Mattias Ohlsson, Carsten Peterson and Bo Söderberg
Neural Networks for Optimization Problems with Inequality Constraints - the Knapsack Problem
Neural Computation 5, 331-339 (1993)

A strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented. The constraints x <= 0 are encoded by x Theta(x) terms in the energy function. A careful treatment of the mean field approximation for the self-coupling parts of the energy is crucial, and results in an essentially parameter-free algorithm. This methodology is extensively tested on the knapsack problem of size up to 103 items. The algorithm scales like NM for problems with N items and M constraints. Comparisons are made with an exact branch and bound algorithm when this is computationally possible (N <= 30). The quality of the neural network solutions consistently lies above 95 % of the optimal ones at a significantly lower CPU expense. For the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach. For "non-homogeneous" problems these produce good solutions, whereas for the more difficult "homogeneous" problems the neural approach is a winner with respect to solution quality and/or CPU time consumption. The approach is of course also applicable to other problems of similar structure, like set covering.

LU TP 92-11