v2007.09.13 - Convex Optimization

280 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.4.3.0.6 Example. Variable-vector normalization.Suppose, within some convex optimization problem, we want vector variablesx, y ∈ R N constrained by a nonconvex equality:x‖y‖ = y (669)id est, ‖x‖ = 1 and x points in the same direction as y ; e.g.,minimize f(x, y)x , ysubject to (x, y) ∈ Cx‖y‖ = y(670)where f is some convex function and C is some convex set. We can realizethe nonconvex equality by constraining rank and adding a regularizationterm to the objective. Make the assignment:⎡ ⎤x [x T y T 1]⎡G = ⎣ y ⎦ = ⎣1X Z xZ Y yx T y T 1⎤⎡⎦=∆ ⎣xx T xy T xyx T yy T yx T y T 1⎤⎦∈ S 2N+1 (671)where X , Y ∈ S N , also Z ∈ S N [sic] . Any rank-1 solution must take theform of (671). (B.1) The problem statement equivalent to (670) is thenwrittenminimizeX , Y , Z , x , ysubject to (x, y) ∈ C⎡f(x, y) + ‖X − Y ‖ FG = ⎣rankG = 1(G ≽ 0)tr(X) = 1δ(Z) ≽ 0X Z xZ Y yx T y T 1⎤⎦(672)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.2.0.0.1)To make this program solvable by convex iteration, as explained before, we