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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 :

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Algorithms and Software for Large-Scale Nonlinear Optimization

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  1. 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)

  2. 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

  3. 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.

  4. 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

  5. Adaptive barrier updates NLP • Functions twice continuously differentiable

  6. Adaptive barrier updates Solve a sequence of barrier subproblems • Approach solution to NLP as

  7. 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)

  8. 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

  9. 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)

  10. Globally Convergent Framework • Official mfor global conv (satisfies barrier stop test) • Trial m for flexibility 1 2 3

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