Transcript Optimization formulation

Optimization formulation
• Optimization methods help us find solutions to problems where we
seek to find the best of something.
• This lecture is about how we formulate the problem mathematically.
• In this lecture we make the assumption that we have choices and that
we can attach numerical values to the ‘goodness’ of each alternative.
• This is not always the case. We may have problems where the only
thing we can do is compare pairs of alternatives and tell which one is
better, but not by how much.
• Can you think of an example?
Problems (optimization formulation)
• Provide two formulations for minimizing the surface area of a cylinder
of a given volume when the diameter and height are the design
variables. One formulation should use the volume as equality
constraint, and another use it to reduce the number of design
variables.
• You need to go from point A to point B in minimum time while
maintaining a safe distance from point C. Formulate an optimization
problem to find the path with no more than three design variables
when A=(0,0), B=(10,10), C=(4,4), and the minimum safe distance is 7.
• Normalize the constraint.
Kriging questions
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What does negative correlation between two random variables mean?
How do you decide whether the data you fit is sparse or dense?
What does this figure show?
What are the key differences between
kriging and linear regression?
What are the similarities?
Two random variables X, Y were sampled
X-sample [ 1 2 3], Y-sample [ 0,2,4]
Calculate correlation coefficient.
EGO questions
• What is “expected improvement” in EGO. How is it different from the
literal definition of the phrase?
• How do you determine whether a point selected by EGO is an
exploration point or an exploitation point? Can it be both?
• EGO shoots from compromise between exploration and exploitation.
What compromise is sought by EGRA?
• Why do we need more accurate constraint when it is near its
boundary?
• What is the meaning of “feasibility” in ‘expected feasibility?’
SOURCE:
http://www.sz-wholesale.com/uploadFiles/041022104413s.jpg
Monte Carlo Simulation
• Given a random variable X and
a limit state function g(X):
sample X: [x1,x2,…,xn];
Calculate [g(x1),g(x2),…,g(xn)]; use to approximate statistics of g.
• Example: X is U[0,1]. Use MCS to find mean of X2
x=rand(10); y=x.^2; %generates 10x10 random matrix
sumy=sum(y)/10
sumy =0.4017 0.5279 0.1367 0.3501 0.3072 0.3362 0.3855
0.3646 0.5033 0.2666
sum(sumy)/10 ans =0.3580
• What is the true mean
SOURCE: http://schools.sd68.bc.ca/ed611/akerley/question.jpg
Top Hat question
• Sampling a distribution with 10,000 points, the mean of the sample
was 6, the standard deviation of the sample was 2, and 100 points
were negative. Estimate the noise (standard deviation) in the mean
and number of negative points over repeated 10,000 samples.
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0.02, 10
0.2,1
0.02,1
0.2,10
FORM questions
• What is the reliability index? If X is the standard normal variable, and
failure means X>2, what is the reliability index? What is approximately
the probability of failure?
• If X, and Y are two standard normal variables, and failure is define as
3x+4y>5, what is the reliability index? What is approximately the
probability of failure?
• Top Hat: For the beam example, the error in estimating the reliability
index was due to non-linarity? Non-normality? Both?
• What is the Most Probable Point? Draw the constraint and MPP for
the constraint of the second bullet.
• What is the objective of the equivalent normal transformation?
Risk allocation and RBDO questions
• What are the considerations in allocating risk between failure modes?
• Given two independent random variables X=N(0,1) and Y=N(0,22) we
have failure when X>1 and when Y>2. Estimate the probability of
failure. If you can change the mean of one of the variables by one
unit, which will you change to achieve the most reduction in failure
probability.
• Explain the difference between the stochastic, analysis, and design
response surfaces (aka surrogates) used in the design of the cryogenic
fuel tank.
• What guideline was used to choose the number of data points used
to fit the surrogates?
Uncertainty budget questions
• In evaluating the strength of a structural element, what uncertainties
are encountered?
• Which are aleatory, and which are epistemic?
• Which uncertainties are addressed by coupon tests, and which
uncertainties by element tests?
• Explain the meaning of terms in the table.