Mattias Ohlsson, Carsten Peterson and Bo Söderberg Neural Networks for Optimization Problems with Inequality Constraints - the Knapsack Problem Neural Computation 5, 331-339 (1993)
Abstract: 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 10^{3} 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 |