Algorithms and Software for LargeScale Nonlinear Optimization

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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
• Adaptive Barrier Updates for NLP
  Interior-Point methods
• (with J. Nocedal, R. Byrd, and A. Waechter)

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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

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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
  constrained QP using constraints in W.
• Set xT xd.

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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

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Adaptive barrier updates

• NLP
• Functions twice continuously differentiable

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Adaptive barrier updates

• Solve a sequence of barrier subproblems
• Approach solution to NLP as

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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)

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Adaptive barrier updates (NLP)

• KNITRO
• Conservative rule
• Initially m0.1
• Decrease m linearly
• Fastlinear decrease near solution
• Globally convergent
• Robust but trade-off some efficiency
• Initial point option

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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)

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Globally Convergent Framework

• Official m for global conv (satisfies barrier
  stop test)
• Trial m for flexibility

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