1. Local and Global Optima
© 2011 Daniel Kirschen and University of Washington

2. Which one is the real maximum?
f(x) A D x
© 2011 Daniel Kirschen and University of Washington

3. Which one is the real optimum?x2 A D x1 B C
© 2011 Daniel Kirschen and University of Washington

4. Local and Global Optima
The optimality conditions are local conditions
They do not compare separate optima
They do not tell us which one is the global optimum
In general, to find the global optimum, we must find and compare all the optima
In large problems, this can be require so much time that it is essentially an impossible task
© 2011 Daniel Kirschen and University of Washington

5. Convexity
If the feasible set is convex and the objective function is convex, there is only one minimum and it is thus the global minimum
© 2011 Daniel Kirschen and University of Washington

6. Examples of Convex Feasible Sets
x1min
x1max
© 2011 Daniel Kirschen and University of Washington

7. Example of Non-Convex Feasible Sets
x1b
x1c
x1d
© 2011 Daniel Kirschen and University of Washington

8. Example of Convex Feasible Sets
A set is convex if, for any two points belonging to the set, all the points on the straight line joining these two points belong to the set x1 x2 x1 x2 x1 x2 x1
x1min
x1max
© 2011 Daniel Kirschen and University of Washington

9. Example of Non-Convex Feasible Sets
x1b
x1c
x1d
© 2011 Daniel Kirschen and University of Washington

10. Example of Convex Function
f(x) x
© 2011 Daniel Kirschen and University of Washington

11. Example of Convex Function
© 2011 Daniel Kirschen and University of Washington

12. Example of Non-Convex Function
f(x) x
© 2011 Daniel Kirschen and University of Washington

13. Example of Non-Convex FunctionD x1 B C
© 2011 Daniel Kirschen and University of Washington

14. Definition of a Convex Function
f(x) z
f(y) xa y xb x
A convex function is a function such that, for any two points xa and xb belonging to the feasible set and any k such that 0 ≤ k ≤1, we have:
© 2011 Daniel Kirschen and University of Washington

15. Example of Non-Convex Function
f(x) x
© 2011 Daniel Kirschen and University of Washington

16. Importance of Convexity
If we can prove that a minimization problem is convex:
Convex feasible set
Convex objective function
Then, the problem has one and only one solution
Proving convexity is often difficult
Power system problems are usually not convex
There may be more than one solution to power system optimization problems
© 2011 Daniel Kirschen and University of Washington