By Dana Bell
Air strength shades Vol 1: 1926-42
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Extra resources for Air Force Colors 1926-1942
If they are not, the optimal point is not a minimal point. Another approach to the conditions for definiteness is to examine the eigenvalues of the second derivative matrix that are the values of ). satisfying 2. Unconstrained Minimization. ,2-). I~ 0. 73) the eigenvalues are real. 66), the second differential necessary conditions for a minimum are equivalent to the eigenvalues being nonnegative. Finally, the sufficient conditions are equivalent to the eigenvalues being positive. 3 Summary In summary, the necessary conditions to be satisifed by an interior minimal point of the performance index J = ¢( x 11 x 2 ) are that the first differential must vanish (dJ = 0) and that the second differential must be nonnegative (d2 J ~ 0).
In this way all of the derivative formulas hold, but dx is not zero. 13) where x is assumed to be the independent variable and y is the dependent variable. 14) where dx is the independent differential. Since differentials of independent differentials are zero, the differential of the first differential becomes · d2 f = fxxdx 2 + 2jxydxdy + jyydy2 + jyd2 y = 0. 15) · Higher-order differentials are obtained in the same way. Note that all the derivatives are evaluated at x, y. 14) can be solved for dy as dy = - ~: dx.
In the light of the above paragraph, interior minimal points can be imbedded in a neighborhood whereas boundary minimal points must be on either side of the neighborhood. Because of this difference, interior minimal points are discussed here, and boundary minimal points are discussed in the chapter on inequality constraints (Chapter 4). 23) takes on an interior minimal value. In this section, conditions on rjJ that must be satisfied at a minimal point are derived. 25) 0 < ix* - xi < c.. This equation can be rewritten as (see Fig.