NLPQL

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NLPQL

• Sequential Quadratic Programming
• 6/17/05

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References

• Schittkowski, K. Solving Nonlinear Least Squares
  Problems by a General Purpose SQP method in
  Trends in Mathematical Optimization.
• Schittkowski, K. NLPQL A fortran subroutine
  solving constrained non linear programming
  problems. Annals of Operations Research Vol 5.
• NLPQL User Manual
• Optimization Concepts and Applications in
  Engineering by Belegundu.

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Sequential Quadratic Programming

• IMHO This is the algorithm of choice for smooth
  constrained optimization problems. (i.e.
  continuously differentiable)
• Prefered by Vanderplaats, Onwubiko, Belegundu,
  Rao, Lasdon
• Works with both feasible and infeasible initial
  design points.
• Works well with equality constraints.
• Requires fewer function evaluations then GRG
• Superior rate of convergence

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NLPQL

• Developed and constantly improved by Prof. Dr. K.
  Schittkowski at University of Bayreuth. First
  version in 1981. Latest version from 2000.
• Probably most heavily tested implementation with
  over 900 test cases.
• Hundreds of commercial applications. Customers
  include General Electric, Rolls-Royce, Siemens,
  BMW, Dow Chemical.

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NLPQL Formulation
iSIGHT needs to translate its constraints from lt
0 as stored internally to gt 0 NLPQL uses F(X)
for Objective value. X for design values G for
constraint values
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Search Direction

• In Steepest Descent and GRG the search direction
  was found using gradient information.
• For SQP, the search direction is found by solving
  a subproblem with a quadratic objective and
  linear constraints. The quadratic objective is an
  approximation to the Lagrangian.

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Getting the Search Direction
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Line Search
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NLPQL Implementation

• Generates a sequence of quadratic programming
  subproblems obtained by a quadratic approximation
  of the Lagrangian function and linearization of
  constraints.
• Second order information is updated by
  quasi-Newton formula (similar to BFGS)
• Line search used to stabilize method. Only two
  user parameters are maximum number of iterations
  and desired final accuracy. The final accuracy
  should not be smaller than minimum absolute
  gradient step.
• As a user, you can not tune algorithm. You can
  primarily tune the problem formulation.
• Insure program is scaled.
• Verify that your gradients are of the right
  tolerance.
• Start the algorithm from multiple points.

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NLPQL within iSIGHT

• Engineous has limited a line search to a maximum
  of 10 evaluations. You cannot change this.
• The print level is 4 for full print out. You
  cannot change this.
• Diagnostics are minimal. Look at
  Karush-Kuhn-Tucker conditions and see if it is
  approaching zero.
• Lagrangian multipliers suggest constraintsto
  relax for greatest gain. Worth investigatingwith
  a Tradeoff Analysis
• If the objective or gradient values are greater
  then SCBOU (1000) the NLPQL automatically scales
  by 1/sqrt(value)

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NLPQL Output File Header
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Typical NLPQL output for an iteration
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Final NLPQL Output Summary
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NLPQL Termination
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NLPQL Termination Reasons

• If you have scaled the model and are using a good
  finite step size and multiple starting points
  then 3 possibilities are
• Termination parameter is too small.
• The constraints are contradicting with an empty
  set of feasible solutions.
• Constraints are feasible but some are degenerate.

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Basic Parameters
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Advanced Parameters
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Lab

• Objectives
• Gain experience using NLPQL
• Analyze NLPQL output for convergence analysis
• Use Lagrangian multipliers for sensitivity
  analysis
• Note iSIGHT Glitch that you need to exit iSIGHT
  to have NLPQL status written to log file. You can
  then view log file in your favorite text editor.

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Lab

• Task 1
• Run the cantilevered beam, beam_Tcl.desc with
  NLPQL. Takeall defaults. Use no problem scaling.
  Answer the following questions
• What was the optimum objective value?
• How many function evaluations did it take?
• How many SQP iterations were needed?
• Plot the Kuhn Tucker value for each iteration.
• What was the reason for termination?

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Lab Continued

• Task 2
• Run the cantilevered bean again but this time
  provide a scale factorfor vol of 100000 and a
  scale factor of 5 for b1-b5 and a scale factorof
  40 for h1-h5. (Note These should have been the
  same settings that you used for exterior
  penalty).
• What was the optimium objective value
• How many function evaluations did it take?
• What was the reason for termination?
• Compare the number of function evaluations with
  that of exteriorpenalty. Which is better?
• Review the Lagrangian Multipliers. Which
  constraint can be relaxedfor the greatest gain
  in an objective?
• Verify your analysis by running a tradeoff
  analysis.

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Lab Continued

• Task 3 Lets try a different problem. The speed
  reducer is anotherclassic standard optimization
  test problem. The problem was takenfrom
  Engineering Optimization Theory and Practice by
  Rao. The problemis described on the next two
  slides.
• The problem has already been coupled in the
  description file,GearTrain_Tcl.desc.
• Review the standard problem formulation in the
  Parameters window.The formulation does not have
  its design variables, objectives andconstraints
  normalized.
• Run the optimization without a scaled formulation
  and with a scaled formulation. Compare the
  optimization value and the number offunction
  evaluations for both. (This problem really shows
  the benefitsof scaling.)
• Review the Lagrangian multipliers of the scaled
  solution. Which are the top two output
  constraints to consider for relaxation and what
  would thebenefit be?
• From the lagrangian multipliers, which two input
  constraints should be consideredfor relaxation
  and what would the benefit be?

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