Unformatted text preview: he simplest is that f is a single, ﬁxed, known loading. In more
sophisticated formulations, the loading f might be a random vector with known distribution,
or known only to lie in some set F , etc.
Show that each of the following four problems is a convex optimization problem, with x as
variable.
• Design for a ﬁxed known loading. The vector f is known and ﬁxed. The design problem
is
minimize E (x, f )
subject to l ≤ xi ≤ u, i = 1, . . . , m
Wtot (x) ≤ W.
• Design for multiple loadings. The vector f can take any of N known values f (i) , i =
1, . . . , N , and we are interested in the worstcase scenario. The design problem is
minimize maxi=1,...,N E (x, f (i) )
subject to l ≤ xi ≤ u, i = 1, . . . , m
Wtot (x) ≤ W.
• Design for worstcase, unknown but bounded load. Here we assume the vector f can take
arbitrary values in a ball B = {f  f ≤ α}, for a given value of α. We are interested in
minimizing the worstcase stored energy, i.e.,
minimize sup f ≤α E (x, f (i) ) subject to l ≤ xi ≤ u, i = 1, . . . , m
Wtot (x) ≤ W.
118 • Design for a random load with known statistics. We can also use a stochastic model of the
uncertainty in the...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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