Remarks on the convergence of Rosen's gradient projection method (Q578857)

The convergence of Rosen's gradient projection method is a long-standing problem in nonlinear programming. Recently, \textit{X. Zhang} [Acta Math. Appl. Sin. 8, 125-128 (1985; Zbl 0566.90077)] proved that it is convergent in the 3-dimensional space; the author and \textit{X. Zhang} [A convergence theorem for Rosen's gradient projection method. MSRI Technical Report 02518-86 (1986)] proved its convergence in n-dimensional space under a restriction on a parameter in Rosen's method. In this paper, we propose a linearly algebraic conjecture which can yield the convergence of Rosen's method without the restriction. By verifying this conjecture for some special cases, we prove that Rosen's method is convergent in 4-dimensional space.