Statistics of partial minima

doi:10.1088/1751-8113/40/47/F02

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Journal of Physics A: Mathematical and Theoretical

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Journal of Physics A: Mathematical and Theoretical Volume 40 Number 47 Create an alert RSS this journal

E Ben-Naim et al 2007 J. Phys. A: Math. Theor. 40 F1021 doi:10.1088/1751-8113/40/47/F02

Statistics of partial minima

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E Ben-Naim¹, M B Hastings¹ and D Izraelevitz²

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We study pseudo-optimal solutions to multi-objective optimization problems by introducing partial minima defined as follows. Point x k-dominates x' when at least k of the coordinates of x are smaller than the corresponding coordinates of x'. A point not k-dominated by any other point in the set is a k-minimum or a partial minimum, generalizing the global minimum. We study statistical properties of partial minima for a set of N points independently distributed inside the d-dimensional unit hypercube using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ~ N^−(d−k)/k, when 1 ≤ k < d. Interestingly, there are k − 1 distinct scaling laws characterizing the largest coordinates: the distribution P(y_j) of the jth largest coordinate, y_j, decays algebraically, ${P(y_j)\sim (y_j)^{-\alpha_j-1}}$ , with ${\alpha_j=j\frac{d-k}{k-j}}$ for 1 ≤ j ≤ k − 1. The average number of partial minima grows logarithmically, ${A\simeq \frac{1}{(d-1)!}(\ln N)^{d-1}}$ , when k = d. The full distribution of the number of minima is obtained in closed form in two dimensions.