Linear Step-adjusting Programming in Factor Space

Abstract

Intelligent behavior that appears 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 faithfully as possible. Since the wall is not able to be represented by a differentiable function, the gradient method cannot be applied to describe the adjusting process. We, therefore, suggest a new model, named linear step-adjusting programming (LSP) in this paper. LSP is similar to a kind of relaxed linear programming (LP). The difference between LP and LSP is that the former aims to find the ultimate optimal point, while the latter just does a direct action in a short period. Where will a blocker encounter? How do you adjust the moving direction? Where further blockers may be encountered next, and how should the direction be adjusted again?[Formula: see text] If the ultimate best is found, that’s a blessing; if not, that’s fine. We request at least an adjustment should be got at the first time. However, the former is idealism, and the latter is realism. In place of a gradient vector, the projection of goal direction g in a subspace plays a core role in LSP. If a hyperplane block y goes ahead along with the direction d, then we must adjust the new direction [Formula: see text] as the projection of g in the blocking plane. Suppose there is only one blocker at a time. In that case, it is straightforward to calculate the projection, but how to calculate the projection when more than one blocker is encountered simultaneously? It is still an open problem for LP researchers. We suggest a projection calculation using the Hat matrix in the paper. LSP will attract interest in economic restructuring, financial prediction, and reinforcement learning.