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Algorithms and Software for Large-Scale 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)

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

- 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

- 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 (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 (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 (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

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

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