# additional_exercises_sol_49.xlsx - (a Using the Schur...

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1 (a) Using the Schur complement theorem we can write the prob minimize t subject to > 0 C 6 with variables rr, t. Note that the two problems are not quite eq the domain, i.e., for points x with F(x) positive semidefinite but linear matrix inequality in the SDP given above is equivalent to F{x) > 0, c e TZ{F{x)\ c r F{x)^c < t where F is the pseudo-inverse (see page 651 of the textboo equivalent to minimize c r F(rr)^c subject to F(x) > 0 c e R(F(z)). If F(x) is positive semidefinite but singular, and c e the o is finite, whereas it is +oo in the original problem. However this value of the problem (unless the set {x \ F(x) >- 0} is empty). As an example, consider c = 0 x 0 0 1 — x Then the problem in the assignment is to minimize 1/rr, with do