Nonlinear Programming Supplementary Notes
Graphics of parent functions
This is just to recall some basic graphs, used to quick test some points.
y=ax+b

y=ax^2+bx+c







Introduction to NLP
Example 1. Profit Maximization


Example 2. Product Maximization




Example 3.


Example 4.


Convex and Concave Functions


Example 5. Check cvx or ccv of

Example 6. check cvx

E7. check cvx ,

E8.
Easy way: sum of convex is convex, so f(x) is convex.

Therorem 1
For maximization problem, if function is concave, local max = global max = optimal solution
For minimization problem, if function is convex, local min = global min = optimal solution
E9. check convexity of

E10. Hessian of of

Definition 1 [Hessian, PM, LPM]



Theorem 2 [convex by PM, concave by PM]


E11. CVX or CCV

E12. CVX or CCV

E13. CVX or CCV

E14. CVX or CCV


E15. CVX/CCV

E16.

E17.

E18.

E19.

E20.

E21.

E22. CVX / CCV

E23. CVX / CCV

Solving NLPs with One Variable


Theorem 3


What happens if ?



Both side neibours must lower or higher

E.24 Profit Maximization by Monopolist



E25. Product Pricing




E26. Monopolist Pricing



E27. find the opt to

E28. Find the optimal solution


Unconstrained NLPs with multiple variables
Theorem 4

This means, if we want to find a local maximum or local minimum, we must first find all points where the
Definition 2

All the points we find by doing are stationary points.
In the single variable case, if we do not has any constraints, we also did this:
- first we calculated then we find several points, let’s say x_1, x_2.
- then we calculated if is local minimum. (if , local max)
- if then it is a saddle point.
Theorem 5



