A Necessary and Sufficient Condition for Minimizing a Convex Fr´echet Differentiable Function on a Certain Hyperplane

A convex Fr´echet differentiable function is minimized subject to a certain hyperplane at a point if the function is minimized in all directions which are defined by a finite set of vectors. The proposed approach is different from the Lagrange multiplier approach. At the end of this paper, a linear program is formulated to solve the case when the above given convex function is quadratic.