Optimization on the Euclidean Unit Sphere
研究在欧几里得单位球面上最小化m个线性形式的连续可微函数，证明该问题等价于在单位球上最小化相关函数，当m远小于n时可大幅降低问题规模，并识别了两类无虚假局部极小值的函数。
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 $m\ll n$ and allows us to solve potentially large scale nonconvex 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.