Local and Global Optima © 2011 Daniel Kirschen and University of Washington 1.
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Transcript of Local and Global Optima © 2011 Daniel Kirschen and University of Washington 1.
Local and Global Optima
© 2011 Daniel Kirschen and University of Washington
1
2
Which one is the real maximum?
© 2011 Daniel Kirschen and University of Washington
x
f(x)
A D
3
Which one is the real optimum?
© 2011 Daniel Kirschen and University of Washington
x1
x2
B
A
C
D
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
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Examples of Convex Feasible Sets
© 2011 Daniel Kirschen and University of Washington
x1
x2
x1
x2
x1x1
x2
x1min x1
max
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Example of Non-Convex Feasible Sets
© 2011 Daniel Kirschen and University of Washington
x1
x2
x1
x2
x1
x2
x1x1a x1
dx1b x1
c
x1
8
Example of Convex Feasible Sets
x1
x2
x1
x2
x1
x2
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
x1x1min x1
max
© 2011 Daniel Kirschen and University of Washington
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Example of Non-Convex Feasible Sets
x1
x2
x1
x2
x1
x2
x1x1a x1
dx1b x1
c
x1
© 2011 Daniel Kirschen and University of Washington
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Example of Convex Function
x
f(x)
© 2011 Daniel Kirschen and University of Washington
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Example of Convex Function
x1
x2
© 2011 Daniel Kirschen and University of Washington
12
Example of Non-Convex Function
x
f(x)
© 2011 Daniel Kirschen and University of Washington
13
Example of Non-Convex Function
x1
x2
B
A
C
D
© 2011 Daniel Kirschen and University of Washington
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Definition of a Convex Function
x
f(x)
xa xby
f(y)
z
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
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Example of Non-Convex Function
x
f(x)
© 2011 Daniel Kirschen and University of Washington
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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
