On the {\L}ojasiewicz Exponent of the Quadratic Sphere Constrained Optimization Problem

In this paper, we prove that the global version of the $\text{L}$ojasiewicz gradient inequality holds for quadratic sphere constrained optimization problem with exponent $\theta =\frac{3}{4}$. An example from Ting Kei Pong shows that $\theta =\frac{3}{4}$ is tight. This is the first $\text{L}$ojasiewicz gradient inequality established for the sphere constrained optimization problem with a linear term.