • Hongbo Dong
  • Jeff Linderoth
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Quadratic Programming Tags ,

    In this paper we study valid inequalities for a fundamental set that involves a continuous vector variable x in [0,1]^n, its associated quadratic form x x’ and its binary indicators. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). We treat valid inequalities for this set as lifted from QPB, which … Read more