STAT6545_F18_OCT3. Memorial University of Newfoundland STAT 6545

Numerical Optimization
OCT 3, 2018
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Optimization of a Function
Definition - The function f is said to be a local minimum
value at x = p, if there exist an open interval I containing p
so that f (p) ≤ f (x) for all x 2 I.
Similarly, f is said to have a local maximum value at x = p, if
there exist an open interval I containing p so that
f (x) ≤ f (p) for all x 2 I.
Definition - Increasing or Decreasing
Assume that f (x) is defined on the interval I
If x1 < x2 implies that f (x1) < f (x2) for all x1; x2 2 I, then f is
said to be increasing on I.
If x1 < x2 implies that f (x1) > f (x2) for all x1; x2 2 I, then f is
said to be decreasing on I.
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Optimization of a Function
Theorem - Suppose that f (x) is continuous on I = [a; b] and
is differentiable on (a; b).
(i) If f 0(x) > 0 for all x 2 (a; b) , then f (x) is increasing on I.
(i) If f 0(x) < 0 for all x 2 (a; b) , then f (x) is decreasing on I.
Theorem - Assume that f (x) is defined on I = [a; b] and has a
local extremum at an interior point p 2 (a; b). If f (x) is
differentiable at x = p, then f 0(p) = 0