Local and Global Optima
© 2011 Daniel Kirschen and University of Washington
1
Which one is the real maximum?
f(x)
A
D
x
df
d2 f
For x = A and x = D, we have:
= 0 and
<0
2
dx
dx
© 2011 Daniel Kirschen and University of Washington
2
Which one is the real optimum?
x2
x1
A, B,C and D are all minima because we have:
¶f
¶f
¶2 f
¶2 f
¶2 f
= 0; = 0 and
< 0; 2 < 0; <0
2
¶x1
¶x2
¶x1
¶x2
¶x1 ¶x2
© 2011 Daniel Kirschen and University of Washington
3
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
4
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
5
Examples of Convex Feasible Sets
x2
x2
x1
x1
x2
x1min
x1max
© 2011 Daniel Kirschen and University of Washington
x1
x1
6
Example of Non-Convex Feasible Sets
x2
x2
x1
x1
x1a x1b x1c x1d
© 2011 Daniel Kirschen and University of Washington
x1
x1
x2
x1
7
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
x2
x2
x1
x1
x2
x1
min
x1
max
© 2011 Daniel Kirschen and University of Washington
x1
x1
8
Example of Non-Convex Feasible Sets
x2
x2
x1
x1
x1
a
x1
b
c
x1 x1
d
© 2011 Daniel Kirschen and University of Washington
x1
x1
x2
x1
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Example of Convex Function
f(x)
x
© 2011 Daniel Kirschen and University of Washington
10
Example of Convex Function
x2
x1
© 2011 Daniel Kirschen and University of Washington
11
Example of Non-Convex Function
f(x)
x
© 2011 Daniel Kirschen and University of Washington
12
Example of Non-Convex Function
x2
x1
© 2011 Daniel Kirschen and University of Washington
13
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:
z = kf ( x a ) + (1- k ) f ( x b ) ³ f ( y ) = f [ kx a + ( 1- k ) x b
© 2011 Daniel Kirschen and University of Washington
]
14
Example of Non-Convex Function
f(x)
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
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