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<br />N <br />00 <br />..... <br />...... <br /> <br />1. Necessary conditions prerequisite for a minimum must <br />consist of the following: <br /> <br />~ > 0, d, < 0, and ~ d, = O,j = 1,2,...,D .. (14) <br />lid. J - lid, J <br />J J <br />and <br />* > 0, CPt ?. 0, and liy CPt = 0, t = l,2,...,T ... (l5) <br />- 84i"t <br /> <br />2, If Eqs. 14 and 15 are satisfied, then sufficient <br />conditions for a minimum are: <br /> <br />~,> 0 j = 1,2,...,D ..............................(l6) <br />J <br /> <br />and <br /> <br />~d > 0 t = 1,2,...,T ..............................(17) <br />uCPt <br /> <br />The minimum has been reached when both the necessary and <br />sufficient conditions have been satisfied. However, if for <br />example, liy/lidj equals zero and dj > 0, the tests are incon- <br />clusive since the sufficient conditTons have not been met. In <br />this case, it is necessary to take the second derivatives of <br />the objective function with respect to the x-vector. This <br />analysis yields a square matrix of second order partial deriva- <br />tives called the Hessian matrix written mathematically as: <br /> <br />H = 1]2 Y ........................................................................................... (18) <br />-x <br /> <br />In order for the stationary point to be a minimum (local or <br />global) the value of the Hessian matrix must be positive- <br />definite, and since the properties of positive-definite matrices <br />can be found in most texts on linear algebra, no further <br />description will be given here. <br /> <br />Evaluation of Optimal Direction <br /> <br />In addition to the description of the fundamental elim- <br />ination technique of this optimizing technique, the preceding <br />sections also provided the definition of the constrained de- <br />rivatives of the objective function in terms of the decision <br />and slack variables. Furthermore, criteria were given with <br />which these parameters can also be evaluated to see when the <br />minimum is achieved. In this section, these same derivatives <br />will be used to determine the direction a particular decision <br />variable, d or CPp' must be "moved" in order to create the <br />maximum redgction in the value of the objective function during <br />each iterative step. Among the nonlinear programming tech- <br />niques for optimization, several essentially alter all of the <br />decision variables at each iteration. In the Jacobian <br /> <br />13 <br /> <br />, ~,', <br /> <br /> <br />,._,,"-;,- <br /> <br />