Nonlinear Programming

1
Nonlinear Programming
Consider the following math program Max Z
KL st 4K L lt 8 K,L
gt 0
place Winston Fig 2 here Where
does the optimal solution fall within the
feasible region?
2
Nonlinear Programming
Consider the following math program Max Z
f(x) st 0 lt x lt 1 where f(x) is
pictured at right.
place Winston Fig 3
here Where does the optimal solution fall
within the feasible region?
3
Nonlinear Programming
Consider the following math program Max Z
f(x) st 0 lt x lt 10 where f(x) is
pictured at right.
place Winston Fig 4
here Note there are three local optima, but
only one optimal solution at point C.
4
Nonlinear Programming
Formulate the following problem as an
NLP. Truckco is trying to determine where they
should locate a single warehouse. The positions
in the x-y plane (in miles) of their four
customers and the number of shipments made
annually to each customer is given below. Locate
the warehouse to minimze the total distance
trucks must travel annually from the ware house
to the four customers. Customer X Coord.
Y Coord. Number of Shipments 1
5 10
200 2 10
5 150
3 0 12
200 4 12
0 300
5
Nonlinear Programming
Truckco formulation Let X,Y be the coordinates
of the warehouse. Min Z 200 D1 150 D2
200 D3 300 D4 s.t. D1 ( ( X- 5)2
(Y-10 )2 ) .5 D2 ( ( X- 10)2 (Y-5 )2
) .5 D3 ( ( X- 0)2 (Y-12 )2 ) .5
D4 ( ( X- 12)2 (Y-0 )2 ) .5 D1,
D2, D3, D4, X, Y gt 0
6
Nonlinear Programming
One-Variable Unconstrained Example Max Z
f(x) s.t. nothing Or
Max Z 10 3x - (2x2 x4) First determine if
the concave (or convex for Min problem). f(x) is
concave if the second derivative is lt 0 for all
values of x. f(x) 3 4x 4x3 f(x) 4
12x2 Where does optimum occur? 3 4x 4x3
0
7
Nonlinear Programming
One-Variable Unconstrained An approximate graph
of x and Z is as follows Z 10 3x - (2x2
x4)
Z
x
8
Nonlinear Programming
One-Variable Unconstrained How do you solve for
x? 3 4x 4x3 0
9
Nonlinear Programming
Multi-Variable Unconstrained Example Max Z
f(x) s.t. nothing Or
Max Z 2x1x2 2x2 x12 2x22 For a concave
function (convex when minimizing), use an
approach whereby you move in a direction of
steepest ascent. In other words, if f(x) is the
surface area of a 3 dimensional graph then move
in a direction that has the steepest angle.
Move as far in the steepest direction until you
are no longer ascending. Then re-evaluate the
steepest ascent at that point. This procedure is
called the gradient search procedure.
10
Nonlinear Programming

• Multi-Variable Unconstrained
• Gradient Search procedure
• Initialization select some initial point x, and
  some e. Go to the stopping rule.
  
• Set xj xj t(df / dxj)xx for all values
  of j
• then substitute these expressions into the
  expression f(x).
• This expression of f(x) is in terms of t.
• Find the value of t that maximizes f(x)
  over t gt 0.
• ReSet x x t (df / dxj)xx . Go to
  stopping rule.
• Stopping rule evaluate if (df / dxj)xx
  lt e for each j. If so, stop. Else, go to step 1.

11
Nonlinear Programming

• Multi-Variable Unconstrained
• Example Gradient Search procedure Z 2x1x2
  2x2 x12 2x22
• df / dx1 2x2 2x1
• df / dx2 2x1 2 4x2
• Initialization set x1, x2 0.
  
• Set x1 x1 t(df / dx1) 0 t(20 20)
  0.
• Set x2 x2 t(df / dx2) 0 t(20 2
  40) 2t.
• f(x1 , x2) 20 2t 22t 00
  22t2t 4t 8t2.
• f(x1 , x2) 4 16t. Let 4 16t 0 then
  t ¼.
• ReSet x1 x1 t(df / dx1) 0 ¼(20
  20) 0.
• x2 x2 t(df / dx2) 0
  ¼(20 2 40) ½.
• Stopping rule df / dx1 1, df / dx2 0.

12
Nonlinear Programming

• Multi-Variable Unconstrained
• Example Gradient Search procedure Z 2x1x2
  2x2 x12 2x22
• df / dx1 2x2 2x1
• df / dx2 2x1 2 4x2
• Iteration 2 x1 0 x2 ½ .
• Set x (0,1/2) t(1,0) (t,1/2).
• f(t,1/2) 2t 1/2 21/2 tt
  21/21/2 t t2 ½.
• f(t,1/2) 1 - 2t. Let 1 2t 0 then t
  ½ .
• ReSet x (0,1/2) ½ (1,0) (½, ½).
• Stopping rule df / dx1 0, df / dx2 1.

13
Nonlinear Programming
Multi-Variable Unconstrained Example Gradient
Search procedure Z 2x1x2 2x2 x12
2x22 This iterative process continues until the
values of the gradient are lt some e.
14
Nonlinear Programming
Multi-Variable Constrained Category - Convex
Programming Example Max Z 5x1 8x2 x12
2x22 Non-linear
s.t. 3x1 2x2 lt 6
Convex feasible region
x1 , x2 gt 0