A finite method to find a point in a set defined by a convex differentiable functional (Q1118803)

It is required to find a point in the set \(Q=\{x\in H:\) \(\phi\) (x)\(\leq 0\}\), where \(\phi\) is a convex continuously Frechet-differentiable functional defined on the real Hilbert space H. We apply the following method to solve this problem: \[ (1)\quad x^{(k+1)}=x^{(k)}-\gamma_ k\frac{\phi (x^{(k)})}{\| g(x^{(k)})\|^ 2}g(x^{(k)}), \] where \(x^{(0)}\) is a point in H, \(g(x^{(k)})\) is the Frechet- derivative of the functional \(\phi\) at the point \(x^{(k)}\), \(\| \cdot \|\) is the norm in the space H induced by the scalar product. We assume that the process stops as soon as \(x^{(k)}\) reaches Q. We show that with \(\gamma_ k\) from the interval \([I+\epsilon,2]\) (\(\epsilon\) is any number from (0;1]), the iterative process (1) is finite and its last point belongs to Q if \(\phi\) satisfies the Slater condition.