Jean-Philippe P. Richard

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 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

    Lifting convex inequalities for bipartite bilinear programs

  • Santanu S. Dey
  • Jean-Philippe P. Richard
  • Xiaoyi Gu
    • Categories Cutting Plane Approaches, Global Optimization Theory, Nonlinear Optimization Tags

    The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality that is valid … Read more

    Network Models with Unsplittable Node Flows with Application to Unit Train Scheduling

  • Danial Davarnia
  • Jean-Philippe P. Richard
  • Ece Icyuz-Ay
  • Bijan Taslimi
    • Categories Cutting Plane Approaches, Network Optimization, Transportation

    We study network models where flows cannot be split or merged when passing through certain nodes, i.e., for such nodes, each incoming arc flow must be matched to an outgoing arc flow of identical value. This requirement, which we call “no-split no-merge” (NSNM), appears in railroad applications where train compositions can only be modified at … Read more

    Simultaneous convexification of bilinear functions over polytopes with application to network interdiction

  • Danial Davarnia
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Network Optimization

    We study the simultaneous convexification of graphs of bilinear functions that contain bilinear products between variables x and y, where x belongs to a general polytope and y belongs to a simplex. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied … Read more

    On the Polyhedral Structure of Two-Level Lot-Sizing Problems with Supplier Selection

  • Ayse Nur Arslan
  • Yongpei Guan
  • Jean-Philippe P. Richard
    • Categories (Mixed) Integer Linear Programming, Production and Logistics Tags , , ,

    In this paper, we study a two-level lot-sizing problem with supplier selection (LSS). This NP-hard problem arises in different production planning and supply chain management applications. We first present a dynamic programming algorithm for LSS that is polynomial when the number of plants is fixed. We use this algorithm to describe the convex hull of … Read more

    Deriving the convex hull of a polynomial partitioning set through lifting and projection

  • Trang T. Nguyen
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , ,

    Relaxations of the bilinear term, $x_1x_2=x_3$, play a central role in constructing relaxations of factorable functions. This is because they can be used directly to relax products of functions with known relaxations. In this paper, we provide a compact, closed-form description of the convex hull of this and other more general bivariate monomial terms (which … Read more

    Lifted Inequalities for 0−1 Mixed-Integer Bilinear Covering Sets

  • Kwanghun Chung
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Global Optimization Theory Tags , ,

    In this paper, we study 0-1 mixed-integer bilinear covering sets. We derive several families of facet-defining inequalities via sequence-independent lifting techniques. We then show that these sets have polyhedral structures that are similar to those of certain fixed-charge single-node flow sets. As a result, we obtain new facet-defining inequalities for these sets that generalize well-known … Read more

    Explicit Convex and Concave Envelopes through Polyhedral Subdivisions

  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
  • Chuanhui Xiong
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Polyhedra Tags , ,

    In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of … Read more

    Strong Valid Inequalities for Orthogonal Disjunctions and Bilinear Covering Sets

  • Kwanghun Chung
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Global Optimization Theory Tags , , , ,

    In this paper, we develop a convexification tool that enables the construction of convex hulls for orthogonal disjunctive sets using convex extensions and disjunctive programming techniques. A distinguishing feature of our technique is that, unlike most applications of disjunctive programming, it does not require the introduction of new variables in the relaxation. We develop and … Read more