Linear Adjusting Programming in Factor Space

TL;DR

This paper introduces Linear Adjusting Programming (LAP), a new optimization model that dynamically adjusts decision directions in factor space to handle constraints without differentiable functions, offering a potential polynomial solution to linear programming problems.

Contribution

The paper proposes LAP, a novel model that differs from traditional LP by focusing on short-term adjustments using projections, and introduces a method to handle multiple simultaneous constraints with the Hat matrix.

Findings

LAP provides a new approach to linear programming with potential polynomial complexity.

The projection method using the Hat matrix effectively handles multiple constraints.

LAP offers a dynamic decision process suitable for intelligent behavior modeling.

Abstract

The definition of factor space and a unified optimization based classification model were developed for linear programming. Intelligent behaviour appeared in a decision process can be treated as a point y, the dynamic state observed and controlled by the agent, moving in a factor space impelled by the goal factor and blocked by the constraint factors. Suppose that the feasible region is cut by a group of hyperplanes, when point y reaches the region's wall, a hyperplane will block the moving and the agent needs to adjust the moving direction such that the target is pursued as faithful as possible. Since the wall is not able to be represented to a differentiable function, the gradient method cannot be applied to describe the adjusting process. We, therefore, suggest a new model, named linear adjusting programming (LAP) in this paper. LAP is similar as a kind of relaxed linear programming (LP), and the difference between LP and LAP is: the former aims to find out the ultimate optimal point, while the latter just does a direct action in short period. You may ask: Where will a blocker encounter? How can the moving direction be adjusted? Where further blockers may be encountered next, and how should the direction be adjusted again? We request at least an adjusting should be achieved at the first time. If a hyperplane blocks y going ahead along with the direction d, then we must adjust the new direction d' as the projection of g in the blocking plane. If there is only one blocker at a time, it is straightforward to calculate the projection, but how to calculate the projection when there are more than one blocker encountered simultaneously? We suggest a projection calculation by means of the Hat matrix in this paper. Linear adjusting programming will attract interest in many fields. It might bring a new light to solve the linear programming problem with a strong polynomial solution.