bilinear programming

On Piecewise Linear Approximations of Bilinear Terms: Structural Comparison of Univariate and Bivariate Mixed-Integer Programming Formulations

  • Lukas Hager
  • Robert Burlacu
  • Andreas Bärmann
  • Thomas Kleinert
    • Categories (Mixed) Integer Linear Programming, Global Optimization Theory, Quadratic Programming Tags , , , ,

    Bilinear terms naturally appear in many optimization problems. Their inherent nonconvexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of … Read more

    A Finitely Convergent Disjunctive Cutting Plane Algorithm for Bilinear Programming

  • Hamed Rahimian
  • Sanjay Mehrotra
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Quadratic Programming Tags , , , ,

    In this paper we present and analyze a finitely-convergent disjunctive cutting plane algorithm to obtain an \(\epsilon\)-optimal solution or detect infeasibility of a general nonconvex continuous bilinear program. While the cutting planes are obtained in a manner similar to Saxena, Bonami, and Lee [Math. Prog. 130: 359–413, 2011] and Fampa and Lee [J. Global Optim. … Read more

    Sequential Convexification of a Bilinear Set

  • Hamed Rahimian
  • Sanjay Mehrotra
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming Tags , , , ,

    We present a sequential convexification procedure to derive, in the limit, a set arbitrary close to the convex hull of $\epsilon$-feasible solutions to a general nonconvex continuous bilinear set. Recognizing that bilinear terms can be represented with a finite number nonlinear nonconvex constraints in the lifted matrix space, our procedure performs a sequential convexification with … Read more

    Facets of a mixed-integer bilinear covering set with bounds on variables

  • Ashutosh Mahajan
  • Hamidur Rahman
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , , ,

    We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation … Read more

    A Copositive Approach for Two-Stage Adjustable Robust Optimization with Uncertain Right-Hand Sides

  • Samuel Burer
  • Guanglin Xu
    • Categories Dynamic Programming, Linear, Cone and Semidefinite Programming, Robust Optimization Tags , , , , ,

    We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which … Read more

    Relaxations and discretizations for the pooling problem

  • Shabbir Ahmed
  • Myun-Seok Cheon
  • Santanu S. Dey
  • Akshay Gupte
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , , , ,

    The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment, and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives unifying arguments and new insights on prevalent techniques. We also present new ideas for computing … Read more

    Analysis of MILP Techniques for the Pooling Problem

  • Santanu S. Dey
  • Akshay Gupte
    • Categories (Mixed) Integer Nonlinear Programming, Approximation Algorithms, Global Optimization Tags , , , , ,

    The $pq$-relaxation for the pooling problem can be constructed by applying McCormick envelopes for each of the bilinear terms appearing in the so-called $pq$-formulation of the pooling problem. This relaxation can be strengthened by using piecewise-linear functions that over- and under-estimate each bilinear term. The resulting relaxation can be written as a mixed integer linear … Read more

    Exact and heuristic solutions of the global supply chain problem with transfer pricing

  • Pierre Hansen
  • Sébastien Le Digabel
  • Sylvain Perron
  • Nenad Mladenovic
    • Categories Applications - OR and Management Sciences, Nonlinear Optimization, Supply Chain Management Tags , , , , ,

    We examine the example of a multinational corporation that attempts to maximize its global after tax profits by determining the flow of goods, the transfer prices, and the transportation cost allocation between each of its subsidiaries. Vidal and Goetschalckx (2001) proposed a bilinear model of this problem and solved it by an Alternate heuristic. We … Read more

    The Inverse Optimal Value Problem

  • Shabbir Ahmed
  • Yongpei Guan
    • Categories Global Optimization, Linear Programming, Linear, Cone and Semidefinite Programming Tags , , ,

    This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost coefficients, determine a cost-coefficient vector such that the corresponding optimal objective value of the linear program is closest to the given value. The above problem, referred here as the inverse optimal value … Read more