Properties of a feasible direction algorithm which finds search directions by approximately solving a subproblem

This paper is concerned mostly with feasible direction methods for which the search direction is obtained by solving approximately a subproblem defined at iteration j by: (SP)j min{fs(zj, y): y??Q} where fs(zj, y) is a lower approximation to the objective function f(y5) and Q is not describable in the conventional form, Q = {x??Rn : g(x) ?? 0, h(x) = 0}, so most of the existing optimization algorithms are not applicable. An existing algorithm obtains search directions by solving exactly (SP)j. However exact solution of the subproblem is not possible in many cases regardless of their simple nature. The effects on convergence rate of approximate minimization of fs are investigated and new results derived. Application to a classical optimal control problem is considered.