# Interior point approach to linear programming: theory, algorithms & parametric analysis

Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 81)

## Abstract

The classical theory of linear programming strongly depends on the fact that among the optimal solutions of an LP-problem there is always a vertex solution. In many situations the analysis is complicated by the fact that this vertex solution may not be unique. The recent research in the field of interior point methods for LP has made clear that every (solvable) LP-problem has a unique socalled central solution, namely the analytic center of the optimal facet of the problem. In this paper we reconsider the theory of LP by using central solutions. The analysis is facilitated by the unicity of the central solution of an LP-problem. Starting from scratch, using an elementary result from calculus, we present new proofs of the fundamental results of LP. These include the existence of a strictly complementary solution, and the strong duality theorem for LP. The proofs are simpler and often more natural than the ones currently known. It turns out that the central solution of an LP-problem is the limit point of the socalled central path of the problem. Based on this observation an algorithm will be derived which approximately follows the central path. The output of this algorithm is an approximation of the central solution. It also gives us the optimal partition of the problem. We finally deal with the topic of parametric analysis. So we investigate the dependence of the central solution on the right hand side coefficients and/or the coefficients in the objective vector. It turns out that also from the parametric point of view the interior point approach is more natural than the usual simplex based approach.

## Keywords

Interior Point Feasible Region Interior Point Method Central Solution Central Path
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.

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