Matrix transformation method in quadratic binary optimization

• Published in: Optical memory & neural networks Vol. 24; no. 2; pp. 67 - 81
• Main Authors: Karandashev, I., Kryzhanovsky, B.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.04.2015
• Subjects: Complexity; Computer Science; Information Storage and Retrieval; Gradient methods; Minimization methods; Combinatorial mathematics; Hopfield neural networks
• ISSN: 1060-992X; 1934-7898
• DOI: 10.3103/S1060992X1502006X

The paper deals with the binary minimization of a quadratic functional. Typically the problem is NP-hard and the organization of the quadratic functional landscape in space of multiple binary variables even without constraints is not currently understood. We tackle the problem with the classical Hopfield neural network and show theoretically the exponential dependence between the energy and the attraction area of a minimum. As a consequence we propose a new algorithm using matrix transformation to cope with the problem. The matrix that gave birth to the functional is replaced by a new mix-matrix. This change of the objective function brings about a sharp increase in the minimization algorithm efficiency: the chances to find a global minimum grow as exp(α N ) where N is the dimensionality of the problem, and the energy spectrum of minima being found shifts considerably to the deep side so that the average of the spectrum differs from the energy of the global minimum by only 4–6% (subject to the type of matrix).