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Algorithmica

pp 1–53 | Cite as

A Multiplicative Weight Updates Algorithm for Packing and Covering Semi-infinite Linear Programs

  • Khaled Elbassioni
  • Kazuhisa Makino
  • Waleed NajyEmail author
Article
  • 29 Downloads

Abstract

We consider the following semi-infinite linear programming problems: \(\max \) (resp., \(\min \)) \(c^Tx\) s.t. \(y^TA_ix+(d^i)^Tx \le b_i\) (resp., \(y^TA_ix+(d^i)^Tx \ge b_i)\), for all \(y \in {{\mathcal {Y}}}_i\), for \(i=1,\ldots ,N\), where \({{\mathcal {Y}}}_i\subseteq {\mathbb {R}}^m_+\) are given compact convex sets and \(A_i\in {\mathbb {R}}^{m_i\times n}_+\), \(b=(b_1,\ldots b_N)\in {\mathbb {R}}_+^N\), \(d^i\in {\mathbb {R}}_+^n\), and \(c\in {\mathbb {R}}_+^n\) are given non-negative matrices and vectors. This general framework is useful in modeling many interesting problems. For example, it can be used to represent a sub-class of robust optimization in which the coefficients of the constraints are drawn from convex uncertainty sets \({{\mathcal {Y}}}_i\), and the goal is to optimize the objective function for the worst case choice in each \({{\mathcal {Y}}}_i\). When the uncertainty sets \({{\mathcal {Y}}}_i\) are ellipsoids, we obtain a sub-class of second-order cone programming. We show how to extend the multiplicative weights update method to derive approximation schemes for the above packing and covering problems. When the sets \({{\mathcal {Y}}}_i\) are simple, such as ellipsoids or boxes, this yields substantial improvements in running time over general convex programming solvers. We also consider the mixed packing/covering problem, in which both packing and covering constraints are given, and the objective is to find an approximately feasible solution.

Keywords

Multiplicative weights update Robust optimization Second-order cone programming Packing and covering 

Notes

Supplementary material

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Khaled Elbassioni
    • 1
  • Kazuhisa Makino
    • 2
  • Waleed Najy
    • 3
    Email author
  1. 1.Masdar InstituteKhalifa University of Science and TechnologyAbu DhabiUAE
  2. 2.Research Institute for Mathematical Sciences (RIMS)Kyoto UniversityKyotoJapan
  3. 3.New York UniversityAbu DhabiUAE

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