constraints on a solution has proved to be practical, efficient, and expedient. The choice of the constant C; prescribes how rigid the inequality constraint is to be adhered. A word of caution to the analyst. It is his responsibility to determined that a feasible solution exists, i.e., that contradictory inequality constraints are not imposed. (c) Converging to the Solution Given a reference value for the parameter vectorx (i.e., for RT, RG, and A, - Ak) and a method for computing the cost function which includes the inequality constraint functions h(x), a gradient vector at the reference point is computed. An accurate one-dimensional search is performed to determine a directional minimum point. Using this information and the values of the gradient vectors in two successive iterations, a correction is computed to the approximate inverse of the Hessian matrix. The parameter vector is updated by using (a) the search parameter or minimum points, (b) the approximate inverse of the Hessian matrix, and (c) the gradient vector. A new iteration is then initiated. The solution is terminated when either [1] the magnitude of the gradient vector is less than a given tolerance, [2] the difference in the function in two successive iterations is less than a given tolerance, or [3] the value of the derivative of the penalty function with respect to the one-dimensional parameter at the origin is less than a given tolerance. These conditions imply that the function cannot be further reduced with the specified constants and constraints. For condition 2, it is sometimes possible to further reduce the cost function by restarting the solution with the inverse of the approximate Hessian matrix reset to the identity matrix. ANALYSIS OF RESULTS (a) Effects of Antenna Thermal Limits The antenna thermal limitation is due to waste heat rejection by the d.c.-to-RF power converter tubes, i.e., klystrons. The present SPS guideline is 23 kW/m2 for the maximum of RF power density that can be radiated. However, subsequent investigations on the referenced system thermal radiators indicated the design was conservative and improvements in the amount of heat rejection may be possible. These improvements would be made by using graphite composite materials with a high emissivity coating for the radiators. The reduction is electricity costs as a function of increasing the antenna thermal limit (To) is shown in Fig. 3. Each point on the curve represents a unique taper optimized at its allowable thermal limit. The data indicate large cost advantages by increasing To above 23 kW/m2. The reason is that more power can be delivered since the ionospheric limit has not yet been reached for these particular configurations. The overall cost reduction per satellite for effect of a 1 -mill per kWh reduction in the electricity rate is approximately $260 million. This reduction would be taken from a total satellite cost of approximately $12.4 billion per system. (b) Effects of Sidelobe Power Density Limits The effects upon electricity costs of increasing the maximum allowable sidelobe
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