# Condition and Linear Programming Optimization

## Abstract

*d*=(

*A*,

*b*,

*c*) specifying a pair of primal and dual linear programming problems in standard form,

*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## Bibliography

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