ESI 4313 Operations Research 2

1
ESI 4313Operations Research 2

• Nonlinear Programming Models
• Lecture 4

2
Excel Solver

• You can solve nonlinear optimization problems
  with the Microsoft Excel Solver in the same way
  as LP problems.
• You may now use nonlinear functions of the
  decision variables!

3
Excel Solver

• If you use the standard solver
• make sure you do not check the box Assume Linear
  Model in the options
• If you use the Premium solver
• Choose Standard GRG Nonlinear solver

4
LINGO

• You can install LINGO from the CD-ROM included
  with IPM
• LINGO is somewhat similar to LINDO
• It can be used to solve nonlinear optimization
  problems

5
LINGO

• Major differences
• Objective is of the form MIN (note the
  sign)
• All statements end with a semi-colon
• Start the constraints immediately after the
  objective, without ST/SUCH THAT/SUBJECT TO
• Use an asterisk () to denote multiplication
• You may/should use parentheses to define the
  order of mathematical operations
• As a default LINGO, like LINDO, includes
  nonnegativity constraints on all variables!

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Example 2

• Model
• LINGO
• min200((5-x)2(10-y)2).5
• 150((10-x)2(2-y)2).5
• 200((0-x)2(12-y)2).5
• 300((1-x)2(1-y)2).5

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Example 2

• Solution

Local optimal solution found at iteration
22 Objective value
5058.530
Variable Value Reduced Cost
X 3.200526
0.000000 Y
6.220236 0.000000
Row Slack or Surplus Dual
Price 1
5058.530 -1.000000
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Example 4

• Model
• LINGO
• max-(5/36)Q2(160/9)Q-(125/9)
• Qgt10
• Qlt100

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Example 4

• Solution

Local optimal solution found at iteration
8 Objective value
555.0000
Variable Value Reduced Cost
Q 63.99998
0.000000 Row
Slack or Surplus Dual Price
1 555.0000
1.000000 2
53.99998 0.000000
3 36.00002 0.000000
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Example 5a

• Model

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Example 5a

• LINGO
• min50S100F
• 5S.517F.5gt40
• 20S.57F.5gt60
• Fgt0
• Sgt0

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Example 5a

• Solution

Local optimal solution found at iteration
21 Objective value
563.0744
Variable Value Reduced Cost
S 5.886590
0.000000 F
2.687450 0.000000
Row Slack or Surplus Dual
Price 1
563.0744 -1.000000
2 0.000000 -15.93120
3 0.000000
-8.148348 4
2.687450 0.000000
5 5.886590 0.000000
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Example 5b

• Model

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Example 5b

• LINGO
• min50S100F
• 5S.517F.5gt40
• 20S.57F.5-0.2(FS).5gt60
• Fgt0
• Sgt0

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Example 5b

• Solution

Local optimal solution found at iteration
6 Objective value
569.7406
Variable Value Reduced Cost
S 6.105958
0.000000 F
2.644427 0.000000
Row Slack or Surplus Dual
Price 1
569.7406 -1.000000
2 0.000000 -15.86829
3 0.000000
-8.526706 4
2.644427 0.000000
5 6.105958 0.000000
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Differences betweenLP and NLP

• The feasible region of an LP is always a convex
  set.
• in fact, a polyhedron
• Isocost/profit curves are straight lines (or
  planes)
• an LP always has an extreme point optimal
  solution (if an optimal solution exists)

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Finding the optimal solution for an LP
x2
20x110x2160
10
x16
z40000
(4½,7)
z20000
10x115x2150
5
z50500
x1
8
6
4
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Differences betweenLP and NLP

• These results do not generalize to NLP
• Consider the following example
• A company can produce a good using capital and
  labor.
• K units of capital and L units of labor yields KL
  units of the good
• Capital can be purchased at 200/unit, while
  labor can be purchased at 1/unit
• A total budget of 40 is available

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Example 1

• NLP model
• maximize z KL
• subject to
• 200K L ? 40
• K, L ? 0

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Example 1

• Ignoring the budget constraint, we can draw the
  isoprofit lines for the objective function
• Rewrite z KL as L z/K
• Plot this function for different values of z

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Isoprofit Lines
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Example 1

• Now also plot the feasible region of the NLP
  problem

L 40 200 K
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Differences betweenLP and NLP

• We see that for an NLP the optimal solution is
  not necessarily an extreme point of the feasible
  region even if the feasible region is a
  polyhedron
• But in the example the optimal solution is on
  the boundary of the feasible region
• Is this true in general?

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Example 2

• Consider a simplified version of the location
  problem with Euclidean distances
• In particular, let there be only 1 customer
  location, (2,2)
• But suppose that the facility location is
  constrained to be in a particular area

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Example 2a

• Suppose that we require that the facility is in
  the square 0,1?0,1, i.e., we have the
  constraints
• so the full problem is

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Example 2a

• Graphically

? y
(2,2)
x ?
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Example 2a

• Again, the optimal solution is on the boundary of
  the feasible region
• Again, is this true in general?

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Example 2b

• Suppose that we require that the facility is in
  the square 0,3?0,3, i.e., we have the
  constraints
• so the full problem is

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Example 2b

• Graphically

? y
(2,2)
x ?
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Example 2b

• In this case, the optimal solution is in the
  interior of the feasible region
• Conclusion
• For general NLP problems, we have lost the
  structure of optimal solutions that we found for
  LP problems
• We need different approaches, concepts, etc. to
  help us solve such problems

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Example 3

• We could have found an even simpler example
• minimize f(x) x2
• subject to
• -1 ? x ? 1