v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 3614.6.0.0.7 Example. Higher-order polynomials.Consider nonconvex problem from Canadian Mathematical Olympiad 1999:findx , y , z∈Rx, y , zsubject to x 2 y + y 2 z + z 2 x = 223 3x + y + z = 1x, y , z ≥ 0(803)We wish to solve for what is known to be a tight upper bound 223 3on the constrained polynomial x 2 y + y 2 z + z 2 x by transformation to arank-constrained semidefinite program; identify⎡ ⎤⎡⎤x [x y z 1] x 2 xy zx xG = ⎢ y⎥⎣ z ⎦ = ⎢ xy y 2 yz y⎥⎣ zx yz z 2 z ⎦ ∈ S4 (804)1x y z 1⎡ ⎤⎡⎤x 2 [ x 2 y 2 z 2 x y z 1] x 4 x 2 y 2 z 2 x 2 x 3 x 2 y zx 2 x 2y 2x 2 y 2 y 4 y 2 z 2 xy 2 y 3 y 2 z y 2z 2z 2 x 2 y 2 z 2 z 4 z 2 x yz 2 z 3 z 2X =x=x 3 xy 2 z 2 x x 2 xy zx x∈ S 7⎢ y⎥⎢ x 2 y y 3 yz 2 xy y 2 yz y⎥⎣ z ⎦⎣ zx 2 y 2 z z 3 zx yz z 2 z ⎦1x 2 y 2 z 2 x y z 1(805)then apply convex iteration (4.4.1) to implement rank constraints:findA , C∈S , bbsubject to tr(XE) = 22[3 3] A bG =b T (≽ 0)1⎡ [ ] ⎤ δ(A)X = ⎣ C[bδ(A) T b ] ⎦(≽ 0)T 1(806)1 T b = 1b ≽ 0rankG = 1rankX = 1