Papers › Minimizing Quadratic Functions in Constant Time
Minimizing Quadratic Functions in Constant Time
Kohei Hayashi, Yuichi Yoshida
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^*=min_(𝐯 ∈ℝⁿ)⟨𝐯, A 𝐯⟩+ n⟨𝐯, diag(𝐝)𝐯⟩+ n⟨𝐛, 𝐯⟩, where A ∈ℝ^(n ×n) is a matrix and 𝐝,𝐛 ∈ℝⁿ are vectors. Our theoretical analysis specifies the number of samples k(δ, ϵ) such that the approximated solution z satisfies |z - z^*| = O(ϵn²) with probability 1-δ. The empirical performance (accuracy and runtime) is positively confirmed by numerical experiments.
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