Minimizing Quadratic Functions in Constant Time

08/25/2016

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by   Kohei Hayashi, et al.

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A sampling-based optimization method for quadratic functions is proposed. Our method approximately solves the following n-dimensional quadratic minimization problem in constant time, which is independent of n: z^*=_v∈R^n〈v, A v〉 + n〈v, diag(d)v〉 + n〈b, v〉, where A ∈R^n × n is a matrix and d,b∈R^n are vectors. Our theoretical analysis specifies the number of samples k(δ, ϵ) such that the approximated solution z satisfies |z - z^*| = O(ϵ n^2) with probability 1-δ. The empirical performance (accuracy and runtime) is positively confirmed by numerical experiments.