Finding ε-Global Optimal Value of a One-dimensional Periodic Function in an Interval Based on Fourier Series and Semi-definite Programming

One-dimensional global optimization of a function f(x) in an interval D is still a difficult problem. In this paper, we pose a new method for finding the ε-global optimal value of f(x) in D. We first approximate the function f (x) via its partial sum of its Fourier series. We show that for given ε, we can find a partial sum s_n(x) n of its Fourier series such that |s_n(x) - f(x)| ≤ ε for all x ∈ D when n is larger than some positive number. Then we consider finding the ε-global optimal value of this partial sum, which turns out to be able to be converted into a semi-definite programming problem via some transformation, hence is able to be solved by interior point method in polynomial time.