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Unformatted text preview: (1) minimize Ax − b 2
1− x ∞ (2) and the very closely related problem In both problems, the variable is x ∈ Rn , and the data are A ∈ Rm×n and b ∈ Rm . Note that
the only diﬀerence between problem (1) and (2) is the square in the numerator. In both problems,
the constraint x ∞ < 1 is implicit. You can assume that b ∈ R(A), in which case the constraint
x ∞ < 1 can be replaced with x ∞ ≤ 1.
Answer the following two questions, for each of the two problems. (So you will answer four questions
all together.) (a) Is the problem, exactly as stated (and for all problem data), convex? If not, is it quasiconvex?
Justify your answer.
(b) Explain how to solve the problem. Your method can involve an SDP solver, an SOCP solver,
an LP solver, or any combination. You can include a one-parameter bisection, if necessary.
(For example, you can solve the problem by bisection on a parameter, where each iteration
consists of solving an SOCP feasibility problem.)
Give the best method you can. In judging best, we...
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