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In this problem we study a natural method for blending the two norms, by using the ℓ1.5 norm,
deﬁned as
2/3 k z 1.5 =
i=1  zi  3/2 for z ∈ Rk . We will consider the simplest approximation or regression problem:
minimize Ax − b 1.5 , with variable x ∈ Rn , and problem data A ∈ Rm×n and b ∈ Rm . We will assume that m > n and
the A is full rank (i.e., rank n). The hope is that this ℓ1.5 optimal approximation problem should
share some of the good features of ℓ2 and ℓ1 approximation.
(a) Give optimality conditions for this problem. Try to make these as simple as possible.
(b) Explain how to formulate the ℓ1.5 norm approximation problem as an SDP. (Your SDP can
include linear equality and inequality constraints.)
(c) Solve the speciﬁc numerical instance generated by the following code:
randn(’state’,0);
A=randn(100,30);
b=randn(100,1);
Numerically verify the optimality conditions. Give a histogram of the residuals, and repeat
for the ℓ2 norm and ℓ1 norm approximations. You can use any method you like to solve the...
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 Fall '13
 F.Borrelli
 The Aeneid

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