Global Optimization General issues in global optimization Classification of algorithms The DIRECT algorithm – Divided rectangles – Exploration and Exploitation.

global optimization algorithms

Global OptimizationGeneral issues in global optimizationClassification of algorithmsThe DIRECT algorithmDivided rectanglesExploration and Exploitation as bi-objective optimizationApplication to High Speed Civil Transport

This lecture will present issues associated with global optimization and then describe as an example the DIRECT algorithm, which is one of my favorites.

When it comes to DIRECT we will go over the basic concept of divided rectangles or boxes that is responsible for its name. We will discuss how it balances exploration in new region vs. exploitation near good known designs, and we will go over an application to the optimization of a supersonic transport.1Global optimization issuesOptimization problem is NP-hardNo-free-lunch theorem (Wolpert and Macready)No single algorithm can do well on all problemsIf an algorithm is improved for one problem, it will suffer for others.Great opportunity for engineers to use problem knowledge to tailor algorithms.Big headache for journals because they get many worthless new algorithms.

The cost of local optimization usually increase somewhere between linearly and quadratically in the number of variables. In contrast global optimization is NP hard, that is, its cost increases faster than any polynomial in the number of variables. That is, the cost increases exponentially.

For local optimization there is a small number of algorithms, such as Nelder Mead and Quasi-Newton methods that are recognized as the best. For global optimization, on the other hand, there the no-free-lunch theorem that states no single algorithm will do well on all problems. Furthermore, if you tune an algorithm to do well on one problem, you will generally make it perform more poorly on other problems.

This is a great opportunity for engineers who can use problem specific knowledge to tailor algorithms to their problems, something that they can rarely do for local optimization algorithms. On the other hand, it is responsible for huge proliferation of papers on algorithms, which are often poorly tested.2Global Optimization

Global optimization algorithms by Thomas Weise

Thomas Weise uses these figures to portray optimization problems of increasing challenge. In Figure a we have a very benign situation, where local minima are very shallow and most local optimization routines may be able to find the global minimum, while in Figure h, even a good global optimizer may take forever to find the optimum because it is hidden in a very narrow region in a region which overall is high (that is, has poor objective).

However, scenarios (a), (b), (c), and (e), are all handled very well by multiple-start local optimizers, and (d) is a typical case for a global optimizers. Cases (f), (g) would present difficulties for most global optimizers, and case (h) is practically hopeless.

3Classification of global optimization algorithmsThe most popular algorithms imitate natural processes, including genetic algorithms, particle swarm optimization, ant colony optimization, and simulated annealing.They rely on randomness for exploration, so every time you run them you may get a different result.DIRECT is an example of a systematic deterministic exploration of the design space.Adaptive sampling algorithms based on surrogates, such as EGO, are gaining popularity.

The most popular global optimization algorithms imitate optimization in nature. We will study genetic algorithms which imitate genetic evolution and particle swarm optimization which imitate bird-swarm search for food. Other algorithms of this type are simulated annealing which imitates atoms reaching the lowest energy state in an annealing metal.

Nature inspired algorithm usually rely on randomness for part of their exploration of the design space, like mutations in genetic evolution. This means that every time you run an optimization you may get a different answer, which is something that I do not particularly like.

DIRECT, on the other hand is an example of systematic exploration of the design space by dividing it into regions, and then into sub-regions. It is purely deterministic.

Adaptive sampling algorithms is another class of deterministic algorithms. They usually fit a surrogate to an initial sample, and use the surrogate to decide where to sample next. We will study the EGO algorithm from that class.4Problems global optimizationWhat does it mean that global optimization is NP hard?What is the no-free-lunch principle, and how does it affect engineering optimization.When should we use local optimizers to solve global optimization problems and when we should not?

5Lipschitzian OptimizationDIRECT was inspired by Lipschitzian optimization. Optimizer divides space into boxes and samples the vertices of eachA box is further divided based on estimated maximum rate of change of the function, K (Lipschitz constant)

Lipschitzian optimization is a predecessor of DIRECT that divides the space into boxes, samples at the vertices and divides boxes further based on the functions Lipschitz constant. The Lipschitz constant is an upper bound on the rate of change of the function, and it can be used even for functions that are not even differentiable 9just Lipschitz continuous). It is named after Rudolf Otto Sigismund Lipschitz(14 May 1832 7 October 1903) was aGermanmathematicianand professor at theUniversity of Bonnfrom 1864.

The process is illustrated in the figures on the right. In the top left figure we have the value of the functions and the two ends. Based on the Lipschitz constant K we find the point x1, where the function could be at its lowest if it decreased by a rate of K from both ends. Note that given K the function cannot be possibly lower than the intersection value at x1.

We now repeat the process at the two intervals that were created by adding x1, that is (a,x1) and (x1,b). The second figure shows that the process predicts a lower bound on the function in (a,x1) that is better than the one in (x1,b), so we select the intersection there and get x2. And so we continue.

6DIRECTThe function value at the middle of each box and its largest dimension are used to determine potentially optimal boxesEach potentially optimal box is dividedLipschitzian optimization for all possible Lipschitz constants

The DIRECT algorithm was developed by Jones and colleagues (Jones, D.R., Perttunen, C.D. and Stuckman, B.E. (1993), Lipschitzan optimization without the Lipschitz constant, Journal of Optimization Theory and Application 79, 157181.)

At a given iteration the design space is divided into boxes, and we have the value of the function at the center of each box. We then look at a plot of box sizes (determined by the largest dimension of the box) vs. function value and look for boxes that should be divided.

On the one hand we would like to divide boxes that have small values, that is exploitation of promising regions. On the other hand we would like to divide large boxes, which is an exploration activity. So we sample at all the points that are on the Pareto front of size and function value, which includes all the points that are on the bottom part of the convex hull.

A good Matlab implementation is available at: http://www4.ncsu.edu/~ctk/Finkel_Direct/

7DIRECT Box Division.

This is an illustration of the progress of DIRECT

1st Iteration: Single box is divided first into three vertically and then because it is a square, the middle box is also divided into three horizontally.

Second iteration: The lowest function value is in the bottom box. Because it is the best and the largest at the same time, it is the only one that is divided.

Third iteration: The best point remains where it was before, but now the top box is also candidate for division because it is larger than the best box. The bottom box will be divided in both direction because it is square.

Fourth iteration: The best value is in the rightmost small box, and two of the larger boxes have the same value of the objective, so they are both going to be divided.8Exploration vs. ExploitationDIRECT uses convex hull of box sizes to balance exploitation vs. explorationWith enough function evaluations every region in design space will be exploredThis is clearly not feasible for high dimensional spacesCoxs paper compares DIRECT to repeated local optimization with random start

DIRECT uses the convex hull of box sizes to generate the Pareto front of function value versus size. Low function values are associated with exploitation, large boxes with exploration.

If we wait long enough, every region in design space will be eventually divided into a small box, so the algorithm is guaranteed to find the global optimum even if the problem is of the nightmare variety. However, practically we will run out of computer time.

Since the number of boxes increases exponentially with dimension, there is a serious question as to how practical is DIRECT for high dimensional problem. The following paper probed this question;

S. E. COX, R. T. HAFTKA, C. A. BAKER, B. GROSSMAN, W. H. MASON and L. T. WATSON A Comparison of Global Optimization Methods for the Design of a High-speed Civil Transport, Journal of Global Optimization 21: 415433, 2001.

9Problems DIRECTWhat is the Lipschtiz constant? What value would be appropriate for the function x2+x in the interval (-2,2)?Invent function values at every point where the function is evaluated in Slide 8 that are consistent with the diagram.What are the meanings of the term exploration and exploitation in the context of global optimization?Optimization of a High Speed Civil Transport26 design variables defining the shape of the wing and fuselage and tail and the flight trajectoryLong range (NY Tokyo type flight) with 250 passengers for minimum takeoff gross weightConstraints on takeoff and landing maneuvers, engine out conditions.CFD analysis plus structural optimization needed for each design.

The optimization of the high speed civil transport (HSCT) was a NASA sponsored project in the mid and late 1990s the revived the idea of a supersonic transport that had been considered and rejected in the 1970s (Europeans pursued it then leading to the Concorde).

With new technology it appeared that this time around the project was feasible, but because it had to fly above 60,000 feet, its environmental impact was high, and this was one of the reasons it never made it beyond design studies.

The paper by Cox described a joint research project between the University of Florida and Virginia Tech to optimize the aerodynamics and structure of the airplane using 26 design variables to describe the shape and its trajectory, and many more to define the structural sizes.

The structural optimization, though, was done for each aerodynamic shape, so that the overall optimization involved only 26 variables. It required very expensive function evaluation consisting of structural optimization and CFD analysis.

The objective function was the gross take-off weight, and constraints were on takeoff and landing maneuvers, flying without one engine, and range requirements.11Results

The three algorithms that were compared included a commercial local optimization code DOT as well as DIRECT and a nature inspired leap-frog algorithm.

The figure compares the best take-off gross weight obtained by the three algorithms for 5, 10, and 26 design variables. The 5-variable case was easy, so all three algorithms obtained the same result. For the 10 variable case, DIRECT had an advantage, while for the 26-variable case DOT had a slight edge.12Results

In terms of function evaluations DIRECT looked best, but this depends on convergence criteria, so it may not be significant. It is clear, though that the optimization required tens of thousands of function evaluations, and this was made possible only by fitting a surrogate (an approximation) to the aerodynamic and structural performance as function of the design variables.13Results

The DOT results were obtained with 100 local optimization from random starting points, and the figure shows the weights obtained for each one of the 100 runs. It is clear that there is some luck involved in this approach.14Results

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DOT Optimum ValuesOptimization #Weight

DIRECT

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26DV CompLFOPCV3DOTDIRECT0.0005FunctionCallsFunctionCallsFunctionCalls10.8035918360.81783119710130.813163144572481.83788428920.83107612420.813595140210680.810726258569516.59815035230.84316322680.81417943858720.812379174569925.60698278440.80834429700.81703277949580.812369232571922.94560908350.82847416200.81737139578730.812822145572159.97697039860.87859416470.8164087290.812707146571485.670.83378128620.81645731878690.812762144571520.12311200180.8346738640.8149955029870.812634144570496.85141599690.80431412960.81802084889300.81279145572614.594137557100.79566413500.81174764968430.812801145568223.354749402110.83846723220.81494785048670.812823145570463.495272211120.81210318360.818774610910180.812823145573142.227658673130.8442118090.82290449048730.812823145576033.143255196140.7916458320.81397215037260.812823145569780.505203333150.80948720790.82217036286420.812823145575519.253928897160.82630318090.8182432497870572770.274814807170.80551814850.817028395756571919.876527657180.86124811880.8190701404127741519573349.098308244190.81263816740.8165036156811571552.530926817200.80446635100.8244685914789577128.013998339210.82330417820.8196429649785573750.075416459220.825223313770.8169124481811571838.713701412230.86618413230.8176006471900572320.452958468240.82816916200.8132674049961569287.183396587250.85431823220.8149532765991570467.293561644260.80510128620.8169937823962571895.647590877270.83823615120.8138260303899569678.221235689280.81021922680.8158884803874571121.936197627290.81446614850.8137067128989569594.698957392300.81620912150.8159564575731571169.520251436310.80182131050.8191566449668573409.651460367320.8093219440.8150458483843570532.093799774330.85654214850.8153873574899570771.1502133340.80498611880.8179174084898572542.185889117350.81802835640.8179976721839572598.370462364360.81821235640.8143924527785570074.716896187370.84725517550.81290168451018569031.179156821380.81241440230.8154367116840570805.698142317390.81595613500.8127175597840568902.291808992400.81205723220.818871292573209.7410.81360537260.8215100689927575057.048226779420.80669425920.8156229101871570936.03709067430.8026631860.8164490334813571514.323389837440.84441416200.8196284043957573739.883022055450.83376815930.8172322813783572062.596890843460.84232611340.81741391171535572189.738168497470.80385228620.8116875111902568181.257773652480.81490838880.8197135779731573799.504548622490.79001835100.8110230124990567716.108677323500.83385717280.81832159081307572825.113594304510.86044632400.816380242959571466.169428669520.80030614850.8150302287896570521.160065352530.8237518370.8148386199926570387.033959184540.80987424300.8140379041928569826.53289462550.87679219980.81063824621268567446.772333597560.86855712690.8174569427723572219.859861847570.82904115930.8141665214840569916.564983016580.8020698640.815686672813570980.670377018590.85427519170.81673610841014571715.275848286600.8222631050.81292150431041569045.053012555610.79963825920.8141851642962569929.614963335620.88051538880.8182959923867572807.194601383630.80816313770.8211014045784574770.983170515640.8349097290.8138343591867569684.051396025650.80923234020.8135642871955569495.000996508660.81468916470.8166150606814571630.542401249670.8077515390.8152375611983570666.292750665680.83081422950.8161747959814571322.357159569690.82205939960.8184548153783572918.370736529700.82160621060.81625312541068571377.187800226710.82410212150.8173236291102572126.540276607720.84305315390.81603048311101571221.33818633730.820075314850.81551615491020570861.308415778740.84247719170.81714681661097572002.771605812750.8054137800.8217908415904575253.589058296760.81827819980.8193869994870573570.899593994770.80946618360.8134999069841569449.934838936780.80158241580.8148874027812570421.181887516790.83911215660.8207814603786574547.022176235800.81773210800.818032183900572622.528131354810.83620627270.8131113951075569177.97651146820.82570124840.822562713785575793.899084821830.83348617280.8147607271132570332.508928349840.85116913500.81026981791048567188.872530209850.802528260.8136224685929569535.727939185860.80579622410.8166284729757571639.931041187870.82370215660.8164574929957571520.245039813880.8192914040.8184974811839572948.23676424890.79806924030.81406617861364182569846.325013876900.83346314580.8151999765756570639.983522204910.87693923760.8170789973929571955.298131466920.86819215930.8190631053810573344.173692373930.80432728080.8167803461071571746.242167929940.8180297560.817087245755571961.071477898950.80203911070.8143144733988570020.131338204960.81418717820.81423119921045569961.83946402970.88445813770.8137847316784569649.312138846980.848909118880.8148818156809570417.270917645990.80206712420.8141686991814569918.0893886471000.86577114310.81250740771011568755.185418922TotalTotalTotalFunc.2188349080844103Min0.7900180.81026981790.810726Average0.8254420860.81627977750.8126178667WeightMin553012.6567188.872530209567508.2Average577809.4602571395.844248077568832.506666667

26DV Comp0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

DOT Optimum Values

PLOTDOT PlanformDIRECT Planform000011.542.38211.542.36623.083.67523.083.63534.624.53134.624.46346.165.06746.164.98557.7015.36557.7015.33765.935.49465.935.49269.2415.52169.2415.52680.7815.50880.7815.52392.3215.38492.3215.412103.865.406103.865.48115.45.507115.45.57126.945.517126.945.548138.485.451138.485.458150.025.51150.025.541175.0255.52175.0255.51200.035.038200.034.914207.935.119207.934.986225.0325.524225.0325.48250.0365.438250.0365.499275.0394.03275.0394.1173000300066.191671.685678.4199.41282.2329.41290.64812.82492.77812.824102.87616.235103.32416.236115.10419.647113.87119.648127.33323.059124.41723.06139.56126.471134.96326.472151.7929.882145.5129.884164.01833.294156.05633.296176.24636.706166.60336.707180.84140.118177.14940.119183.67843.529183.12443.531186.51546.941185.64946.943189.35150.353188.17550.355192.18853.765190.753.767195.02557.176193.22557.179197.86260.588195.7560.591200.69964198.27564.003213.21264209.07264.003212.98460.588209.66360.591212.75757.176210.25457.179212.52953.765210.84453.767212.30150.353211.43550.355212.07446.941212.02546.943211.84643.529212.61643.531211.62640.118213.20740.119212.87636.706213.79736.707214.77133.294214.38833.296216.66529.882214.97929.884218.55926.471215.56926.472220.45423.059216.16423.06222.34819.647218.38719.648224.24316.235221.36416.236226.13712.824224.34112.824228.0319.412227.3189.412229.9266230.2956197.75131.803196.74327.429201.38231.892200.37427.518205.01331.982204.00527.608208.53432.023207.52627.649212.05432.065211.04627.691215.3332.058214.32227.684218.60632.051217.59827.677221.77231.997220.76427.623224.93731.942223.92927.568224.93727.406223.92923.032221.77227.351220.76422.977218.60627.297217.59822.923215.3327.29214.32222.916212.05427.283211.04622.909208.53427.325207.52622.951205.01327.366204.00522.992201.38227.456200.37423.082197.75127.545196.74323.171197.75131.803196.74327.429193.15940.711193.58545.668196.7940.8197.21645.757200.42140.89200.84745.847203.94240.931204.36845.888207.46240.973207.88845.93210.73840.966211.16445.923214.01440.959214.4445.916217.1840.905217.60645.862220.34540.85220.77145.807220.34536.314220.77141.271217.1836.259217.60641.216214.01436.205214.4441.162210.73836.198211.16441.155207.46236.191207.88841.148203.94236.233204.36841.19200.42136.274200.84741.231196.7936.364197.21641.321193.15936.453193.58541.41193.15940.711193.58545.668000011.542.38411.542.36823.083.67823.083.63834.624.53434.624.46746.165.07146.164.98957.7015.3757.7015.34165.935.49865.935.49769.2415.52669.2415.5380.7815.51280.7815.52792.3215.38892.3215.417103.865.411103.865.484115.45.512115.45.574126.945.522126.945.553138.485.455138.485.462150.025.515150.025.545175.0255.524175.0255.515200.035.042200.034.918207.935.123207.934.99225.0325.529225.0325.484250.0365.443250.0365.503275.0394.034275.0394.12130003000000011.54-2.38211.54-2.36623.08-3.67523.08-3.63534.62-4.53134.62-4.46346.16-5.06746.16-4.98557.701-5.36557.701-5.33765.93-5.49465.93-5.49269.241-5.52169.241-5.52680.781-5.50880.781-5.52392.321-5.38492.321-5.412103.86-5.406103.86-5.48115.4-5.507115.4-5.57126.94-5.517126.94-5.548138.48-5.451138.48-5.458150.02-5.51150.02-5.541175.025-5.52175.025-5.51200.03-5.038200.03-4.914207.93-5.119207.93-4.986225.032-5.524225.032-5.48250.036-5.438250.036-5.499275.039-4.03275.039-4.1173000300066.191-671.685-678.419-9.41282.232-9.41290.648-12.82492.778-12.824102.876-16.235103.324-16.236115.104-19.647113.871-19.648127.333-23.059124.417-23.06139.561-26.471134.963-26.472151.79-29.882145.51-29.884164.018-33.294156.056-33.296176.246-36.706166.603-36.707180.841-40.118177.149-40.119183.678-43.529183.124-43.531186.515-46.941185.649-46.943189.351-50.353188.175-50.355192.188-53.765190.7-53.767195.025-57.176193.225-57.179197.862-60.588195.75-60.591200.699-64198.275-64.003213.212-64209.072-64.003212.984-60.588209.663-60.591212.757-57.176210.254-57.179212.529-53.765210.844-53.767212.301-50.353211.435-50.355212.074-46.941212.025-46.943211.846-43.529212.616-43.531211.626-40.118213.207-40.119212.876-36.706213.797-36.707214.771-33.294214.388-33.296216.665-29.882214.979-29.884218.559-26.471215.569-26.472220.454-23.059216.164-23.06222.348-19.647218.387-19.648224.243-16.235221.364-16.236226.137-12.824224.341-12.824228.031-9.412227.318-9.412229.926-6230.295-6197.751-31.803196.743-27.429201.382-31.892200.374-27.518205.013-31.982204.005-27.608208.534-32.023207.526-27.649212.054-32.065211.046-27.691215.33-32.058214.322-27.684218.606-32.051217.598-27.677221.772-31.997220.764-27.623224.937-31.942223.929-27.568224.937-27.406223.929-23.032221.772-27.351220.764-22.977218.606-27.297217.598-22.923215.33-27.29214.322-22.916212.054-27.283211.046-22.909208.534-27.325207.526-22.951205.013-27.366204.005-22.992201.382-27.456200.374-23.082197.751-27.545196.743-23.171197.751-31.803196.743-27.429193.159-40.711193.585-45.668196.79-40.8197.216-45.757200.421-40.89200.847-45.847203.942-40.931204.368-45.888207.462-40.973207.888-45.93210.738-40.966211.164-45.923214.014-40.959214.44-45.916217.18-40.905217.606-45.862220.345-40.85220.771-45.807220.345-36.314220.771-41.2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PLOT0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

DOT DesignDIRECT Design

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DOTDIRECT26 DV case

DIRECT

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DOT DesignDIRECT Design

26DV CompLFOPCV3DOTDIRECTFunctionCallsFunctionCallsFunctionCalls10.8035918360.81783119710130.81316314420.83107612420.813595140210680.81072625830.84316322680.81417943858720.81237917440.80834429700.81703277949580.81236923250.82847416200.81737139578730.81282214560.87859416470.8164087290.81270714670.83378128620.81645731878690.81276214480.8346738640.8149955029870.81263414490.80431412960.81802084889300.81279145100.79566413500.81174764968430.812801145110.83846723220.81494785048670.812823145120.81210318360.818774610910180.812823145130.8442118090.82290449048730.812823145140.7916458320.81397215037260.812823145150.80948720790.82217036286420.812823145160.82630318090.8182432497870170.80551814850.817028395756180.86124811880.8190701404127741519190.81263816740.8165036156811200.80446635100.8244685914789210.82330417820.8196429649785220.825223313770.8169124481811230.86618413230.8176006471900240.82816916200.8132674049961250.85431823220.8149532765991260.80510128620.8169937823962270.83823615120.8138260303899280.81021922680.8158884803874290.81446614850.8137067128989300.81620912150.8159564575731310.80182131050.8191566449668320.8093219440.8150458483843330.85654214850.8153873574899340.80498611880.8179174084898350.81802835640.8179976721839360.81821235640.8143924527785370.84725517550.81290168451018380.81241440230.8154367116840390.81595613500.8127175597840400.81205723220.818871292410.81360537260.8215100689927420.80669425920.8156229101871430.8026631860.8164490334813440.84441416200.8196284043957450.83376815930.8172322813783460.84232611340.81741391171535470.80385228620.8116875111902480.81490838880.8197135779731490.79001835100.8110230124990500.83385717280.81832159081307510.86044632400.816380242959520.80030614850.8150302287896530.8237518370.8148386199926540.80987424300.8140379041928550.87679219980.81063824621268560.86855712690.8174569427723570.82904115930.8141665214840580.8020698640.815686672813590.85427519170.81673610841014600.8222631050.81292150431041610.79963825920.8141851642962620.88051538880.8182959923867630.80816313770.8211014045784640.8349097290.8138343591867650.80923234020.8135642871955660.81468916470.8166150606814670.8077515390.8152375611983680.83081422950.8161747959814690.82205939960.8184548153783700.82160621060.81625312541068710.82410212150.8173236291102720.84305315390.81603048311101730.820075314850.81551615491020740.84247719170.81714681661097750.8054137800.8217908415904760.81827819980.8193869994870770.80946618360.8134999069841780.80158241580.8148874027812790.83911215660.8207814603786800.81773210800.818032183900810.83620627270.8131113951075820.82570124840.822562713785830.83348617280.8147607271132840.85116913500.81026981791048850.802528260.8136224685929860.80579622410.8166284729757870.82370215660.8164574929957880.8192914040.8184974811839890.79806924030.81406617861364900.83346314580.8151999765756910.87693923760.8170789973929920.86819215930.8190631053810930.80432728080.8167803461071940.8180297560.817087245755950.80203911070.8143144733988960.81418717820.81423119921045970.88445813770.8137847316784980.848909118880.8148818156809990.80206712420.81416869918141000.86577114310.81250740771011TotalTotalTotalFunc.2188349080843921Min0.7900180.81026981790.810726Average0.8254420860.81627977750.8126178667WeightMin553012.6567188.872530209567508.2Average577809.4602571395.844248077568832.506666667