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Module 3 : Problem Discretization using Approximation Theory

Section 8 : Appendix: Necessary and Sufficient Conditions for Unconstrained Optimality

8.2 Necessary Condition for Optimality
The necessary condition for optimality, which can be used to establish whether a given point is a stationary (maximum or minimum) point, is given by the following theorem.

Theorem 43
If $\phi (\QTR{bf}{x)}$ is continuous and differentiable and has an extreme (or stationary) point (i.e. maximum or minimum ) point at then
		---- (335)
Proof: Suppose is a minimum point and one of the partial derivatives, say the $k^{th}$ one, does not vanish at then by Taylor's theorem
		---- (336)
		---- (337)
Since is of order ( $\Delta x_{i}$ ) $^{2}\,,$ the terms of order $\Delta x_{i}$ will dominate over the higher order terms for sufficiently small Thus, sign of is decided by sign of

Suppose,
		---- (338)
then, choosing $\Delta x_{k}<0$ implies
		---- (339)
and $\phi (\QTR{bf}{x)}$ can be further reduced by reducing $\Delta x_{k}.\,$ This contradicts the assumption that is a minimum point. Similarly, if
		---- (340)
then, choosing $\Delta x_{k}>0$ implies
		---- (341)
and $\phi (\QTR{bf}{x)}$ can be further reduced by increasing $\Delta x_{k}.\,$ This contradicts the assumption that is a minimum point. Thus, will be a minimum of $\phi (\QTR{bf}{x)}$ only if
		---- (342)
Similar arguments can be made if is a maximum of