Optimization formulation Optimization methods help us find solutions to problem where we seek to...
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Optimization formulation • Optimization methods help us find solutions to problem where we seek to find the best of something. • This lecture is about how we formulate the problem mathematically. • In this course we make the assumption that we have choices and that we can attach numerical values to the ‘goodness’ of each choice. • 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?
Optimization formulation Optimization methods help us find solutions to problem where we seek to find the best of something. This lecture is about how
Slide 2 Optimization formulation Optimization methods help us
find solutions to problem where we seek to find the best of
something. This lecture is about how we formulate the problem
mathematically. In this course we make the assumption that we have
choices and that we can attach numerical values to the goodness of
each choice. 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? Slide 3 Young Modulus Example Slide 4 Direct unconstrained
formulation Minimize maximum difference eps=[1 2 4]; sig=[1 2 3]
e=linspace(0.5,1,101); sigmodel=e'*eps; sigr=ones(101,1)*sig;
diff=abs(sigr-sigmodel); maxdiff=max(diff'); plot(e,maxdiff);
xlabel('E'); ylabel('maximum difference'); Non-smooth function
Difficult to optimize numerically. Slide 5 Constrained formulation
To avoid a non-smooth function, a standard trick is to add a design
variable b that bounds the difference, as well as two constraints
Note that the objective function is equal to one design variable,
and the other variable E appears only in the constraints. Not only
is the new objective and constraints smooth, but they are also
linear. Slide 6 Standard notation The standard form of an
optimization problem used by most textbooks and software is
Standard form of Youngs modulus fit Slide 7 Multiple objectives It
is common to have multiple objectives. For fitting Youngs modulus,
we may have also wanted to minimize the average absolute
difference. avgdiff=mean(diff'); plot(e,maxdiff,avgdiff,'r')
xlabel('E') legend('maxdiff','avgdiff','Location','north') Interval
of interest is (0.75,0.83) Slide 8 Column design example
Vanderplaats, Numerical optimization techniques for engineering
design, 3 rd edition. Slide 9 Design formulation. Slide 10 In
praise of normalization The column design formulation has
normalized constraints Improves understanding: When equal to 0.1 we
know it is 10% violation. Improve numerical conditioning, because
all constraints are of the same order of magnitude. For same
reason, it is good to normalize design variables when they are not
all of the same order, like diameter and thickness.
