v2010.10.26 - Convex Optimization

360 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.6.0.0.5 Example. Tractable polynomial constraint.The set of all coefficients for which a multivariate polynomial were convex isgenerally difficult to determine. But the ability to handle rank constraintsmakes any nonconvex polynomial constraint transformable to a convexconstraint. All optimization problems having polynomial objective andpolynomial constraints can be reformulated as a semidefinite program witha rank-1 constraint. [285] Suppose we requireIdentify3 + 2x − xy ≤ 0 (798)⎡ ⎤ ⎡ ⎤x [ x y 1] x 2 xy xG = ⎣ y ⎦ = ⎣ xy y 2 y ⎦∈ S 3 (799)1x y 1Then nonconvex polynomial constraint (798) is equivalent to constraint settr(GA) ≤ 0G 33 = 1(G ≽ 0)rankG = 1(800)with direct correspondence to sense of trace inequality where G is assumedsymmetric (B.1.0.2) and⎡ ⎤0 − 1 12A = ⎣ − 1 0 0 ⎦∈ S 3 (801)21 0 3Then the method of convex iteration from4.4.1 is applied to implement therank constraint.4.6.0.0.6 Exercise. Binary Pythagorean theorem.Technique of Example 4.6.0.0.5 is extensible to any quadratic constraint; e.g.,x T Ax + 2b T x + c ≤ 0 , x T Ax + 2b T x + c ≥ 0 , and x T Ax + 2b T x + c = 0.Write a rank-constrained semidefinite program to solve (Figure 105){ x + y = 1x 2 + y 2 (802)= 1whose feasible set is not connected. Implement this system in cvx [167] byconvex iteration.