Landscape analyses and global search of knapsack problems

This paper shows statistical analyses of the search-space landscape of knapsack problems in due consideration of stochastic optimization. It is known from existing works that travelling salesman problems have a landscape called a rugged landscape. We deal with the 1000 knapsack problems where the values and the weights of the objects are arranged randomly. It was revealed that the landscape of the knapsack problems is a rugged landscape similar to that of travelling salesman problems by introducing proper estimate values and topology. It is assumed that the rugged landscape is a combination of the global valley-like structure and the local noise-like structure. We propose a new algorithm to estimate the optimum point by introducing the least mean square method to fit the global structure at some points selected randomly in the search space. The method does not contradict No Free Lunch Theorems because of availing of the feature of the landscape. It is forecasted that not only knapsack problems but also many practical problems have the structure which is characterized with the same measure. These results are useful to compose more effective optimization methods without trial and error.