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Condition pp 201-222 | Cite as

Condition and Linear Programming Optimization

  • Peter Bürgisser
  • Felipe Cucker
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 349)

Abstract

In the previous chapter we analyzed an algorithm deciding feasibility for a triple d=(A,b,c) specifying a pair of primal and dual linear programming problems in standard form,
$$ \min c^{\mathrm {T}}x \quad \mbox{subject to}\quad Ax=b,\quad x\geq 0 , $$
(SP)
and
$$ \max b^{\mathrm {T}}y \quad \mbox{subject to}\quad A^{\mathrm {T}}y\leq c. $$
(SD)
If such an algorithm decides that a triple d is feasible, we may want to compute the optimizers x and y , as well as the optimal value v , of the pair (SP)–(SD). To do so is the goal of this chapter.

An approach to this problem is to apply the interior-point algorithm along with its basic analysis as provided in Chap.  9. A possible obstacle is the fact that the feasible point z=(x,y,s) returned by the algorithm of Chap.  10 does not necessarily belong to the central neighborhood \(\mathcal {N}(\frac{1}{4})\).

Another obstacle, now at the heart of this book’s theme, is how to deduce, at some iteration, the optimizers x and y . Without doing so, we will increasingly approach these optimizers without ever reaching them. It is not surprising that a notion of condition should be involved in this process. This notion follows lines already familiar to us. For almost all feasible triples d, a small perturbation of d will produce a small change in x and y . For a thin subset of data, instead, arbitrarily small perturbations may substantially change these optimizers. The central character of this chapter, the condition number  Open image in new window , measures the relative size of the smallest perturbation that produces such a discontinuous change in the optimizers.

In this chapter, we describe and analyze algorithms solving the optimal basis problem, which, we recall, consists in, given a feasible triple d, finding an optimal basis for it.

Keywords

Feasible Point Optimal Basis Feasibility Problem Central Neighborhood Continuous Segment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. 58.
    D. Cheung and F. Cucker. Solving linear programs with finite precision: II. Algorithms. Journal of Complexity, 22:305–335, 2006. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongHong KongHong Kong SAR

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