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<br />tv <br />QO <br />.... <br />.~ <br /> <br />Differential Algorithm, one decision variable (d or ~ ) is <br />selected from among the set which when moved wi1~ resu~t in the <br />most progress toward the minimum. If an individual term from <br />Eq, II is written in discrete element form, the new value of the <br />decision variable (or slack variable) can be determined, <br /> <br />yV _ yO =(*JO (diV - diO) ...........................(19) <br /> <br />or, <br /> <br />v 0 _ (OY )0 <br />y-y-8(j) <br />~t <br /> <br />where the reader is reminded that the superscripts 0 and v refer <br />to the functional evaluations made at the old and new feasible <br />solutions. It may also be worth mentioning that ~p can only <br />be increased, whereas dp can be also decreased (assuming the <br />non-negativity constraints are not violated). As a result, the <br />increase in a slack variable is in reality a loosening of an <br />active constraint. <br /> <br />v <br /><Pt ................................. (20) <br /> <br />The choice of the decision variable or the slack variable <br />to be modified is primarily made on the basis of largest <br />absolute value among the respective constrained derivatives. <br />Three general categories are examined. To begin with, the <br />largest positive valued derivative with which the associated <br />decision variable is greater than zero is determined and the <br />Kuhn-Tucker conditions are checked according to the previous <br />section. Mathematically, this first alternative can be written, <br /> <br />find: max [%a-/ 0 I di> 0, i = 1,2,... ,D] ............. (21) <br /> <br /> <br />where the notation I di > 0 means "subject to the value of di <br />being positive." <br /> <br />The second alternative selection for the step direction <br />is in the negative constrained derivatives. In this case, the <br />specific decision variable will be increased and unless an <br />upper bound on the variable is imposed, no examination of the <br />decision need be made. Symbolically then, <br /> <br /> <br />find: min [~t::. 0, i = l,2,. . ., D] ,...............(22) <br /> <br /> <br />Finally, the largest reduction in the objective function <br />may be facilitated by loosening a particular active constraint. <br />Unless the constrained derivative of y with respect to the <br />slack variable is negative, the Kuhn-Tucker conditions are <br />satisfied. Therefore, this solution can be expressed as: <br /> <br />14 <br />