A projection cutting plane algorithm for convex programming problems

The solution of a convex programming problem of the form min{phi(0)(x)/phi(j)(x, y) less than or equal to 0, j = 1,..., k} = min{phi(0)(x)\(x, y) is an element of Omega subset of R(n) X R(m)} is considered. The complexity of cutting plane methods for these problems depends on m + n. In this paper, the problem is reformulated as min{phi(o)(x)/x is an element of omega}, where omega = proj(R)n Omega, and the complexity depends only on n. How the cutting plane is found in the new formulation is discussed, together with a complexity analysis of this operation. Finally, an example where the projection cutting plane algorithm is more efficient than standard cutting plane methods is presented.