Surrogate-based constrained multi-objective optimization

Aerospace design is synonymous with the use of long running and computationally intensive simulations, which are employed in the search for  optimal designs in the presence of multiple, competing objectives and constraints. The difficulty of this search is often exacerbated by numerical `noise' and inaccuracies in simulation data and the frailties of complex simulations, that is they often fail to return a result. Surrogate-based optimization methods can be employed to solve, mitigate, or circumvent problems associated with such searches. This presentation gives an overview of constrained multi-objective optimization using Gaussian process based surrogates, with an emphasis on dealing with real-world problems.

Alex Forrester3rd July 2009

Coming up:• Surrogate model based optimization – the basic idea

• Gaussian process based modelling

• Probability of improvement and expected improvement

• Missing data

• Noisy data

• Constraints

• Multiple objectives

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Surrogate model based optimization

• Surrogate used to expedite search for global optimum

• Global accuracy of surrogate not a priority

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SAMPLING PLAN

OBSERVATIONS

CONSTRUCT SURROGATE(S)

design sensitivities available?

multi-fidelity data?

SEARCH INFILL CRITERION(optimization using the

surrogate(s))

constraints present?

noise in data?

multiple design objectives?

ADD NEW DESIGN(S)

PRELIMINARY EXPERIMENTS

Gaussian process based

modelling4

Building Gaussian process models, e.g. Kriging

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• Sample the function to be predicted at a set of points

• Correlate all points using a Gaussian type function

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• 20 Gaussian “bumps” with appropriate widths (chosen to maximize likelihood of data) centred around sample points

• Multiply by weightings (again chosen to maximize likelihood of data)

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Add together to predict function

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Kriging prediction True function

Optimization

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Polynomial regression based search (as Devil’s advocate)

Gaussian process prediction based optimization

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Gaussian process prediction based optimization (as Devil’s advocate)

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But, we have error estimates with Gaussian processes

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Error estimates used to construct improvement criteria

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Probability of improvement

Expected improvement

Probability of improvement

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• Useful global infill criterion

• Not a measure of improvement, just the chance there will be one

Expected improvement

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• Useful metric of actual amount of improvement to be expected

• Can be extended to constrained and multi-objective problems

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Missing Data

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What if design evaluations fail?• No infill point augmented to the surrogate

– model is unchanged

– optimization stalls

• Need to add some information or perturb the model

– add random point?

– impute a value based on the prediction at the failed point, so EI goes to zero here?

– use a penalized imputation (prediction + error estimate)?

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Aerofoil design problem• 2 shape functions

(f1,f2) altered

• Potential flow solver (VGK) has ~35% failure rate

• 20 point optimal Latin hypercube

• max{E[I(x)]} updates until within one drag count of optimum

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Results

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A typical penalized imputation based optimization

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Four variable problem

• f1,f2,f3,f4 varied

• 82% failure rate

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A typical four variable penalized imputation based optimization• Legend as for two

variable

• Red crosses indicate imputed update points.

• Regions of infeasible geometries are shown as dark blue.

• Blank regions represent flow solver failure

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‘Noisy’ Data

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‘Noisy’ data

• Many data sets are corrupted by noise

• We are usually interested in deterministic ‘noise’

• ‘Noise’ in aerofoil drag data due to discretization of Euler equations

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Failure of interpolation based infill• Surrogate becomes

excessively snaky

• Error estimates increase

• Search becomes too global

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Regression improves model

• Add regularization constant to correlation matrix

• Last plot of previous slide improved

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Failure of regression based infill• Regularization assumes

error at sample locations (brought in through lambda in equations below)

• Leads to expectation of improvement here

• Ok for stochastic noise

• Search stalls for deterministic simulations

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Use “re-interpolation”

• Error due to noise ignored using new variance formulation (equation below)

• Only modelling error

• Search proceeds as desired

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Two variable aerofoil example

• Same parameterization as missing data problem

• Course mesh causes ‘noise’

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Interpolation – very global

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Regression - stalls

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Re-interpolation – searches local basins, but finds global optimum

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Constrained EI

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Probability of constraint satisfaction

• g(x) is the constraint function

• F=G(x)-gmin is a measure of feasibility, where G(x) is a random variable

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It’s just like the probability of improvement, but with a limit, not a minimum

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Probability of satisfaction

Prediction of constraint function

Constraint function

Constraint limit

Constrained probability of improvement• Probability of

improvement conditional upon constraint satisfaction

• Simply multiply the two probabilities:

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Constrained expected improvement

• Expected improvement conditional upon constraint satisfaction

• Again, a simple multiplication:

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A 1D example

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After one infill point

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A 2D example

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Multi-objective EI

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Pareto optimization• We want to identify a

set of non-dominated solutions

• These define the Pareto front

• We can formulate an expectation of improvement on the current non-dominated solutions

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Multi-dimensional Gaussian process

• Consider a 2 objective problem

• The random variables Y1 and Y2 have a 2D probability density function:

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Probability of improving on one point• Need to integrate

the 2D pdf:

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• Integrating under all non-dominated solutions:

• The EI is the first moment of this integral about the Pareto front (see book)

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A 1D example

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Matlab demo

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Nowacki beam• Fixed length steel cantilever beam under 5kN load

• Variables:

– height

– width

• Objectives:

– minimize cross section area

– minimize bending moment

• Constraints:

– area ratio

– Bending moment

– buckling

– deflection

– Shear

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Problem setup• 10 point optimal Latin hypercube

• Kriging model of each objective and constraint

– Parameters tuned with GA + SQP (using adjoint of likelihood)

• 20 points added at the maximum constrained multi-objective expected improvement

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Sampling plan

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Initial trade off

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5 updates

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10 updates

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15 updates

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20 updates

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Final trade off

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Summary• Surrogate based optimization offers answers to, or

ways to get round, many problems associated with real world optimization

• This seemingly blunt tool must, however, be used with precision as there are many traps to fall into

• In a multi-objective context, the use of surrogates is particularly promising

• There has not been time to cover new surrogate methods (e.g. blind Kriging), multi-fidelity modelling or enhancements to EI, in terms of its exploitation/exploration tradeoff properties

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References• A. I. J. Forrester, A. Sóbester, A. J. Keane, Engineering Design via Surrogate

Modelling: A Practical Guide, John Wiley & Sons, Chichester, 240 pages, ISBN 978-0-470-06068-1.  

• A. I. J. Forrester, A. J. Keane, Recent advances in surrogate-based optimization, Progress in Aerospace Sciences, 45, 50-79, (doi:10.1016/j.paerosci.2008.11.001)

•  A. I. J. Forrester, A. Sóbester, A. J. Keane, Optimization with missing data, Proc. R. Soc. A, 462(2067), 935-945, (doi:10.1098/rspa.2005.1608).

• A. I. J. Forrester, N. W. Bressloff, A. J. Keane, Design and analysis of ‘noisy’ computer experiments, AIAA journal, 44(10), 2331-2339, (doi:10.2514/1.20068).

• All code at www.wiley.com/go/forrester

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Gratuitous publicity

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