The Constrained Economic Dispatch Problem was formulated and a detailed method of solving it using a Lagrange Multiplier approach was developed. The method developed is a non-linear, primal-dual technique that uses forward elimination and back substitution applied to the Hessian matrix and gradient vector of the Lagrangian in conjunction with: (1) Constraint Linearization; (2) Traumatic Events; (3) Relaxable Constraints; (4) Implicit Dual Variables; (5) Discrimination; (6) Tracking Starts; and (7) Most Violated Constraint Logic, to obtain a reliable solution in a time frame suitable for real-time economic dispatch. A fast method of calculating the true system incremental cost and the incremental costs of all binding constraints in the presence of a relaxed constraint was also developed. The optimization theory was converted to software on Entergy Corporation's Bulk Power Management System and extensive tests were conducted which verified the expected high performance of the methodology Constrained optimum dispatch solutions were obtained for network security constraints in the form of area import, line mva, line mw and line group mw constraints. Using special techniques under a traumatic event framework, power system regulating margin constraints and generating unit ramp rate constraints were formulated and added to the software developed for honoring network constraints. In addition, the network security, the regulating margin and the ramp rate constraints were combined in 1 and 2-stage constrained dynamic economic dispatch formulations of the problem. Solutions were routinely obtained for overconstrained and parallel constraint cases Using Entergy data, dynamic load tests showed that a static dispatch violates the generating unit ramp rate limits and therefore is not truly an optimum dispatch. Consequently, the unit output power dynamic trajectories are significantly different from the static trajectories. Performance benefits produced by a dynamic constrained dispatch accrue also from the calculation of the incremental costs of holding generating unit ramp rate limits and regulating margins and the implementation of a multi-step relaxable regulating margin constraint. Additional benefits accrue by decomposing the regulating margin into one component which is specified by the system operator and a second component which is estimated automatically A mathematical proof of convergence of the methodology and mathematical proofs of all optimization techniques developed are presented