The quartic minimization over the sphere can be reformulated as a nonlinear nonconvex semidefinite program over the spectraplex. In this paper, under mild assumptions, we show that the reformulated nonlinear semidefinite program possesses the quadratic growth property at a rank one critical point which is a local minimizer of the quartic minimization problem. The quadratic growth property further implies the strong metric subregularity of the subdifferential of the objective function of the unconstrained reformulation of the nonlinear semidefinite program, from which we can show that the objective function is a Łojasiewicz function with exponent \(\frac{1}{2}\) at the corresponding critical point. With these results, we can establish the linear convergence of an efficient DCA method proposed for solving the nonlinear semidefinite program.

球体上的四次最小化可以重新表示为光谱复合体上的非线性非凸半定程序。在本文中，在温和的假设下，我们证明了重新表述的非线性半定规划在一级临界点处具有二次增长性质，这是四次最小化问题的局部极小值。二次增长性质进一步暗示了非线性半定规划无约束重构的目标函数次微分的强度量次正则性，由此我们可以证明目标函数是指数为 \(\frac{1} { 2}\)在相应的临界点。利用这些结果，我们可以建立一种用于求解非线性半定规划的高效 DCA 方法的线性收敛性。