Geometric and game approaches for some discrete optimization problems

Abstract

We consider in this paper the adaptation of heuristics used for programming nondeterministic games to the problems of discrete optimization. In particular, we use some "game" heuristic methods of decision-making in various discrete optimization problems. The object of each of these problems is programming anytime algorithms. Among the problems described in this paper, there are the classical traveling salesman problem and some connected problems of minimization for nondeterministic finite automata. The first of the considered methods is the geometrical approach to some discrete optimization problems. For this approach, we define some special characteristics relating to some initial particular case of considered discrete optimization problem. For instance, one of such statistical characteristics for the traveling salesman problem is a significant development of the so-called "distance functions" up to the geometric variant such problem. And using this distance, we choose the corresponding specific algorithms for solving the problem. Besides, other considered methods for solving these problems are constructed on the basis of special combination of some heuristics, which belong to some different areas of the theory of artificial intelligence. More precisely, we shall use some modifications of unfinished branchand-bound method; for the selecting immediate step using some heuristics, we apply dynamic risk functions; simultaneously for the selection of coefficients of the averaging-out, we also use genetic algorithms; and the reductive self-learning by the same genetic methods is also used for the start of unfinished branch-and-bound method again. This combination of heuristics represents a special approach to construction of anytime-algorithms for the discrete optimization problems. This approach can be considered as an alternative to application of methods of linear programming, and to methods of multi-agent optimization, and also to neural networks.

An Algorithm for the Inverse Problem of Matrix Processing: DNA Chains, Their Distance Matrices and Reconstructing

“…The minimization method is given in the next section, where the justification for the possibility of piecemeal filling of the matrix is given, to obtain a value of badness close to optimal (i.e., in terminology of [13], "to obtain a pseudo-optimal solution"). Simplifying it, we can say that that, taking the average values of the maximum sides of the formed triangles (i.e., α in previous formulas; note that in [9], this value was counted final) as the beginning of the iterative process, we get a pseudo-optimal value in a few iterations.…”