Introduction to the Theory of Optimization in Euclidean Space by Challal Samia;

Author:Challal, Samia;
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2019-10-22T00:00:00+00:00

So, by applying the method of Lagrange Multipliers, we introduce the Lagrangian

and look for the interior extreme points as solutions of the system

So, the point (1, 1, 1) is a stationary point for the Lagrangian . But, it is not an extreme point for . Indeed, the second test derivative gives

The leading principal minors of at (1, 1, 1) are

Hence, (1, 1, 1) is a saddle point.

It remains to show that the point (1, 1) is the maximum point for the problem; see Figure 3.14 for a graphical solution using level curves. Indeed, since it is the only interior point to the segment, it suffices to compare the value of f at (1, 1) with its value at the end points of the segment. We have

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