Optimization on the Euclidean Unit Sphere

Abstract

We consider the problem of minimizing a continuously differentiable function f of m linear forms in n variables on the Euclidean unit sphere. We show that this problem is equivalent to minimizing the same function of related m linear forms (but now in m variables) on the Euclidean unit ball. When the linear forms are known, this results in a drastic reduction in problem size whenever m ≪ n and allows to solve potentially large scale non-convex such problems. We also provide a test to detect when a polynomial is a polynomial in a fixed number of forms. Finally, we identify two classes of functions with no spurious local minima on the sphere: (i) quasi-convex polynomials of odd degree and (ii) nonnegative and homogeneous functions. Finally, odd degreed forms have only nonpositive local minima and at most (d − 1) m are strictly negative.