(Mixed) Integer Nonlinear Programming

Computing an enclosure for multiobjective mixed-integer nonconvex optimization problems using piecewise linear relaxations

  • Moritz Link
  • Stefan Volkwein
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization, Network Optimization Tags , , , ,

    In this paper, a new method for computing an enclosure of the nondominated set of multiobjective mixed-integer problems without any convexity requirements is presented. In fact, our criterion space method makes use of piecewise linear relaxations in order to bypass the nonconvexity of the original problem. The method chooses adaptively which level of relaxation is … Read more

    Piecewise Polyhedral Relaxations of Multilinear Optimization

  • Jongeun Kim
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , , ,

    In this paper, we consider piecewise polyhedral relaxations (PPRs) of multilinear optimization problems over axis-parallel hyper-rectangular partitions of their domain. We improve formulations for PPRs by linking components that are commonly modeled independently in the literature. Numerical experiments with ALPINE, an open-source software for global optimization that relies on piecewise approximations of functions, show that … Read more

    A Gauss-Newton-based Decomposition Algorithm for Nonlinear Mixed-Integer Optimal Control Problems

  • Adrian Bürger
  • Clemens Zeile
  • Angelika Altmann-Dieses
  • Sebastian Sager
  • Moritz Diehl
    • Categories (Mixed) Integer Nonlinear Programming, Control Applications, Systems governed by Differential Equations Optimization Tags , ,

    For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is … Read more

    Automatic Reformulations for Convex Mixed-Integer Nonlinear Optimization: Perspective and Separability

  • Meenarli Sharma
  • Ashutosh Mahajan
    • Categories (Mixed) Integer Nonlinear Programming, Optimization Software and Modeling Systems Tags , , , ,

    Tight reformulations of combinatorial optimization problems like Convex Mixed-Integer Nonlinear Programs (MINLPs) enable one to solve these problems faster by obtaining tight bounds on the optimal value. We consider two techniques for reformulation: perspective reformulation and separability detection. We develop routines for the automatic detection of problem structures suitable for these reformulations and implement new … Read more

    Advancements in the computation of enclosures for multi-objective optimization problems

  • Gabriele Eichfelder
  • Leo Warnow
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    A central goal for multi-objective optimization problems is to compute their nondominated sets. In most cases these sets consist of infinitely many points and it is not a practical approach to compute them exactly. One solution to overcome this problem is to compute an enclosure, a special kind of coverage, of the nondominated set. One … Read more

    A branch-and-prune algorithm for discrete Nash equilibrium problems

  • Stefan Schwarze
  • Oliver Stein
    • Categories (Mixed) Integer Nonlinear Programming, Game Theory Tags , , ,

    We present a branch-and-prune procedure for discrete Nash equilibrium problems with a convex description of each player’s strategy set. The derived pruning criterion does not require player convexity, but only strict convexity of some player’s objective function in a single variable. If satisfied, it prunes choices for this variable by stating activity of certain constraints. … Read more

    An approximation algorithm for optimal piecewise linear approximations of bounded variable products

  • Lukas Hager
  • Robert Burlacu
  • Andreas Bärmann
  • Katja Kutzer
    • Categories (Mixed) Integer Nonlinear Programming, Approximation Algorithms, Quadratic Programming Tags , , ,

    We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as … Read more

    A Reciprocity Between Tree Ensemble Optimization and Multilinear Optimization

  • Jongeun Kim
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming Tags , , ,

    In this paper, we establish a polynomial equivalence between tree ensemble optimization and optimization of multilinear functions over the Cartesian product of simplices. We use this insight to derive new formulations for tree ensemble optimization problems and to obtain new convex hull results for multilinear polytopes. A computational experiment on multi-commodity transportation problems with costs … Read more

    Mixed-Integer Programming Techniques for the Minimum Sum-of-Squares Clustering Problem

  • Jan Pablo Burgard
  • Carina Moreira Costa
  • Christopher Hojny
  • Kleinert Thomas
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Data-Mining Tags , , ,

    The minimum sum-of-squares clustering problem is a very important problem in data mining and machine learning with very many applications in, e.g., medicine or social sciences. However, it is known to be NP-hard in all relevant cases and to be notoriously hard to be solved to global optimality in practice. In this paper, we develop … Read more

    An SDP Relaxation for the Sparse Integer Least Square Problem

  • Alberto Del Pia
  • Dekun Zhou
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Semi-definite Programming Tags , , ,

    In this paper, we study the polynomial approximability or solvability of sparse integer least square problem (SILS), which is the NP-hard variant of the least square problem, where we only consider sparse {0, ±1}-vectors. We propose an l1-based SDP relaxation to SILS, and introduce a randomized algorithm for SILS based on the SDP relaxation. In … Read more