Abstract
Optimization problems and their solution by symbolic regression methods are considered. The search is performed on non-Euclidean space. In such spaces it is impossible to determine a distance between two potential solutions and, therefore, algorithms using arithmetic operations of multiplication and addition are not used there. The search of optimal solution is performed on the space of codes. It is proposed that the principle of small variations of basic solution be applied as a universal approach to create search algorithms. Small variations cause a neighborhood of a potential solution, and the solution is searched for within this neighborhood. The concept of inheritance property is introduced. It is shown that for non-Euclidean search space, the application of evolution and small variations of possible solutions is effective. Examples of using the principle of small variation of basic solution for different symbolic regression methods are presented.
Highlights
Optimization Problems by SymbolicAll optimization problems can be divided into two large classes
We propose to search for the optimal solution in the neighborhood of good basic solutions
To expand the area of their application and, in particular, to simplify the execution of the operations of the genetic algorithm, a universal approach was developed based on the principle of small variations of the basic solution
Summary
All optimization problems can be divided into two large classes. One class includes the problems, where a target function is calculated on values of elements of search space. All methods of symbolic regression search for optimal solutions on a space of codes. Even when searching for solutions on a vector space with a numerical metrics crossover, mutation operations are applied to Gray codes of possible solutions. There are many symbolic regression methods, such as grammatical evolution [2], Cartesian GP [3], analytic programming [4], network operator method [5], parser-matrix evolution [6], complete binary GP [7] including sparse regression [8,9,10], and others [11,12,13,14,15], for finding solutions to various non-numerical optimization problems in which it is necessary to find optimal structures, graphs, constructions, formulas, mathematical expressions, schemes, etc. To avoid the problem of constructing rules for crossover of complex codes of symbolic regression methods, the principle of small variations of the basic solution was formulated in [24].
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