Optimization Techniques for Design Problems
OPTIMIZATION
TECHNIQUES
Objective Function,
Maxima,
Minima and Saddle Points, Convexity
and Concavity
Objectives
Study the basic components of an
optimization problem
.
Formulation of design problems as mathematical programming
problems.
Define
stationary points
Necessary and sufficient conditions for the relative maximum of a
function of a single variable and for a function of two variables.
Define the global optimum in comparison to the relative or local
optimum
Determine
the convexity or concavity of functions
3
Introduction - Preliminaries
Basic components of an optimization problem :
An
objective function
expresses the main aim of the model
which is
either to be minimized or maximized.
Set of
unknowns
or
variables
which control the value of the objective
function.
Set of
constraints
that allow the unknowns to take on certain values
but exclude others.
Optimization problem is then to:
4
Find values of the
variables
that minimize or maximize the
objective
function
while satisfying the
constraints
.
Objective Function
•
Many optimization problems have a single objective function
•
The two interesting exceptions are:
•
No objective function:
User does not particularly want to optimize anything
so there is no reason to define an objective function. Usually called a
feasibility problem.
•
Multiple objective functions
.
In practice, problems with multiple objectives
are reformulated as single-objective problems by either forming a weighted
combination of the different objectives or by treating some of the objectives
by constraints.
5
Statement of an optimization problem
To find
X =
which
maximizes
f
(
X
)
Subject to the constraints
g
i
(
X
) <= 0 , i = 1, 2,
…
.,m
l
j
(
X
) = 0 , j = 1, 2,
…
.,p
6
Statement of an optimization problem
…
where
X
is an
n
-dimensional vector called the design vector
f
(
X
) is called the
objective function
g
i
(
X
) and
l
j
(
X
) are inequality and equality constraints, respectively.
This type of problem is called a
constrained optimization problem
.
Optimization problems can be defined without any constraints as well
U
nconstrained optimization problems.
7
Objective Function Surface
•
If the locus of all points satisfying
f
(
X
) = a constant
c
is considered, it can
form a family of surfaces in the design space called the
objective function
surfaces
.
•
When drawn with the constraint surfaces as shown in the figure we can
identify the optimum point (maxima).
•
Possible graphically only when the number of design variable is two.
•
When we have three or more design variables because of complexity in the
objective function surface we have to solve the problem as a mathematical
problem and this visualization is not possible.
8
Objective function surfaces to find the
optimum point (maxima)
9
Variables and Constraints
Variables
These are essential. If there are no variables, we cannot define the objective
function and the problem constraints.
Constraints
Even though constraints are not essential, it has been argued that almost all
problems really do have constraints.
In many practical problems, one cannot choose the design variable
arbitrarily.
Design constraints
are restrictions that must be satisfied to
produce an acceptable design.
10
Constraints (contd.)
•
Constraints can be broadly classified as :
•
Behavioral or Functional constraints
:
Represent limitations on the
behavior and performance of the system.
•
Geometric or Side constraints
: Represent physical limitations on
design variables such as availability, fabricability, and transportability.
11
Constraint Surfaces
Consider the optimization problem presented earlier with only inequality
constraints
g
i
(
X
) . Set of values of
X
that satisfy the equation
g
i
(
X
) forms a
boundary surface in the design space called a
constraint surface.
Constraint surface divides the design space into two regions: one with
g
i
(
X
) < 0
(feasible region) and the other in which
g
i
(
X
) > 0 (infeasible region). The points
lying on the hyper surface will satisfy
g
i
(
X
) =0.
12
The figure shows a hypothetical two-dimensional design
The figure shows a hypothetical two-dimensional design
space where the feasible region is denoted by hatched
space where the feasible region is denoted by hatched
lines
lines
13
Stationary Points: Functions of
Single and Two Variables
14
Stationary points
For a continuous and differentiable function
f
(
x
) a
stationary
point
x
*
is a point at which the function vanishes,
i.e.
f
’
(
x
) = 0 at
x = x
*
. x
*
belongs to its domain of definition.
Stationary point may be a minimum, maximum or an
inflection (saddle) point
15
Stationary points…
(a) Inflection point (b) Minimum (c) Maximum
Relative and Global Optimum
•
A function is said to have a
relative
or
local
minimum at
x = x
*
if
for all sufficiently small positive and negative values of
h,
i.e. in the near vicinity of
the point
x
.
•
A point
x
*
is called a
relative
or
local
maximum if
for all values of
h
sufficiently close to zero.
•
A function is said to have a
global
or
absolute
minimum at
x = x
*
if
for all
x
in the
domain over which
f
(
x
) is defined.
•
A function is said to have a
global
or
absolute
maximum at
x = x
*
if
for all
x
in the
domain over which
f
(
x
) is defined.
17
Relative and Global Optimum…
18
Functions of a single variable
Consider the function
f
(
x
) defined for
To find the value of
x
*
such that
x
*
maximizes
f
(
x
) we need to
solve a
single-variable optimization
problem.
19
Functions of a single variable …
Necessary condition :
For a single variable function
f
(
x
) defined for
x
which has a relative maximum at
x = x
*
,
x
*
if the
derivative
f
‘
(
x
) =
df
(
x
)
/dx
exists
as a finite number at
x = x
*
then
f
‘
(
x*
) = 0.
Above theorem holds good for relative minimum as well.
Theorem only considers a domain where the function is continuous and
derivative.
It does not indicate the outcome if a maxima or minima exists at a point
where the derivative fails to exist. This scenario is shown in the figure
below, where the slopes m
1
and m
2
at the point of a maxima are unequal,
hence cannot be found as depicted by the theorem.
20
Functions of a single variable ….
21
Salient features:
Theorem does not consider if the
maxima or minima occurs at the
end point of the interval of
definition.
Theorem does not say that the
function will have a maximum or
minimum at every point where
f
’
(
x
) = 0, since this condition
f
’
(
x
) = 0 is for stationary points
which include inflection points
which do not mean a maxima or
a minima.
Sufficient condition
•
For the same function stated above let
f
’
(
x
*
) =
f
”
(
x
*
) = . . . =
f
(n-1)
(
x
*
) =
0, but
f
(n)
(
x
*
) ≠ 0, then it can be said that
f
(
x
*
) is
•
(a) a minimum value of
f
(
x
) if
f
(n)
(
x
*
) > 0 and
n
is even
•
(b) a maximum value of
f
(
x
) if
f
(n)
(
x
*
) < 0 and
n
is even
•
(c) neither a maximum or a minimum if
n
is odd
22
Example 1
Find the optimum value of the function
and also state if the function attains a maximum or a
minimum
.
Solution
For maxima or minima,
or
x
*
= -3/2
is positive .
Hence the point
x
*
= -3/2 is a point of minima and the function attains a
minimum value of -29/4 at this point.
23
Example 2
24
Find the optimum value of the function
and also state if
the function attains a maximum or a minimum
Solution:
or
x
=
x
*
= 2 for maxima or minima.
at
x
*
=
2
at
x
*
=
2
at
x
*
=
2
Hence
f
n
(
x
) is positive and
n
is even hence the point
x = x*
= 2 is a point of
minimum and the function attains a minimum value of 0 at this point.
More examples can be found in the lecture notes
Functions of two variables
Concept discussed for one variable functions may be easily extended to functions of
multiple variables.
Functions of two variables are best illustrated by contour maps, analogous to
geographical maps
.
25
A contour is a line
representing a constant value
of
f
(
x
) as shown in the
following figure. From this
we can identify
maxima,
minima
and
points of
inflection.
A contour plot
Necessary conditions
From the above contour map, perturbations from points of local minima in any
direction result in an increase in the response function
f
(
x
), i.e.
the slope of the
function is zero at this point of local minima.
Similarly, at
maxima
and
points of inflection
as the slope is zero, the first derivative
of the function with respect to the variables are zero.
Which gives us
at the stationary points. i.e. the
gradient vector of
f
(
X
), at
X = X
*
=
[x
1
, x
2
]
defined as follows, must equal
zero:
26
Sufficient conditions
Consider the following second order derivatives:
Hessian matrix defined by
H
is formed using the above second order
derivatives.
27
Sufficient conditions …
The value of determinant of the
H
is calculated and
if
H
is positive definite then the point
X
= [x
1
, x
2
]
is a point of local
minima.
if
H
is negative definite then the point
X
= [x
1
, x
2
]
is a point of local
maxima.
if
H
is neither then the point
X =
[x
1
, x
2
]
is neither a point of maxima nor
minima.
28
Example
Locate the stationary points of f(X) and classify them as
relative maxima, relative minima or neither based on the
rules discussed in the lecture.
Solution
Example …
30
From
,
So the two stationary points are
X
1
= [-1,-3/2] and X
2
= [3/2,-1/4]
Example …
31
The Hessian of
f
(
X
) is
At
X
1
= [-1,-3/2]
,
Since one eigen value is positive and
one negative,
X
1
is neither a relative
maximum nor a relative minimum
Example …
Example …
At X
2
= [3/2,-1/4]
Since both the eigen values are positive,
X
2
is a local minimum.
Minimum value of
f(x)
is -0.375
More examples can be found in the lecture notes
Convex and Concave Functions
33
Convex Function (Function of one variable)
A real-valued function
f
defined on an interval (or on any convex subset
C
of some vector space) is called
convex
, if for any two points
x
and
y
in its
domain
C
and any
t
in [0,1], we have
A function is convex if and only if its epigraph (the set of points lying on
or above the graph) is a convex set. A function is also said to be
strictly
convex
if
for any
t
in (0,1) and a line connecting any two points on the function lies
completely above the function.
34
A convex function
35
Testing for convexity of a single variable
function
A function is convex if its slope is non decreasing or
0
It is strictly convex if its slope is continually increasing
or
0 throughout the function.
36
Properties of convex functions
•
A convex function
f
, defined on some convex open interval
C
, is
continuous on
C
and differentiable at all or at many points. If
C
is closed,
then
f
may fail to be continuous at the end points of
C
.
•
A continuous function on an interval
C
is convex if and only if
for all a and b in
C
.
•
A differentiable function of one variable is convex on an interval if and
only if its derivative is monotonically non-decreasing on that interval.
37
Properties of convex functions …
•
A continuously differentiable function of one variable is convex on an interval if and
only if the function lies above all of its tangents: for all
a
and
b
in the interval.
•
A twice differentiable function of one variable is convex on an interval if and only if
its second derivative is non-negative in that interval; this gives a practical test for
convexity.
•
A continuous, twice differentiable function of several variables is convex on a
convex set if and only if its Hessian matrix is positive semi definite on the interior of
the convex set.
•
If two functions
f
and
g
are convex, then so is any weighted combination
af
+
b
g
with non-negative coefficients
a
and
b
. Likewise, if
f
and
g
are convex, then the
function max{f
,
g
} is convex.
38
Concave function (Function of one
variable)
•
A differentiable function
f
is
concave
on an interval if its derivative
function
f
′ is decreasing on that interval: a concave function has a
decreasing slope.
•
A function
f
(
x
) is said to be
concave
on an interval if, for all
a
and
b
in
that interval,
•
Additionally,
f
(
x
) is
strictly concave
if
39
A concave function
40
Testing for concavity of a single variable
function
•
A function is convex if its slope is non increasing or
0.
•
It is strictly concave if its slope is continually
decreasing or
0 throughout the function.
41
Properties of concave functions
•
A continuous function on
C
is concave if and only if
for any
a
and
b
in
C
.
•
Equivalently,
f
(
x
) is concave on [
a
,
b
] if and only if the function −
f
(
x
) is convex on
every subinterval of [
a
,
b
].
•
If
f
(
x
) is twice-differentiable, then
f
(
x
) is concave if and only if
f
′′(
x
) is non-positive. If its second derivative is negative then it is strictly concave,
but the opposite is not true, as shown by
f
(
x
) = -
x
4
.
42
Example
Locate the stationary points of
and find out if
the function is convex, concave or neither at the points of optima based on the
testing rules discussed above.
Solution
:
Consider the point
x
=
x*
= 0
at x
*
= 0
at x
*
= 0
43
Example …
44
Since the third derivative is non-zero,
x = x* = 0 is neither a point of maximum
or minimum , but it is a point of inflection. Hence the function is neither convex
nor concave at this point.
Consider the point
x
=
x*
= 1
at x
*
= 1
Since the second derivative is negative, the point x
=
x*
= 1 is a point of local
maxima with a maximum value of
f(x)
= 12 – 45 + 40 +5 = 12. At the point the
function is concave since
Example …
45
Consider the point
x
=
x*
= 2
at x
*
= 2
Since the second derivative is positive, the point x
=
x*
= 2 is a point of local
minima with a minimum value of
f(x)
= -11. At the point the function is
concave since
Functions of two variables
•
A function of two variables,
f
(
X
) where X is a vector = [x
1
,x
2
], is strictly
convex if
•
where
X
1
and
X
2
are points located by the coordinates given in their
respective vectors. Similarly a two variable function is strictly concave
if
46
Contour plot of a convex function
47
Contour plot of a concave function
48
Sufficient conditions
•
To determine convexity or concavity of a function of multiple variables,
the eigen values of its Hessian matrix is examined and the following
rules apply.
•
If all eigen values of the Hessian are positive the function is strictly
convex.
•
If all eigen values of the Hessian are negative the function is strictly
concave.
•
If some eigen values are positive and some are negative, or if some
are zero, the function is neither strictly concave nor strictly convex.
49
Example
Locate the stationary points of
and find out if the function is convex, concave or neither at the points of optima
based on the rules discussed in this lecture.
Solution:
Solving the above the two stationary points are
X
1
= [-1,-3/2] and X
2
= [3/2,-1/4]
Example
…
Example
…
•
Since one eigen value is positive and one negative, X
1
is neither a relative maximum
nor a relative minimum. Hence at X
1
the function is neither convex nor concave.
•
At X
2
= [3/2,-1/4]
•
Since both the eigen values are positive, X
2
is a local minimum
•
Function is convex at this point as both the eigen values are positive.
Lagrange Multipliers and
Kuhn-tucker Conditions
Objectives
54
To study the optimization with multiple decision variables
and equality constraint : Lagrange Multipliers.
To study the optimization with multiple decision variables
and inequality constraint : Kuhn-Tucker (KT) conditions
Constrained optimization with equality
constraints
55
A function of multiple variables,
f
(
x
), is to be optimized subject to one or
more equality constraints of many variables. These equality constraints,
g
j
(
x
),
may or may not be linear. The problem statement is as follows:
Maximize (or minimize)
f
(
X
), subject to
g
j
(
X
) = 0, j = 1, 2, … ,
m
where
(1)
Constrained optimization…
With the condition that ; or else if
m > n
then the problem
becomes an over defined one and there will be no solution. Of the
many available methods, the method of constrained variation and the
method of using Lagrange multipliers are discussed.
Solution by method of Lagrange
multipliers
57
•
Continuing with the same specific case of the optimization problem with
n
= 2 and
m
= 1 we define a
quantity
λ
, called the
Lagrange multiplier
as
(2)
•
Using this in the constrained variation of
f
[ given in the previous lecture]
And (2) written as
(3)
(4)
Solution by method of Lagrange
multipliers…
58
•
Also, the constraint equation has to be satisfied at the extreme point
(5)
•
Hence equations (2) to (5) represent the necessary conditions for the point
[
x
1
*, x
2
*
] to be an extreme point.
•
λ
could be expressed in terms of as well has to be non-
zero.
•
These necessary conditions require that at least one of the partial
derivatives of
g
(
x
1
, x
2
) be non-zero at an extreme point.
59
•
The conditions given by equations (2) to (5) can also be generated by
constructing a functions
L,
known as the Lagrangian function, as
(6)
•
Alternatively, treating
L
as a function of x
1
,x
2
and
, the necessary
conditions for its extremum are given by
(7)
Solution by method of Lagrange
multipliers…
Necessary conditions for a general problem
60
•
For a general problem with
n
variables and
m
equality constraints the
problem is defined as shown earlier
•
Maximize (or minimize)
f
(
X
), subject to
g
j
(
X
) = 0,
j
= 1, 2,
…
,
m
where
•
In this case the Lagrange function,
L
, will have one Lagrange multiplier
j
for each constraint as
(8)
61
•
L
is now a function of
n + m
unknowns,
, and the
necessary conditions for the problem defined above are given by
(9)
•
which represent
n + m
equations in terms of the
n + m
unknowns,
x
i
and
j
. The
solution to this set of equations gives us
and (10)
Necessary conditions for a general problem…
Sufficient conditions for a general problem
62
•
A sufficient condition for
f
(
X
) to have a relative minimum at
X
*
is that each
root of the polynomial in
Є
, defined by the following determinant equation
be positive.
(11)
63
where
(12)
Similarly, a sufficient condition for
f
(
X
) to have a relative maximum at
X
*
is that each root of the polynomial in
Є
, defined by equation (11)
be
negative.
•
If equation (11), on solving yields roots, some of which are positive and
others negative, then the point
X
*
is neither a maximum nor a minimum.
Sufficient conditions for a general problem…
Example
64
Minimize
,
Subject to
Solution
with
n
= 2 and
m
= 1
L =
or
Example…
65
Hence,
Example…
66
The determinant becomes
or
Since is negative,
X
*, correspond to a maximum
Kuhn – Tucker Conditions
67
KT condition: Both necessary and sufficient if the objective function is
concave and each constraint is linear or each constraint function is
concave, i.e., the problems belongs to a class called the convex
programming problems.
Kuhn-Tucker Conditions: Optimization
Model
Consider the following optimization problem
Minimize
f(X)
subject to
g
j
(X)
≤
0
for j=1,2,…,p
where the decision variable vector
X=[x
1
,x
2
,…,x
n
]
68
Kuhn-Tucker Conditions
Kuhn-Tucker conditions for
X
*
= [
x
1
*
, x
2
*
, . . . x
n
*
]
to be a local minimum are
69
Kuhn Tucker Conditions …
In case of minimization problems, if the constraints are of
the form
g
j
(X) ≥ 0, then
λ
j
have to be non-positive
If the problem is one of maximization with the constraints
in the form
g
j
(X) ≥ 0, then
λ
j
have to be nonnegative.
70
Example 1
71
Minimize
subject to
72
Kuhn – Tucker Conditions
Example 1…
From (17)
either = 0 or ,
Case 1:
= 0
From (14), (15) and (16) we have
x
1
= x
2
= and
x
3
=
Using these in (18) we get
From (22)
,
, therefore, =0,
Therefore,
X
*
= [ 0, 0, 0 ]
This solution set satisfies all of (18) to (21)
Example 1…
73
Case 2:
Using (14), (15) and (16)
,
we have
or
But conditions (21)
and (22) give us and
simultaneously, which cannot be possible with
Hence the solution set for this optimization problem is
X
*
= [ 0 0 0 ]
Example 1…
74
Minimize
subject to
Example 2
75
Kuhn – Tucker Conditions
Example 2…
76
From (25)
either = 0 or ,
Case 1
From (23) and (24) we have and
Using these in (26) we get
Considering ,
X
*
= [ 30, 0]. But this solution set violates (27)
and (28)
For ,
X
*
= [ 45, 75]. But this solution set violates (27)
Example 2…
77
Case 2:
Using in (23) and (24), we have
Substitute (31) in (26), we have
For this to be true, either
Example 2…
78
For ,
This solution set violates
(27)
and
(28)
For ,
This solution set is satisfying all equations from (27) to (31) and hence
the desired
Thus, the solution set for this optimization problem is
X
*
= [ 80, 40 ]
Example 2…
79
BIBLIOGRAPHY / FURTHER READING
1.
Rao S.S.,
Engineering Optimization – Theory and Practice
, Fourth
Edition, John Wiley and Sons, 2009.
2.
Ravindran A., D.T. Phillips and J.J. Solberg,
Operations Research –
Principles and Practice
, John Wiley & Sons, New York, 2001.
3.
Taha H.A.,
Operations Research – An Introduction
, 8
th
edition, Pearson
Education India, 2008.
4.
Vedula S., and P.P. Mujumdar,
Water Resources Systems: Modelling
Techniques and Analysis
, Tata McGraw Hill, New Delhi, 2005.
80
(ii) Constrained and Unconstrained
Optimization
Objectives
82
To study the optimization of functions of multiple variables without
optimization.
To study the above with the aid of the gradient vector and the Hessian matrix.
To discuss the implementation of the technique through examples
To study the optimization of functions of multiple variables subjected to equality
constraints using
the method of constrained variation
Unconstrained optimization
83
If a convex function is to be minimized, the stationary point is
the global minimum and analysis is relatively straightforward
A similar situation exists for maximizing a concave variable
function.
Necessary and sufficient conditions for the optimization of
unconstrained function of several variables
Necessary condition
84
•
In case of multivariable functions a necessary condition for a stationary
point of the function
f
(
X
) is that each partial derivative is equal to zero.
•
Each element of the gradient vector defined below must be equal to
zero. i.e. the gradient vector of
f
(
X
), at
X=X
*
, defined as follows,
must be equal to zero:
Sufficient condition
85
For a stationary point
X
* to be an extreme point, the matrix of second
partial derivatives (Hessian matrix) of
f
(
X
) evaluated at
X
* must be:
positive definite when
X
* is a point of relative minimum, and
negative definite when
X
* is a relative maximum point.
When all eigen values are negative for all possible values of
X
, then
X
* is
a global maximum, and when all eigen values are positive for all possible
values of
X
, then
X
* is a global minimum.
If some of the eigen values of the Hessian at
X
* are positive and some
negative, or if some are zero, the stationary point,
X
*, is neither a local
maximum nor a local minimum.
Example
86
Analyze the function
and classify the stationary points as maxima, minima and points of inflection
Solution
Example …
87
Solving these simultaneous equations we get
X
*
=[1/2, 1/2, -2]
88
Hessian of
f
(
X
) is
Example …
89
or
Since all eigen values are negative the function attains a maximum at the point
X
*
=[1/2, 1/2, -2]
or
Example …
Constrained optimization
90
A function of multiple variables,
f
(
x
), is to be optimized subject to one or
more equality constraints of many variables. These equality constraints,
g
j
(
x
), may or may not be linear. The problem statement is as follows:
Maximize (or minimize)
f
(
X
), subject to
g
j
(
X
) = 0, j = 1, 2,
…
,
m
where
91
With the condition that ; or else if
m > n
then the problem becomes
an over defined one and there will be no solution. Of the many available
methods, the method of constrained variation is discussed
Another method using Lagrange multipliers will be discussed in the next
lecture.
Constrained optimization…
Solution by Method of Constrained
Variation
92
•
For the optimization problem defined above, consider a specific case with
n
= 2 and
m
= 1
•
The problem statement is as follows:
Minimize
f
(
x
1
,x
2
), subject to
g
(
x
1
,x
2
) = 0
•
For
f
(
x
1
,x
2
) to have a minimum at a point X
*
= [
x
1
*
,x
2
*
], a necessary
condition is that the total derivative of
f
(
x
1
,x
2
) must be zero at [
x
1
*
,x
2
*
].
(1)
Method of Constrained Variation…
93
•
Since
g
(
x
1
*
,x
2
*
) = 0 at the minimum point, variations
dx
1
and
dx
2
about the
point [
x
1
*
, x
2
*
] must be
admissible variations
, i.e. the point lies on the
constraint:
g
(
x
1
*
+ dx
1
, x
2
*
+ dx
2
) = 0
(2)
assuming
dx
1
and
dx
2
are small the Taylor’s series expansion of this gives us
(3)
94
or
at [
x
1
*
,x
2
*
]
(4)
which is the condition that must be satisfied for all
admissible
variations
.
Assuming , (4)
can be rewritten as
(5)
Method of Constrained Variation…
95
(5)
indicates that once variation along
x
1
(
dx
1
) is chosen arbitrarily, the variation along
x
2
(
dx
2
) is decided automatically to satisfy the condition for the
admissible variation.
Substituting equation (5)
in
(1) we have:
(6)
The equation on the left hand side is called the constrained variation of
f
. Equation (5)
has
to be satisfied for all
dx
1
, hence we have
(7)
This gives us the necessary condition to have [
x
1
*
, x
2
*
] as an extreme point (maximum or
minimum)
Method of Constrained Variation…
Explore the basic components of optimization problems, such as objective functions, constraints, and global vs. local optima. Learn about single vs. multiple objective functions and constrained vs. unconstrained optimization problems. Dive into the statement of optimization problems and the concept of objective function surfaces in a graphical context.
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Presentation Transcript
OPTIMIZATION TECHNIQUES
Objective Function, Maxima, Minima and Saddle Points, Convexity and Concavity
3 Objectives Study the basic components of an optimization problem. Formulation of design problems as mathematical programming problems. Define stationary points Necessary and sufficient conditions for the relative maximum of a function of a single variable and for a function of two variables. Define the global optimum in comparison to the relative or local optimum Determine the convexity or concavity of functions
4 Introduction - Preliminaries Basic components of an optimization problem : An objective function expresses the main aim of the model which is either to be minimized or maximized. Set of unknowns or variables which control the value of the objective function. Set of constraints that allow the unknowns to take on certain values but exclude others. Optimization problem is then to: Find values of the variables that minimize or maximize the objective function while satisfying the constraints.
5 Objective Function Many optimization problems have a single objective function The two interesting exceptions are: No objective function: User does not particularly want to optimize anything so there is no reason to define an objective function. Usually called a feasibility problem. Multiple objective functions. In practice, problems with multiple objectives are reformulated as single-objective problems by either forming a weighted combination of the different objectives or by treating some of the objectives by constraints.
6 Statement of an optimization problem x 1 x 2 . To find X = which maximizes f(X) . nx Subject to the constraints gi(X) <= 0 , i = 1, 2, .,m lj(X) = 0 , j = 1, 2, .,p
7 Statement of an optimization problem where X is an n-dimensional vector called the design vector f(X) is called the objective function gi(X) and lj(X) are inequality and equality constraints, respectively. This type of problem is called a constrained optimization problem. Optimization problems can be defined without any constraints as well Unconstrained optimization problems.
8 Objective Function Surface If the locus of all points satisfying f(X) = a constant c is considered, it can form a family of surfaces in the design space called the objective function surfaces. When drawn with the constraint surfaces as shown in the figure we can identify the optimum point (maxima). Possible graphically only when the number of design variable is two. When we have three or more design variables because of complexity in the objective function surface we have to solve the problem as a mathematical problem and this visualization is not possible.
9 Objective function surfaces to find the optimum point (maxima)
10 Variables and Constraints Variables These are essential. If there are no variables, we cannot define the objective function and the problem constraints. Constraints Even though constraints are not essential, it has been argued that almost all problems really do have constraints. In many practical problems, one cannot choose the design variable arbitrarily. Design constraints are restrictions that must be satisfied to produce an acceptable design.
11 Constraints (contd.) Constraints can be broadly classified as : Behavioral or Functional constraints : Represent limitations on the behavior and performance of the system. Geometric or Side constraints : Represent physical limitations on design variables such as availability, fabricability, and transportability.
12 Constraint Surfaces Consider the optimization problem presented earlier with only inequality constraints gi(X) . Set of values of X that satisfy the equation gi(X) forms a boundary surface in the design space called a constraint surface. Constraint surface divides the design space into two regions: one with gi(X) < 0 (feasible region) and the other in which gi(X) > 0 (infeasible region). The points lying on the hyper surface will satisfy gi(X) =0.
13 The figure shows a hypothetical two-dimensional design space where the feasible region is denoted by hatched lines Behavior constraint g2 0 Infeasible region Side constraint g3 0 Feasible region Behavior constraint g1 0 . . Bound acceptable point. Free acceptable point Bound unacceptable point. Free unacceptable point
14 Stationary Points: Functions of Single and Two Variables
15 Stationary points For a continuous and differentiable function f(x) a stationary point x*is a point at which the function vanishes, i.e. f (x) = 0 at x = x*. x*belongs to its domain of definition. Stationary point may be a minimum, maximum or an inflection (saddle) point
Stationary points ( ) 0 x = f ( ) ( ) 0 , f x ( ) 0 x ( ) 0 x f f 0 f x ( ) ( ) 0 x 0 , f x ( ) 0 x ( ) ( ) ( ) 0 x = 0 , f f x f f ( ) 0 x = f = 0 f x (a) Inflection point (b) Minimum (c) Maximum
17 Relative and Global Optimum Afunction is said to have a relative or local minimum at x = x*if for all sufficiently small positive and negative values of h, i.e. in the near vicinity of + * the point x. ( ) f x ( ) f x h A point x*is called a relative or local maximum if for all values of h sufficiently close to zero. + * ( ) f x ( ) f x h A function is said to have a global or absolute minimum at x = x*if for all x in the domain over which f(x) is defined. A function is said to have a global or absolute maximum at x = x*if ( ) ( ) f x f x * for all x in the domain over which f (x) is defined. * ( ) f x ( ) f x
18 Relative and Global Optimum A1, A2, A3= Relative maxima A2= Global maximum B1, B2= Relative minima B1= Global minimum . Relative minimum is also global optimum A2 . f(x) f(x). . A3 A1 . B2 . B1 x a a b b x
19 Functions of a single variable a x b Consider the function f(x) defined for [ , ] a b To find the value of x* such that x*maximizes f(x) we need to solve a single-variable optimization problem.
20 Functions of a single variable Necessary condition : For a single variable function f(x) defined for x which has a relative maximum at x = x*, x* derivative f (x) = df(x)/dx exists as a finite number at x = x*then [ , ] a b [ , ] a b if the f (x*) = 0. Above theorem holds good for relative minimum as well. Theorem only considers a domain where the function is continuous and derivative. It does not indicate the outcome if a maxima or minima exists at a point where the derivative fails to exist. This scenario is shown in the figure below, where the slopes m1 and m2at the point of a maxima are unequal, hence cannot be found as depicted by the theorem.
21 Functions of a single variable . Salient features: Theorem does not consider if the maxima or minima occurs at the end point of the interval of definition. Theorem does not say that the function will have a maximum or minimum at every point where f (x) = 0, since this condition f (x) = 0 is for stationary points which include inflection points which do not mean a maxima or a minima.
22 Sufficient condition For the same function stated above let f (x*) = f (x*) = . . . = f(n-1)(x*) = 0, but f(n)(x*) 0, then it can be said that f (x*) is (a) a minimum value of f (x) if f(n)(x*) > 0 and n is even (b) a maximum value of f (x) if f(n)(x*) < 0 and n is even (c) neither a maximum or a minimum if n is odd
23 Example 1 = + Find the optimum value of the function and also state if the function attains a maximum or a minimum. 2 ( ) f x 3 5 x x Solution = 3 0 + = '( ) f x 2 x For maxima or minima, or x*= -3/2 ''( *) x = 2 f is positive . Hence the point x*= -3/2 is a point of minima and the function attains a minimum value of -29/4 at this point.
24 Example 2 Find the optimum value of the function and also state if the function attains a maximum or a minimum = 4 ( ) f x ( 2) x Solution: f x = = 3 '( ) 4( 2) 0 x or x = x*= 2 for maxima or minima. ''( *) 12( * 2) x = = 2 0 f x at x*= 2 = = '''( *) x ( ) x 24( * 2) x 0 f at x*= 2 *= 24 f at x*= 2 Hence fn(x) is positive and n is even hence the point x = x* = 2 is a point of minimum and the function attains a minimum value of 0 at this point. More examples can be found in the lecture notes
25 Functions of two variables Concept discussed for one variable functions may be easily extended to functions of multiple variables. Functions of two variables are best illustrated by contour maps, analogous to geographical maps. A contour is a line representing a constant value of f(x) as shown in the following figure. From this we can identify maxima, minima and points of inflection. A contour plot
26 Necessary conditions From the above contour map, perturbations from points of local minima in any direction result in an increase in the response function f(x), i.e. the slope of the function is zero at this point of local minima. Similarly, at maxima and points of inflection as the slope is zero, the first derivative of the function with respect to the variables are zero. Which gives us at the stationary points. i.e. the f x f = = 0; 0 x 1 2 gradient vector of f(X), at X = X*= [x1 , x2] defined as follows, must equal zero: f x f f x xf ( *) 1 = = 0 x ( *) 2
27 Sufficient conditions Consider the following second order derivatives: 2 2 2 f f f ; ; x x 2 1 2 2 x x 1 2 Hessian matrix defined by H is formed using the above second order derivatives. 2 2 f f x x 2 1 f x 1 2 2 = H 2 f x x 2 2 x 1 2 [ , ] x x 1 2
28 Sufficient conditions The value of determinant of the H is calculated and if H is positive definite then the point X = [x1, x2] is a point of local minima. if H is negative definite then the point X = [x1, x2] is a point of local maxima. if H is neither then the point X = [x1, x2] is neither a point of maxima nor minima.
Example Locate the stationary points of f(X) and classify them as relative maxima, relative minima or neither based on the rules discussed in the lecture. = X /3 2 + + + 3 1 2 2 ( ) 2 5 2 4 5 f x 1 2 x x x x x 1 2 Solution
30 Example f x From , = (X) 0 1 ( ) 2 + = 2 2 2 2 5 0 x x 2 2 + + = 2 2 8 14 3 0 x x 2 + + = (2 3)(4 1) 0 x x 2 2 3/2 or = = 1/4 x x 2 2 So the two stationary points are X1= [-1,-3/2] and X2= [3/2,-1/4]
31 Example 2 2 2 2 f f f f = = = = 4 ; x 4; 2 The Hessian of f(X) is 1 x x x x 2 2 x x 1 2 1 2 2 1 4 2 x 1 = H 2 4 4 2 x 1 = I-H 4 2 4 + 2 At X1 = [-1,-3/2] , = = + 4) 4 = I-H ( 4)( 0 2 4 16 4 = 2 0 Since one eigen value is positive and one negative, X1is neither a relative maximum nor a relative minimum =+ = 12 12 1 2
Example At X2= [3/2,-1/4] 6 2 = = 4) 4 = I-H ( 6)( 0 2 4 = + = 5 5 5 5 1 2 Since both the eigen values are positive, X2is a local minimum. Minimum value of f(x) is -0.375 More examples can be found in the lecture notes
33 Convex and Concave Functions
34 Convex Function (Function of one variable) A real-valued function f defined on an interval (or on any convex subset C of some vector space) is called convex, if for any two points x and y in its domain C and any t in [0,1], we have + + ( (1 ) ) t b ( ) (1 tf a ) ( )) t f b f ta A function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set. A function is also said to be strictly convex if + + ( (1 ) ) t b ( ) (1 tf a ) ( ) t f b f ta for any t in (0,1) and a line connecting any two points on the function lies completely above the function.
35 A convex function
36 Testing for convexity of a single variable function A function is convex if its slope is non decreasing or 2 2 / f x 0 It is strictly convex if its slope is continually increasing 2 2 / f x or 0 throughout the function.
37 Properties of convex functions A convex function f, defined on some convex open interval C, is continuous on C and differentiable at all or at many points. If C is closed, then f may fail to be continuous at the end points of C. A continuous function on an interval C is convex if and only if ( ) 2 a b + + ( ) f b f a f for all a and b in C. 2 A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval.
38 Properties of convex functions A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: for all a and b in the interval. A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative in that interval; this gives a practical test for convexity. A continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semi definite on the interior of the convex set. If two functions f and g are convex, then so is any weighted combination af + b g with non-negative coefficients a and b. Likewise, if f and g are convex, then the function max{f, g} is convex.
39 Concave function (Function of one variable) A differentiable function f is concave on an interval if its derivative function f is decreasing on that interval: a concave function has a decreasing slope. A function f(x) is said to be concave on an interval if, for all a and b in that interval, + + [0,1], ( (1 ) ) t b ( ) (1 tf a ) ( ) t f b t f ta Additionally, f(x) is strictly concave if + + [0,1], ( (1 ) ) t b ( ) (1 tf a ) ( ) t f b t f ta
40 A concave function
41 Testing for concavity of a single variable function A function is convex if its slope is non increasing or 2 2 / f x 0. It is strictly concave if its slope is continually / f x 2 2 decreasing or 0 throughout the function.
42 Properties of concave functions A continuous function on C is concave if and only if a b + + ( ) f a ( ) f b for any a and b in C. f 2 2 Equivalently, f(x) is concave on [a, b] if and only if the function f(x) is convex on every subinterval of [a, b]. If f(x) is twice-differentiable, then f(x) is concave if and only if f (x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x4.
43 Example = + + 5 4 3 ( ) 12 f x 45 40 5 x x x Locate the stationary points of the function is convex, concave or neither at the points of optima based on the testing rules discussed above. Solution: and find out if = + = 4 3 2 '( ) f x 60 = 180 3 0,1,2 120 0 x x + x = 4 3 2 or 2 0 x x x x = Consider the point x = x* = 0 = + = '' * * 3 x * 2 x * ( ) x 240( ) 540( ) 240 0 f x at x *= 0 = + = ''' * * 2 x * ( ) x 720( ) 1080 240 240 f x at x *= 0
44 Example Since the third derivative is non-zero, x = x* = 0 is neither a point of maximum or minimum , but it is a point of inflection. Hence the function is neither convex nor concave at this point. Consider the point x = x* = 1 ( ) 240 = x x f ( ) ( ) x ( ) x 3 2 + = at x *= 1 * * * * 540 240 60 Since the second derivative is negative, the point x = x* = 1 is a point of local maxima with a maximum value of f(x) = 12 45 + 40 +5 = 12. At the point the 2 x f function is concave since 0 2
45 Example Consider the point x = x* = 2 ( ) x ( ) x ( ) x ( ) 240 = x 3 2 = + * * * * 240 540 240 f at x *= 2 Since the second derivative is positive, the point x = x* = 2 is a point of local minima with a minimum value of f(x) = -11. At the point the function is 2 x f concave since 0 2
46 Functions of two variables A function of two variables, f(X) where X is a vector = [x1,x2], is strictly convex if + + ( (1 ) ) ( ) (1 ) ( t f ) f t t tf 1 2 1 2 where X1and X2are points located by the coordinates given in their respective vectors. Similarly a two variable function is strictly concave if + + ( (1 ) ) ( ) (1 ) ( t f ) f t t tf 1 2 1 2
47 Contour plot of a convex function
48 Contour plot of a concave function
49 Sufficient conditions To determine convexity or concavity of a function of multiple variables, the eigen values of its Hessian matrix is examined and the following rules apply. If all eigen values of the Hessian are positive the function is strictly convex. If all eigen values of the Hessian are negative the function is strictly concave. If some eigen values are positive and some are negative, or if some are zero, the function is neither strictly concave nor strictly convex.
Example Locate the stationary points of 3 1 ( ) 2 /3 2 f x = X + + + 2 2 5 2 4 5 1 2 x x x x x 1 2 and find out if the function is convex, concave or neither at the points of optima based on the rules discussed in this lecture. Solution: f x f x ( *) + + 2 1 x 0 0 2 2 4 5 4 x x x 1 = = = 2 f x 2 ( *) 1 2 2 Solving the above the two stationary points are X1= [-1,-3/2] and X2= [3/2,-1/4]
