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The convex variational programming problem has been found of practical importance to power system scheduling and control. Convex variational programming problems are closely related to the convex programming problems treated by Kuhn and Tucker . Among their positive attributes are the absence of difficulties with weak relative minima and with saddle-point or conjugate-point conditions. As a result it is feasible to develop dual-variational problems and to provide estimates for bracketing the extreme value of the functional of the variational problem when gradient or successive approximations methods are employed for determination of near optimal control schedules. This paper develops necessary and sufficient conditions for the optimal control schedule and a means for numerically determining bounds to the extreme value of the functional. Applications are given to the problem of hydrothermal power system coordination.