Induced optimal partition invariancy in linear optimization: constraints perturbation

TL;DR

This paper introduces the concept of induced optimal partition invariancy in linear optimization under simultaneous perturbations of constraints, providing a computational method to identify invariance intervals and analyzing the optimal value function behavior.

Contribution

It presents a new concept of induced optimal partition, explores its relation to existing partitions, and offers a modified computational method for invariance interval detection.

Findings

The induced optimal partition concept extends traditional invariance analysis.

The method successfully identifies invariance intervals in large-scale problems.

Behavior of the optimal value function is characterized within these intervals.

Abstract

In this paper, we study uni-parametric linear optimization problems, in which simultaneously the right-hand-side and the left-hand-side of constraints are linearly perturbed with identical parameter. In addition to the concept of change point, the induced optimal partition is introduced, and its relation to the well-known optimal partition is investigated. Further, the concept of free variables in each invariancy intervals is described. A modified generalized computational method is provided with the capability of identifying the intervals for the parameter value where induced optimal partitions are invariant. The behavior of the optimal value function is described in its domain. Some concrete examples depict the results. We further implement the methodology on some test problems to observe its behavior on large scale problems.