552 research outputs found

    Fitting piecewise linear continuous functions

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    Abstract We consider the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective. We review some fitting problems that can be modeled as convex programs, and then introduce mixed-binary generalizations that allow variability in the regions defining the best-fit function's domain. We also study the additional constraints required to impose convexity on the best-fit function

    Mixed Integer Linear Programming Formulation Techniques

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    A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques. However, in some cases the resulting MIP can be either too weak or too large to be effectively solved by state of the art solvers. In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems

    Valid inequalities for the problem of optimizing a nonseparable piecewise linear function on 0-1 variables

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    Ankara : The Department of Industrial Engineering and the Graduate School of Engineering and Science of Bilkent University, 2011.Thesis (Master's) -- Bilkent University, 2011.Includes bibliographical references leaves 34-36.In many procurement and transportation applications, the unit prices depend on the amount purchased or transported. This results in piecewise linear cost functions. Our aim is to study the structure that arises due to a piecewise linear objective function and to propose valid inequalities that can be used to solve large procurement and transportation problems. We consider the problem of optimizing a nonseparable piecewise linear function on 0-1 variables. We linearize this problem using a multiple-choice model and investigate properties of facet defining inequalities. We propose valid inequalities and lifting results.Aksüt, Ziyaattin HüsrevM.S

    Generalized decision rule approximations for stochastic programming via liftings

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    Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higher-dimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem
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