Linear Programming Processivity and Structural Optimisation of Intelligent Systems ^†

Abstract

The factor space theory of causal analysis and factor manifolds is essentially an optimisation problem in the application of intelligent systems and is closely related to linear programming. In this paper, based on the organic combination of factor space theory and the linear programming prism cut algorithm, we propose a procedural expansion model of linear programming, which proceduralises the solution method of linear programming by transforming the background relations or conceptual backgrounds with convexity into a set of restricted surfaces with the continuity method of the factor phase domain. The one-time solution is turned into an approximate optimisation-seeking process, and then this processisation is transferred to the framework of the factor space so that the processisation algorithm of linear programming can become a computational tool for the restructuring of intelligent systems.

1. Introduction

Linear programming is a classical optimisation problem, and many algorithms for intelligent optimisation can be used to solve linear programming problems; however, these algorithms are not proposed to solve a specific problem in linear programming. In 2011, Pei Zhuang Wang first proposed a prismatic cutting method for solving linear programming pairwise problems with geometric figurative thinking [1]: a figurative heuristic specifically aimed at solving linear programming intelligent algorithms. In 1982, Pei-Zhuang Wang proposed the theory of factor spaces to study the nature of human knowledge representation and decision-making [2]. In the theory of factor spaces, causal analysis and factor manifestation are the main tools in the application of factor spaces to intelligent systems, which are also essentially optimisation techniques [3].

On the basis of the above research [4], a new mathematical approach to the optimisation of intelligent systems can be provided by transposing a process-oriented extension of linear programming into the theoretical framework of factor spaces. In fact, as long as the discrete factor phase space is embedded in the real number interval after ordering, the procedural linear programming solution can be transferred to the restructuring of intelligent systems in general, i.e., intelligent system restructuring can then be depicted as a proceduraliszation of linear programming [5,6,7,8].

2. Process-Oriented Extension of Linear Programming

In factor space theory, the problem of feature extraction for artificial intelligence can be generalised to factor manifolds, which can be reduced to an optimisation problem. In order to extend the one-off solution of the optimisation problem into a process of structural optimisation, an extension of the ordinary linear programming problem is required, and this extended model can then be placed in the framework of the factor space.

The linear programming and its dual problem can be expressed separately as:

$$(P)\hspace{1em}\hspace{1em}\begin{array}{ll}Max& cx,\\ Sub& Ax\le b.\end{array} \mathrm{and} (D)\hspace{1em}\hspace{1em}\begin{array}{ll}Min& by,\\ Sub& Ay\ge c.\end{array}$$

The process-oriented extension models for linear programming and its dual problem are:

$$(B)\hspace{1em}\hspace{1em}\begin{array}{ll}\mathit{Bigger}& cx,\\ Sub& Ax\le b.\end{array} \mathrm{and} (S)\hspace{1em}\hspace{1em}\begin{array}{ll}\mathit{Smaller}& by,\\ Sub& Ay\ge c.\end{array}$$

Bigger is not about maximising cx at once but about making it as large as possible in steps. When Bigger is replaced by Maximum, problem (B) returns to linear programming (P); Smaller is not about minimising by at once but about making it as small as possible in steps. The core of the process expansion is to extend linear programming from a one-time solution and the optimum to a step-by-step approximation process. This extension is more relevant to the practical needs of decision making.

The simplex method is not valid for solving the process-oriented model of linear programming. Therefore, the process-oriented model solution of linear programming is based on the theory of prismatic cuts, a theory which, in addition to providing a pictorial geometric description of the simplex method in dyadic space, also allows for a simple solution to the optimisation process of linear programming.

In dyadic programming (S), a prismatic cut theory based on the restriction surface yAj = c is applied to Y. The n restriction surface cuts the dyadic space Y into a convex polyhedron, forming a dyadic feasible domain S. The lowest point is approached gradually in the feasible domain from a given initial point downwards in imitation of free fall. The descent is first in the direction of gravity, to the lower wall of the feasible domain, and then on the blocking surface in the direction of the projection of gravity to the surfaces. When the descent is stalled, the descent continues in a new blocking surface in the direction of the projection of gravity until the point at which it is needed can be stopped. Specific procedural algorithms are given in the text as well as arithmetic examples.

3. Linear Programming Models for Structural Optimisation of Intelligent Systems

While linear programming is a continuous variable operation, the phase domain of factors is a qualitative sequence of words whose sample points are discrete in location. This requires a solution to the problem of continuously using the discrete phases of the factors.

In the definition of factor space, the phase domain is ordered under certain objectives, and the ordered phase domain can be embedded in an interval of real numbers. Thus, factors have both discrete phase domains and continuous phase domains. The background relations of R between multiple factors can be continuous by following the principles of real punctuation and convexification: let the discrete phases divide the interval into lattices, and the discrete sample lattices take specific real points as representatives (real punctuation), and then the convex closure formed by these base points can be used as the continuous background relation R (convexification). In general, the background relations can all be reduced to convex sets. With the continuum approach to factors, we can incorporate linear programming into the factor space framework for the optimisation process of intelligent systems.

The most pressing question for a dynamic system with decision goals is, when the system is in a certain state, what direction does it take next? It is not as simple and controllable as free fall, and where the “next turning point” will be is often unpredictable. For system control and optimisation, the projection vector is the most important: it is the direction of change and adjustment to the status quo.

This paper provides definitions of the sample distribution of the concept, the manifold basis of the concept, and proof of the relevant properties. On this basis, a simple orientation adjustment algorithm is given to solve this uncontrollable situation, while the problem of transforming an intelligent system into a linear programming procedural solution is presented in detail through a case study of intelligent decision making for new product development.

4. Short Conclusions

This paper gives a process-oriented extension model of linear programming, which, in turn, transforms a background relation or conceptual background with convexity into a set of restriction surfaces using a continuum of factor phase domains, enabling intelligent systems to be optimised using linear programming. This paper bridges the interconversion between the optimisation of intelligent systems and traditional mathematical optimisation and is of great importance.