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Algorithms and Software for Large-Scale Nonlinear Optimization. OTC day, 6 Nov 2003 Richard Waltz, Northwestern University Project I : Large-scale Active-Set methods for NLP Fact or Fiction ? (with J. Nocedal, R. Byrd and N. Gould) Project II :

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algorithms and software for large scale nonlinear optimization
Algorithms and Software for Large-Scale Nonlinear Optimization

OTC day, 6 Nov 2003

Richard Waltz, Northwestern University

  • Project I:

Large-scale Active-Set methods for NLP Fact or Fiction?

(with J. Nocedal, R. Byrd and N. Gould)

  • Project II:

Adaptive Barrier Updates for NLP Interior-Point methods

(with J. Nocedal, R. Byrd, and A. Waechter)

current active set methods
Current Active-Set Methods
  • Successive Linear Programming (SLP)
    • Inefficient, slow convergence
  • Successively Linearly Constrained (SLC)
    • e.g. MINOS
    • Difficulty scaling up
  • Sequential Quadratic Programming (SQP)
    • e.g. filterSQP, SNOPT
    • Very robust when less than a couple thousand degrees of freedom
    • For larger problems QP subproblems may be too expensive
slp eqp approach
SLP-EQP Approach
  • Fletcher, Sainz de la Maza (1989)

Overview

0. Given: x

  • Solve LP to get working setW.
  • Compute a step, d, by solving an equality constrainedQP using constraints in W.
  • Set: xT= x+d.
slp eqp
SLP-EQP
  • Strengths:
    • Only solve LP and EQP subproblems
    • Early results very encouraging
    • Competitive with SQP – able to solve problems with more degrees of freedom
  • But…
    • Not yet competitive with Interior
    • Difficulties in warm starting LP subproblems
    • How to handle degeneracy?
    • Theory needs more development
adaptive barrier updates
Adaptive barrier updates

NLP

  • Functions twice continuously differentiable
adaptive barrier updates1
Adaptive barrier updates

Solve a sequence of barrier subproblems

  • Approach solution to NLP as
adaptive barrier updates nlp
Adaptive barrier updates (NLP)

Overview of Barrier Strategies:

  • Fixed decrease with barrier stop test (e.g. KNITRO)
  • Centrality-based strategies (e.g. LOQO)
  • Probing strategies (e.g. Mehrotra PC)
adaptive barrier updates nlp1
Adaptive barrier updates (NLP)

KNITRO

  • Conservative rule
    • Initially m=0.1
    • Decrease m linearly
    • Fastlinear decrease near solution
  • Globally convergent
  • Robust but trade-off some efficiency
  • Initial point option
adaptive barrier updates nlp2
Adaptive barrier updates (NLP)
  • Develop a more flexible adaptive rule
    • Allow increases in barrier parameter!
  • q : function of:

Spread of complementarity pairs

Recent steplengths

Ease of meeting a barrier stop test

Probing step (e.g. predictor step)

globally convergent framework
Globally Convergent Framework
  • Official mfor global conv (satisfies barrier stop test)
  • Trial m for flexibility

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