Outline

1. Formulations for Surrogate-Based Optimization Using Data Fit and Multifidelity ModelsDaniel M. Dunlavy and Michael S. EldredSandia National Laboratories, Albuquerque, NMSIAM Conference on Computation Science and EngineeringFebruary 19-23, 2007SAND2007-0813C Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000

2. Outline Introduction to Surrogate-Based Optimization • Data fit case • Model hierarchy case Algorithmic Components • Approximate subproblem • Iterate acceptance logic • Merit functions • Constraint relaxation • Corrections and convergence Computational Experiments • Barnes • CUTE • MEMS design

3. Introduction to Surrogate-Based Optimization (SBO) • Purpose: Reduce the number of expensive, high-fidelity simulations • Use succession of approximate (surrogate) models • Drawback:Approximations generally have a limited range of validity • Trust regions adaptively manage this range based on efficacy of approximations throughout optimization run • Benefit: SBO algorithms can be provably-convergent • E.g., using trust region globalization and local 1st-order consistency • Surrogate Models of Interest • Data fits(e.g., local, multipoint, global) • Multifidelity(e.g., multigrid optimization) • Reduced-order models (e.g., spectral decomposition)

4. Smoothing • O(101) variables • Local consistency f x2 x1 Kriging Quad Poly Neural Network Splines Trust Region SBO – Data Fit Case Sequence of trust regions Data fit surrogates: • Global • Polynomial response surfaces • Artificial neural networks • Splines • Kriging models • Gaussian processes • Local • 1st/2nd-order Taylor series • Multipoint: • Two-Point Exponential Approx. (TPEA) • Two-point Adaptive Nonlinearity Approx. (TANA)

5. Trust Region SBO – Multifidelity Case Multifidelity surrogates: • Loosened solver tolerances • Coarsened discretizations: e.g, mesh size • Omitted physics: e.g., Euler CFD, panel methods Multifidelity models in SBO: • Truth evaluation only at center of trust region • Smoothing character is lost  requires well-behaved low-fidelity model • Correction quality is crucial • Additive: f(x)HF = f(x)LF + a(x) • Multiplicative: f(x)HF = f(x)LFb(x) • Combined (convex combination of add/mult) • May require design vector mapping • Space mapping • Corrected space mapping • POD mapping • Hybrid POD space mapping Sequence of trust regions

6. Outline Intro to Surrogate-Based Optimization • Data fit case • Model hierarchy case Algorithmic Components • Approximate subproblem • Iterate acceptance logic • Merit functions • Constraint relaxation • Corrections and convergence Computational Experiments • Barnes • CUTE • MEMS design

7. SBO Algorithm Components:Approximate Subproblem Formulations (TRAL) (IPTRSAO) (SQP-like) (Direct surrogate)

8. SBO Algorithm Components:Iterate Acceptance Logic • Trust region ratio • Ratio of actual to predicted improvement • Filter method • Uses concept of Pareto optimality applied to objective and constraint violation • Still need logic to adapt TR size for accepted steps

9. SBO Algorithm Components:Merit function selections Penalty: Adaptive penalty:adapts rp using monotonic increases in the iteration offset value in order to accept any iterate that reduces the constraint violation  mimics a (monotonic) filter Lagrangian: Multiplierestimation: Augmented Lagrangian: (derived from elimination of slacks) Multiplierestimation: (drives Lagrangian gradient to KKT)

10. Approximate subproblem Relaxed subproblem Relax nonlinear constraints SBO Algorithm Components:Constraint Relaxation • Composite step: • Byrd-Omojokun • Celis-Dennis-Tapia • MAESTRO • Homotopy: • Heuristic (IPTRSAO-like) • Probability one Estimate t: Leave portion of TR volume feasible for relaxed subproblem

11. Outline Intro to Surrogate-Based Optimization • Data fit case • Model hierarchy case Algorithmic Components • Approximate subproblem • Iterate acceptance logic • Merit functions • Constraint relaxation • Corrections and convergence Computational Experiments • Barnes • CUTE • MEMS design

12. Data Fit SBO Experiment #1Barnes Problem: Algorithm Component Test Matrices • Surrogates • Linear polynomial, quadratic polynomial, 1st-order Taylor series, TANA • Approximate subproblem • Function: f(x), Lagrangian, augmented Lagrangian • Constraints: g(x)/h(x), linearized, none • Iterate Acceptance Logic • Trust region ratio, filter • Merit function • Penalty, adaptive penalty, Lagrangian, augmented Lagrangian • Constraint relaxation • Heuristic homotopy, none

13. Data Fit SBO Experiment #2Barnes Problem: Effect of Constraint Relaxation Sample homotopy constraint relaxation 106 starting points on uniform grid Optimal/feasible directions differ by < 90o. Relaxation beneficial to achieve balance. Optimal/feasible directions opposed. Relaxation harmful since feasibility must ultimately take precedence.

14. Data Fit SBO Experiment #3Selected CUTE Problems: Test Matrix Summaries • Surrogates • TANA better than local • Approximate subproblem • Function: f(x) • Constraints: g(x)/h(x) • Iterate Acceptance Logic • TR ratio varies considerably • Filter more consistent/robust • Merit function • No clear preference • Constraint relaxation • No clear preference

15. 13 design vars: Wi, Li, qi Multifidelity SBO Testing:Bi-Stable MEMS Switch 38.6x

16. NPSOL DOT MEMS Design Results:Single-Fidelity and Multifidelity • Current best SBO approaches carried forward: • Approximate subproblem: original function, original constraints • Augmented Lagrangian merit function • Trust region ratio acceptance Note: Both single-fidelity runs fail with forward differences

17. Concluding Remarks • Data Fit SBO Performance Comparison: • Primary conclusions • Approx. subproblem: Direct surrogate (original obj., original constr.) • Iterate acceptance: TR ratio more variable, filter more robust • Secondary conclusions • Merit function: Augmented Lagrangian, Adaptive Penalty • Constraint relaxation: homotopy, adaptive logic needed • Multifidelity SBO Results: • Robustness: SBO less sensitive to gradient accuracy • Expense: Equivalent high-fidelity work reduced by ~3x • Future Work: • Penalty-free, multiplier-free approaches (filter-based merit fn/TR resizing) • Constraint relaxation linking optimal/feasible directions to a • Investigation of SBO performance for alternate subproblem formulations

18. DAKOTA • Capabilities available in DAKOTA v4.0+: • Available for download:http://www.cs.sandia.gov/DAKOTA • Paper containing more details: • Available for download: http://www.cs.sandia.gov/~dmdunla/publications/Eldred_FSB_2006.pdf • Questions/comments: • Email: dmdunla@sandia.gov

19. Extra Slides

20. Starting point (30,40) Starting point (65,1) Starting point (10,20) Data Fit SBO Experiment #1Barnes Problem: Algorithm Component Test Matrices

21. Starting point (30,40) Starting point (65,1) Starting point (10,20) Data Fit SBO Experiment #1Barnes Problem: Algorithm Component Test Matrices

22. Data Fit SBO Experiment #3Selected CUTE Problems: Test Matrix Summaries 3 top performers for each problem: SP Obj–SP CON—Surr—Merit—Accept—Con Relax

23. SBO Algorithm Components:Corrections and Convergence • Corrections: • Additive • Multiplicative • Combined • Zeroth-order • First-order • Second-order (full, FD, Quasi) • Soft convergence assessment: • Diminishing returns • Hard convergence assessment: • Bound constrained-LLS for updating multipliers for standard Lagrangian minimizes the KKT residual • If feasible and KKT residual < tolerance, then hard convergence achieved

24. Then: Approximate A,B with 2nd-order Taylor series centered at xc: where: Local Correction Derivations Exact additive/multiplicative:

25. Multipoint Correction Derivation Combined additive/multiplicative (notional): Assume a weighted sum of a, b corrections which preserves consistency: Previous xcRejected x* Enforce additional matching condition: where xp Can be used to preserve global accuracy while satisfying local consistency

26. Parameters Responses DAKOTA Framework Model: Iterator Parameters Interface LeastSq DoE Design continuous discrete Functions Application system fork directgrid NLSSOL GN DDACE CCD/BB objectives constraints NL2SOL QMC/CVT Uncertain normal/logn uniform/logu triangular beta/gamma EV I, II, III histogram interval least sq. terms ParamStudy generic UQ Optimizer Gradients Approximationglobal polynomial 1/2/3, NN, kriging, MARS, RBF multipoint – TANA3 local – Taylor series hierarchical DSTE LHS/MC Vector List numericalanalytic SFEM Reliability Center MultiD Hessians numericalanalyticquasi State continuous discrete NLPQL DOT CONMIN NPSOL OPT++ COLINY JEGA Strategy: control of multiple iterators and models Coordination: Strategy Iterator Nested Layered Cascaded Concurrent Adaptive/Interactive Model Optimization Uncertainty Iterator Hybrid Parallelism: OptUnderUnc Asynchronous local Message passing Hybrid 4 nested levels with Master-slave/dynamic Peer/static Model SurrBased UncOfOptima Iterator Pareto/MStart 2ndOrderProb Branch&Bound/PICO Model