Quadratic Growth and Linear Convergence of a DCA Method for Quartic Minimization over the Sphere

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 \(\frac{1}{2}\) $\frac{1}{2}$ 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.