Note on a method of conjugate subgradients for minimizing nondifferentiable functions

An algorithm is described for finding the minimum of any convex, not necessarily differentiable, functionf of several variables. The algorithm yields a sequence of points tending to the solution of the problem, if any; it requires the calculation off and one subgradient off at designated points. Its rate of convergence is estimated for convex and also for twice differentiable convex functions. It is an extension of the method of conjugate gradients, and terminates whenf is quadratic. Editor's note. The communications of Wolfe and Lemarechal which follow — received almost simultaneously — display different points of view, but deal with the same problem and use similar techniques. They are preliminary versions of promising attacks on the problem of minimizing a convex, but not necessarily differentiable, function of many variables. MATHEMATICAL PROGRAMMING STUDY 3 entitledNondifferentiable optimization is to be devoted to this subject.