2011 Daniel Kirschen and University of Washington 1 Which one is the real maximum 2011 Daniel Kirschen and University of Washington 2 x fx A D Which one is the real optimum 2011 Daniel Kirschen and University of Washington ID: 280075
Download Presentation The PPT/PDF document "Local and Global Optima" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Local and Global Optima
© 2011 Daniel Kirschen and University of Washington
1Slide2
Which one is the real maximum?
© 2011 Daniel Kirschen and University of Washington
2
x
f(x)
A
DSlide3
Which one is the real optimum?
© 2011 Daniel Kirschen and University of Washington
3
x
1
x
2
B
A
C
DSlide4
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 optimumIn 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 Washington4Slide5
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
5Slide6
Examples of Convex Feasible Sets
© 2011 Daniel Kirschen and University of Washington
6
x
1
x
2
x
1
x
2
x
1
x
1
x
2
x
1
min
x
1
maxSlide7
Example of Non-Convex Feasible Sets
© 2011 Daniel Kirschen and University of Washington
7
x
1
x
2
x
1
x
2
x
1
x
2
x
1
x
1
a
x
1
d
x
1
b
x
1
c
x
1Slide8
Example of Convex Feasible Sets
x
1
x
2
x
1
x
2
x
1
x
2
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
x
1
x
1
min
x
1
max
© 2011 Daniel Kirschen and University of Washington
8Slide9
Example of Non-Convex Feasible Sets
x
1
x
2
x
1
x
2
x
1
x
2
x
1
x
1
a
x
1
d
x
1
b
x
1
c
x
1
© 2011 Daniel Kirschen and University of Washington
9Slide10
Example of Convex Function
x
f(x)
© 2011 Daniel Kirschen and University of Washington
10Slide11
Example of Convex Function
x
1
x
2
© 2011 Daniel Kirschen and University of Washington
11Slide12
Example of Non-Convex Function
x
f(x)
© 2011 Daniel Kirschen and University of Washington
12Slide13
Example of Non-Convex Function
x
1
x
2
B
A
C
D
© 2011 Daniel Kirschen and University of Washington
13Slide14
Definition of a Convex Function
x
f(x)
x
a
x
b
y
f(y)
z
A convex function is a function such that, for any two points x
a
and x
b
belonging to the feasible set and any
k
such that 0 ≤ k ≤1, we have:
© 2011 Daniel Kirschen and University of Washington
14Slide15
Example of Non-Convex Function
x
f(x)
© 2011 Daniel Kirschen and University of Washington
15Slide16
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 solutionProving convexity is often difficultPower system problems are usually not convex There may be more than one solution to power system optimization problems
© 2011 Daniel Kirschen and University of Washington16