ELEG5481 SIGNAL PROCESSING OPTIMIZATION TECHNIQUES 6. GEOMETRIC PROGRAM

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1 ELEG5481 SIGNAL PROCESSING OPTIMIZATION TECHNIQUES 6. GEOMETRIC PROGRAM Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 1

2 Some Basics A function f : R n R with domf = R n ++, defined as f(x) = cx a 1 1 xa 2 2 xa n n where c > 0 & a i R, is called a monomial function. A sum of monomials f(x) = K k=1 c k x a 1k 1 x a 2k 2 x a nk n where c k > 0 & a ik R, is called a posynomial function. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 2

3 The log-sum-exp function f(x) = log(e x e x n ), where domf = R n. The log-sum-exp function is convex on R n. The following extended log-sum-exp function f(x) = log(e at 1 x+b e at m x+b m ), where a i R n & b i R, is also convex on R n. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 3

4 Geometric Program where D = R n ++. min K 0 k=1 c 0,kx a 0,1k 1 x a 0,2k 2 x a 0,nk n s.t. K i k=1 c i,kx a i,1k 1 x a i,2k 2 x a i,nk n 1, i = 1,..., m d i x g i1 1 x g in n = 1, i = 1,..., p The objective & inequality constraint functions are posynomials. The equality constraint functions are monomials. Not a convex optimization problem in general. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 4

5 Geometric Program in a Convex Form Let x i = e y i. A monomial can be re-expressed as where b = log c. cx a 1 1 xa 2 2 xa n n = ce a 1y1 e a ny n = e at y+b Likewise, a posynomial can be re-expressed as where b k = log c k. K k=1 c k x a 1k 1 x a 2k 2 x a nk n = K k=1 e at k y+b k The geometric program can then be transformed to min K 0 k=1 eat 0k y+b 0k s.t. K i k=1 eat ik y+b ik 1, i = 1,..., m e gt i y+h i = 1, i = 1,..., p Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 5

6 By applying the logarithm to all functions, we obtain an equiv. problem that is convex. min log K 0 k=1 eat 0k y+b 0k s.t. log K i k=1 eat ik y+b ik 0, i = 1,..., m g T i y + h i = 0, i = 1,..., p Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 6

7 Example: Optimal Power Assignment Revisited Recall the power assignment problem in the last lesson: j i min max G ijp j + σi 2 i=1,...,k G ii p i s.t. 0 p i p max,i, i = 1,..., K By epigraph reformulation, we may rewrite the problem as min t j i s.t. G ijp j + σi 2 t, i = 1,..., K G ii p i 0 p i p max,i, i = 1,..., K Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 7

8 The problem can be further reformulated as min t s.t. j i G ij G 1 ii p jp 1 i t 1 + σi 2 G 1 ii p 1 i t 1 1, i = 1,..., K p 1 max,i p i 1, i = 1,..., K with D = R K The problem becomes a geometric program. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 8

9 Example: Gate sizing Consider a digital circuit with gates interconnected by wires. The problem is to determine the size of each gate to minimize the delay in the circuit subject to area and power constraints. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 9

10 The total area of all gates is A = n a i x i, i=1 where x i 1 is the scale factor of gate i, and a i is the area of gate i under unit scaling. The total power consumed by all gates is P = n f i e i x i, i=1 where f i is the transition frequency of the gate, and e i is the energy lost in gate transition. The input capacitance C i and the driving resistance R i can be modeled as C i = α i + β i x i, R i = γ i /x i, for some positive constants α i, β i and γ i. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 10

11 The delay of a gate is R i j F (i) D i = C j, if i is not an output gate R i Ci out, if i is an output gate where F (i) is the set of gates with the inputs connected to gate i, and Ci out load capacitance of an output gate. is the The worst-case delay D is the maximum delay of all paths from inputs to outputs, i.e. D = max D i, I O i I where O is the set of all paths from inputs to outputs. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 11

12 In this example, D is given by D = max{d 1 + D 5, D 1 + D 4 + D 6, D 2 + D 4 + D 6, D 3 + D 6 }. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 12

13 The problem can be formulated as min D s.t. A A max, P P max, x i 1, i = 1,..., n. This problem is not a GP, as D is the maximum of some posynomials. But it can be converted to a GP by adding a slack variable t, min t s.t. D i t, for all I O, i I A A max, P P max, x i 1, i = 1,..., n. Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 13

14 Additional Reading S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi, A tutorial on geometric programming, Optimization and Engineering, 2007 M. Chiang, Geometric programming for communication systems, Foundations and Trends in Communications and Information Theory, M. Chiang, C. W. Tan, D. P. Palomar, D. O Neill, and D. Julian, Power control by geometric programming, IEEE Trans. Wireless Commun., Wing-Kin Ma, Dept. Electronic Eng., The Chinese University of Hong Kong 14