A convergent gradient procedure in prehilbert spaces
Edward K. Blum · Pacific Journal of Mathematics · 1966
In this paper, we present a new method of approximating the minimum of a functional, J, defined on a prehilbert space and subject to constraints of the form φ^x) = 0, 1 ^ i ^ p, where the ψ { are also functional on the space. The method generates a convergent sequence of approximations using the gradients of J and ψim However, it is not a steepest descent procedure with respect to J. A theorem is proven which establishes the convergence of the approximating sequence to the minimum. The literature on extremal problems in abstract spaces is fairly-extensive. We refer to the bibliographies in [1], [5] for a partial list. That part of the literature which deals with approximation procedures for finding extrema subject to various constraints is also extensive. Again, see [1], [5] and also [3] for bibliographies. In particular, gradient-type methods have received considerable attention recently in a variety of contexts as in [2], [7], [6] to mention a few. In this paper, we present a new method of approximating the minimum of a functional, J", defined on a prehilbert space and subject to equality constraints. The method generates a convergent sequence of approximations using gradients but it is not of the steepest descent type with respect to J. 2. Preliminary remarks * Let £Γ be a prehilbert space. For u, v, e H, we denote their inner product by (u, ΐ/> and | | u ||2 = ζu, u). Let J and ψif 1 S i ^ P, be real functionals defined on some subset in H. We shall say that u * e H is a solution of the minimiza-tion problem defined by {J, fa] if ψt (u*) = 0, 1 ^ i ^ p and J(u*) ^ J(u) for those u in a neighborhood of u * which satisfy the con-straints ψi(u) = 0, 1 ^ i ^ p. By the gradient of J at u we mean an element of H, designated by FJ(u), such that for all An in some neighborhood of 0 e H, J(u + Δu)- J{u) = ζFJ(u), Any + e(u, An), where | ε(u, An) |/| | An | | — • 0 as Δu — • 0. Similarly, Fψ^u) denotes the gradient of ψi at u. If / is a real functional defined in a neighborhood of u e H and if df(u; h) = lim^o (f(u + sh) — f(u))/s exists for all h e H, we call df(u; h) the weak differential of / at u with respect to h. The relation