LINEAR, INTEGER, NONLINEAR PROGRAMMING

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LINEAR, INTEGER, NONLINEAR PROGRAMMING

• A practitioners perspective

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OPTIMIZATION

• min f(x)s.t. G(x) b.

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CLASSIFICATION

• Decision Variables
• Functions Fractional Integer
• Linear LP ILP
• Nonlinear NLP INLP

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LP

• SIMPLEX method (linear algebra)
• Corner point optimality
• Move from corner-to-corner, improve f(x)
• Very efficient
• Can solve problems with thousands of variables
  and constraints

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ILP

• BranchBound (divide-and-conquer)
• Solve the LP, ignoring the integrality
• Select a fractional variable, x6 15.7
• Create two new problems x6 lt 15, x6 gt 16
• Solve the new problems
• Continue until all branches exhausted
• No of branches is exponential in no. of var.

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NLP

• Gradient method (uses derivatives)
• Repeat until convergence
• Find an improving direction
• Move in the improving direction
• Can converge to a local optimum
• Multiple starts recommended

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INLP

• Ignore integrality, solve the NLP
• Use Branch Bound
• Solve a series of NLPs
• Computationally demanding
• No guarantee of optimality
• YUCK!

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• Now that we have reduced half of an MSc in
  OR program to 10 slides, we can move on to our
  regularly scheduled program.

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Optimization on Spreadsheets

• Erhan Erkut
• Faculty of Business, University of Alberta
• February 2, 1999
• Session 4, HEC-Montreal

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Spreadsheet solver

• Modeling language
• ... with graphical user interface

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First solver

• Whats Best? (Lotus 1-2-3 add on)
• Birth date 1985
• Father Sam Savage

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Excel solver

• Birth date 1991 (Excel 3.0)
• Producer Frontline Systems
• Current Population 35M
• Optimizers
• Linear simplex w. bounds
• Nonlinear GRG
• Integer BB

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How large a problem?

• Standard 200 variables
• Premium 800
• Platform 16,000

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Speed?

• Excel 3.0
• 5x
• Excel 5.0
• 20x
• Excel 97
• 25x
• Premium

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Bugs?

• Bugs in earlier versions
• Frontline offers 500 for a new bug discovery
• Two of our students received the prize...

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No competition for CPLEX,...

• but real-world problems are within reach.

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Frontline says

• 7M/year savings via a 200-variable application
  (Fortune 50 company)
• No. of applications with lt 200 variables
  (5-10) x (No. of larger applications)
• OR/MS Pros. are involved in lt 1 of all solver
  applications in use

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Excel 97 advertising

• Auto scaling applies to LP
• Can set convergence tolerance for IP
• Improved linearity tests
• Direct input of binary variables (bin)
• Much faster

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Premium advertising

• Altius, fortius, celsius
• Linearity and feasibility reports
• Standard IP breadth-first search
• Premium IP
• depth-first followed by breadth-first
• sophisticated selection rules
• dual simplex

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The solver is..

• not designed for the OR Pros,
• but created with the spreadsheet user in mind.
• wider use
• - wider abuse

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Some of the abuses

• Using IF instead of 0/1 variables
• Not reading the dialogue boxes
• Not scaling the problem
• Using a high tolerance for IP
• Not keeping everything on one sheet
• Finding local optimum for NLP
• f(x)/g(x) lt c
• Assume Linear Model (Y/N)
• 2 tons

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A few tips/reminders

• 1) Use names, color, layout to deal with
  documentability problem

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A few tips/reminders

• 2) Use auto scaling when necessary
• 15000A 20000B lt 160000 (1)
• 3A 4B lt 8 (2)
• Scaling Divide (1) by 10,000

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A few tips/reminders

• 3) Precision
• a b means...
• b ? gt a gt b ?,
• where ? precision

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A few tips/reminders

• 4) Tolerance
• Gap between UB and LB in IP
• (Start high, then reduce)

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A few tips/reminders

• 5) NLP multiple runs with different starting
  points (if OF not convex)

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A few tips/reminders

• 6) Dual variables are formatted the same way as
  variable and constraint values.

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A few curiosity items

• Set target cell equal to ??
• Leftover from Goal Seek

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A few curiosity items

• min (2SUMPRODUCT)
• min (SUMPRODUCT2)
• 2(min SUMPRODUCT)
• Whats the difference?

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A few curiosity items

• Limits report
• (What is the max/min each variable can take?)
• Why have one?
• cause Lotus 1-2-3 had one...