v2010.10.26 - Convex Optimization

364 CHAPTER 4. SEMIDEFINITE PROGRAMMINGterm to the objective. Make the assignment:⎡ ⎤ ⎡ ⎤ ⎡x [ x T y T 1] X Z x xx T xy T xG = ⎣ y ⎦ = ⎣ Z Y y ⎦⎣yx T yy T y1x T y T 1 x T y T 1⎤⎦∈ S 2N+1 (814)where X , Y ∈ S N , also Z ∈ S N [sic] . Any rank-1 solution must take theform of (814). (B.1) The problem statement equivalent to (813) is thenwrittenminimizeX , Y ∈ S , Z , x , ysubject tof(x, y) + ‖X − Y ‖ F(x, y) ∈ C⎡ ⎤X Z xG = ⎣ Z Y y ⎦(≽ 0)x T y T 1rankG = 1tr(X) = 1δ(Z) ≽ 0(815)The trace constraint on X normalizes vector x while the diagonal constrainton Z maintains sign between respective entries of x and y . Regularizationterm ‖X −Y ‖ F then makes x equal to y to within a real scalar; (C.2.0.0.2)in this case, a positive scalar. To make this program solvable by convexiteration, as explained in Example 4.4.1.2.4 and other previous examples, wemove the rank constraint to the objectiveminimizeX , Y , Z , x , ysubject to (x, y) ∈ C⎡f(x, y) + ‖X − Y ‖ F + 〈G, W 〉G = ⎣tr(X) = 1δ(Z) ≽ 0X Z xZ Y yx T y T 1⎤⎦≽ 0by introducing a direction matrix W found from (1700a):(816)minimizeW ∈ S 2N+1 〈G ⋆ , W 〉subject to 0 ≼ W ≼ Itr W = 2N(817)