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Algorithms and Software for Large-Scale

Nonlinear OptimizationOTC 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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1. Successive Linear Programming (SLP)• Inefficient, slow convergence

2. Successively Linearly Constrained (SLC)• e.g. MINOS• Difficulty scaling up

3. 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

Current Active-Set Methods

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Fletcher, Sainz de la Maza (1989)

Overview0. Given: x1. Solve LP to get working setW.2. Compute a step, d, by solving an

equality constrained QP using constraints in W.

3. Set: xT = x+d.

SLP-EQP Approach

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SLP-EQPStrengths:

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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NLP

Functions twice continuously differentiable

n

i

i

x

x

Iixg

Eixh

xf

,0)(

,0)( s.t.

)( min

Adaptive barrier updates

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Solve a sequence of barrier subproblems

Approach solution to NLP as

,0)(

,0)( s.t.

ln)( min,

Iisxg

Eixh

sxf

ii

i

iIi

sx

Adaptive barrier updates

0

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Overview of Barrier Strategies:1. Fixed decrease with barrier stop

test (e.g. KNITRO)2. Centrality-based strategies (e.g.

LOQO)3. Probing strategies (e.g. Mehrotra

PC)

Adaptive barrier updates (NLP)

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KNITROConservative rule Initially Decrease linearly Fastlinear decrease near solution

Globally convergentRobust but trade-off some efficiencyInitial point option

Adaptive barrier updates (NLP)

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Develop a more flexible adaptive rule Allow increases in barrier parameter!

: function of: Spread of complementarity pairsRecent steplengthsEase of meeting a barrier stop testProbing step (e.g. predictor step)

Adaptive barrier updates (NLP)

1

s

T

k n

zs

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1. Official for global conv (satisfies barrier stop test)

2. Trial for flexibility

Globally Convergent Framework