Abstract: This note gives a construction for minimizing certain twice-differentiable functions on a closed convex subset C, of a Hubert Space, H.The algorithm assumes one can constructively "project" points onto convex sets.A related algorithm may be found in Cheney-Goldstein [l], where a constructive fixed-point theorem is employed to construct points inducing a minimum distance between two convex sets.In certain instances when such projections are not too difficult to construct, say on spheres, linear varieties, and orthants, the method can be effective.For applications to control theory, for example, see Balakrishnan [2], and Goldstein [3].In what follows P will denote the "projection" operator for the convex set C. This operator, which is well defined and Lipschitzian, assigns to a given point in H its closest point in C (see, e.g., [l]).Take x£ff and y£C.Then [x -y, P(x) -y]^\\P(x) -y\\ 2 .In the nontrivial case this inequality is a consequence of the fact that C is supported by a hyperplane through P(x) with normal x -P(x).Let ƒ be a real-valued function on H and x 0 an arbitrary point of C. Let 5 denote the level set (xGC:/(x) ^f(x 0 )}, and let S be any open set containing the convex hull of S. Let ƒ'(*, •)= [V/(x), •] signify the Fréchet derivative of ƒ at x.A point zin C will be called stationary if P(z-pVf(z)) =z for all p>0; equivalently, when ƒ is convex the linear functional f (z, •) achieves a minimum on C at z.