Zhen Lu

1
Reduced Hessian Sequential Quadratic
Programming(SQP)

• Zhen Lu
• CPACT
• University of Newcastle
• MDC Technology

2
Background

• BSc in Automatic Control, Tsinghua University,
  China
• MSc in Automatic Control, Tsinghua University,
  China
• The first year PhD student, CPACT, University of
  Newcastle, UK

3
Research area

• My research area Process optimization
• On-line optimization
• Optimizing control
• Research Project - Optimization of Batch Reactor
  Operations (MDC)

4
Introduction

• Disadvantages of SQP
• Advantages of reduced Hessian SQP
• Description of rSQP
• Implementation of rSQP

• Summarize
• Numerical examples
• Conclusion
• Future work

5
Disadvantages of SQP methods

• In the SQP method, large and sparse QP
  sub-problems must be solved at each iteration.
  This can be computationally intensive to solve.
• Many chemical process optimization problems have
  a small number of degrees of freedom.
• A mixture of analytical second derivatives and
  many small, dense quasi-Newton updates are used
  to approximate the Hessian matrix of the
  Lagrangian function in the full space of the
  variables.

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Advantages of reduced Hessian SQP

• The reduced Hessian SQP is designed for large
  Non-linear Programming(NLP) problems with few
  degrees of freedom.
• The approach only requires projected second
  derivative information and this can often be
  approximated efficiently with quasi-Newton update
  formulae.
• This feature makes rSQP especially attractive for
  process systems where second derivative
  information may be difficult or computationally
  intensive to obtain.

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Advantages of reduced Hessian SQP

• Reduced Hessian SQP methods project the quadratic
  programming sub-problem into the reduced space of
  independent variables.
• Refinements of the reduced Hessian SQP approach
  guarantee a one-step super-linear convergence
  rate.

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Description of rSQP

• Optimization problems of the form

9
Description of rSQP

• Quadratic sub-problem

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Description of rSQP

• To compute the search direction , the null-space
  approach is used. The solution is written as
• Where is an matrix
  spanning the null space of , is an
  matrix spanning the range of .
• and

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Description of rSQP

• The QP sub-problem can be expressed by
• The solution is

12
Description of rSQP

• The components of x are grouped into m basic, or
  dependent variables and non-basic or
  control variables. The columns of A are grouped
  accordingly

13
Description of rSQP

• When the number of variables n is large and the
  number of degrees of freedom n-m is small, it
  is attractive to approximate the reduced Hessian
  .
• To ensure that good search directions are always
  generated, the algorithm approximates the cross
  term by a vector

14
Description of rSQP

• is approximated by a quasi-Newton matrix
• The reduced Hessian matrix is
  approximated by a positive definite quasi-Newton
  matrix

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Implementation of rSQP

• Update S

16
Implementation of rSQP

• Update B

17
Summarize

• The algorithm does not require the computation of
  the Hessian of the Lagrangian.
• The algorithm only makes use of first derivatives
  of objective function and constraints.
• The reduced Hessian matrix is approximated by a
  positive definite quasi-Newton matrix.

18
Numerical examples

• Model 1
• degrees of freedom 1

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Numerical examples

• Model 2
• degrees of freedom 50

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Numerical examples

• Model 3
• x01.1, 1.1, , 1.1
• degrees of freedom 1

21
Numerical examples

• Model 3
• x00.1, 0.1, , 0.1
• degrees of freedom 1

22
Numerical examples

• Model 3
• x02.1, 2.1, , 2.1
• degrees of freedom 1

23
Conclusion

• The algorithm is well-suited for large problems
  with few degrees of freedom.
• Reduced Hessian SQP approach saves the time of
  computing Hessian matrix, cuts down the cost of
  computation.
• Reduced Hessian SQP algorithm is at least as
  robust as SQP method.

24
Future work

• Use differential algebraic equations as
  constraints.
• Apply reduced Hessian SQP method to batch and
  continuous processes.