5.24The Lagrangean for this problem is?(x,±)=²21+²22+²23−±(2²1−3²2+5²3−19)A necessary condition forx∗to solve the problem is that the Lagrangean be stationaryatx∗,thatis³?1?=2²∗1−2±=0³?2?²∗2+3±³?3?²∗3−5±which implies²∗1=±²∗2=−32∗2=52±(5.89)It is also necessary that the solution satisfy the constraint, that is2²∗1−3²∗2²∗3=19Substituting (5.89) into the constraint we get2±+92±+252±±which implies±= 1. Substituting in (5.89), the solution isx∗=(1,−32,52). Since theconstraint is aﬃne and the objective (−´) is concave, stationarity of the Lagrangeanis also suﬃcient for global optimum (Corollary 5.2.4).5.25The Lagrangean is?(²1,²2²±1²1−±2−±(µ1²1+µ2²2−¶)The Lagrangean is stationary where³?1?=·²±−11²1−±2−
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