v2010.10.26 - Convex Optimization

236 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.3.1 fractional power[149] To implement an objective of the form x α for positive α , we quantizeα and work instead with that approximation. Choose nonnegative integer qfor adequate quantization of α like so:α k 2 q (543)where k ∈{0, 1, 2... 2 q −1}. Any k from that set may be written∑k= q b i 2 i−1 where b i ∈ {0, 1}. Define vector y=[y i , i=0... q] with y 0 =1:i=13.3.1.1 positiveThen we have the equivalent semidefinite program for maximizing a concavefunction x α , for quantized 0≤α 0x ∈ C≡maximizex∈R , y∈R q+1subject toy q[yi−1 y iy ix b i]≽ 0 ,i=1... qx ∈ C (544)where nonnegativity of y q is enforced by maximization; id est,[ ]x > 0, y q ≤ x α yi−1 y i⇔≽ 0 , i=1... q (545)3.3.1.2 negativeIt is also desirable to implement an objective of the form x −α for positive α .The technique is nearly the same as before: for quantized 0≤α 0x ∈ C≡y iminimizex , z∈R , y∈R q+1subject tox b iz[yi−1 y iy ix b i][ ]z 1≽ 01 y q≽ 0 ,i=1... qx ∈ C (546)