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Optimization formulation

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- 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?

- 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.

- 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
krigingand 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.

- 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:

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- 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

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- 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.
- 0.02, 10
- 0.2,1
- 0.02,1
- 0.2,10

- 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?

- 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?

- 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.