Suppose that you are given an optimization problem as: Maximize F(x), where F(x)

Question

Suppose that you are given an optimization problem as: Maximize F(x), where F(x) is a function of the decision variable x. That is, there are no constraints and there is a single decision variable. Briefly explain your answers to the following questions. (F(x) can be linear or nonlinear) a) Can a local maximum be a global minimum? b) Can a local minimum be greater than a local maximum? c) Is a global maximum always greater than every other local maximum? d) Can a global minimum be greater than a global maximum? e) Is a local minimum always less than a global maximum?

Explanation / Answer

a.

There are no critical points defined in the above diagram. here the local minimum is also a global minimum.

b. Yes, a local minimum can be greater than a local mimum

Reason: Although there is only one global maximum and minum , there can be more than one local maximum and minimum in a function.

c. Yes, A global maximum is always greater than every local maximum

Reason: Although there is only one global maximum and minum , there can be more than one local maximum and minimum in a function.

d. No

Reason: Although there is only one global maximum and minum , there can be more than one local maximum and minimum in a function.

e Yes

Reason: Although there is only one global maximum and minum , there can be more than one local maximum and minimum in a function.

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