Classification of algorithms The DIRECT algorithm Divided rectangles Exploration and Exploitation as biobjective optimization Application to High Speed Civil Transport Global optimization issues ID: 732311
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Slide1
Global Optimization
General issues in global optimizationClassification of algorithmsThe DIRECT algorithmDivided rectanglesExploration and Exploitation as bi-objective optimizationApplication to High Speed Civil TransportSlide2
Global optimization issues
Optimization 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.Slide3
Global Optimization
Global optimization algorithms by Thomas WeiseSlide4
Classification of global optimization algorithms
The 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.Slide5
Problems global optimization
What 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?Answers in the notes page.Slide6
Lipschitzian
Optimization
DIRECT was inspired by
Lipschitzian
optimization.
Optimizer
divides space into boxes and samples the vertices of each
A
box is further divided based on
estimated maximum
rate of change of the function,
K (
Lipschitz
constant)Slide7
DIRECT
The
function value at the middle of each box and it’s largest dimension are used to determine potentially optimal boxes
Each potentially optimal box is divided
Lipschitzian
optimization for all possible
Lipschitz
constantsSlide8
DIRECT Box Division
.Slide9
Exploration vs. Exploitation
DIRECT 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 spaces
Cox’s paper compares DIRECT to repeated local optimization with random startSlide10
Problems DIRECT
What is the Lipschtiz constant? What value would be appropriate for the function x2+x in the interval (-2,2)? Solution
Invent function values at every point where the function is evaluated in Slide 8 that are consistent with the diagram
.
Solution
What are the meanings of the term exploration and exploitation in the context of global optimization
?
SolutionSlide11
Optimization of a High Speed Civil Transport
26 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.Slide12
ResultsSlide13
ResultsSlide14
ResultsSlide15
Results