Showing papers on "Concave function published in 1970"

Some Convex Programs Whose Duals Are Linearly Constrained

Abstract: Let f0, f1, …, fm be convex functions on Rn, and let (P) denote the problem of minimizing f0(x) subject to f1(x) ≤ 0, …, fm(x) ≤ 0. According to the theory of conjugate functions, many different dual problems can be associated with (P), each one corresponding to a particular class of perturbations of (P). Thus, in developing dual methods of solutions of (P), one has considerable flexibility in the choice of the dual problem, and the choice can be made in view of its suitability for a given purpose. This paper treats some simple possibilities in the important case where each of the functions fi satisfies the following condition: fi is not affine along any line segment, unless it is affine along the entire line extending the segment (The latter holds, for example, if fi is analytic.) It is shown that the perturbations can be chosen in this case sothat the corresponding dual problem (P*) consists essentially of maximizing a differentiable concave function subject to linear constraints. The duality theorems applicable to (P) and(P*) are then somewhat more refined than those in the general theory; e.g. the infimum in (P) and the supremum in (P*) are necessarily equal, if (P) is consistent. The duality theory for the geometric programs of Duffin, Peterson and Zener, and the quadratic and lp programs of Peterson and Ecker, is derived as an illustration.

Note---A Note on the Cyclic Coordinate Ascent Method

Abstract: The cyclic coordinate ascent method is a frequently used algorithm in optimization problems. It requires no derivatives and indicates in one iteration if a given point is optimal. It is proved that the cyclic coordinate ascent method will converge for pseudo concave functions, as well as for strictly concave functions as was previously known.