We solve the problem of maximizing a clf over a closed convex set (or, equivalently an infinite dimensional polyhedron), in all three cases: the strictly tangent, the internal, and the external case, under the usual hypothesis that the matrix operators of the involved inequality systems have closed range, generalizing the finite dimensional theory, given in d’Alessandro (A Conical Approach to Linear Programming, Scalar and Vector Optimization Problems (Gordon and Breach Science Publishers, Amsterdam, 1997)). We also generalize to infinite dimension the primal external optimization algorithm, introduced in d’Alessandro (Optimization 25:197–207, 1992) (see also d’Alessandro, A Conical Approach to Linear Programming, Scalar and Vector Optimization Problems (Gordon and Breach Science Publishers, Amsterdam, 1997)). Finally, we show that the solution of the problem can also be achieved solving a sequence of finite dimensional LP problems, because the sequence of their maxima converges to the maximum of the infinite dimensional problem. We also give finite dimensional approximations for the minimum distance problems, which, for example, are involved in the primal external optimization algorithm, generalized to infinite dimensions in this book.