1 Algorithms and Software for Large-Scale Nonlinear Optimization OTC day, 6 Nov 2003 Richard Waltz, Northwestern University Project I: Large-scale Active-Set.

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Presentation transcript:

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 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

3 Fletcher, Sainz de la Maza (1989) Overview 0. Given: x 1. Solve LP to get working set W. 2. Compute a step, d, by solving an equality constrained QP using constraints in W. 3. Set: x T = x+d. SLP-EQP Approach

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 NLP Functions twice continuously differentiable Adaptive barrier updates

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

7 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)

8 KNITRO Conservative rule Initially  Decrease  linearly Fastlinear decrease near solution Globally convergent Robust but trade-off some efficiency Initial point option Adaptive barrier updates (NLP)

9 Develop a more flexible adaptive rule Allow increases in barrier parameter!   : function of: Spread of complementarity pairs Recent steplengths Ease of meeting a barrier stop test Probing step (e.g. predictor step) Adaptive barrier updates (NLP)

10 1. Official  for global conv (satisfies barrier stop test) 2. Trial  for flexibility    Globally Convergent Framework