MIT6_079F09_lec10

MIT6_079F09_lec10 - Convex optimization examples...

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Unformatted text preview: Convex optimization examples multi-period processor speed scheduling minimum time optimal control grasp force optimization optimal broadcast transmitter power allocation phased-array antenna beamforming optimal receiver location 1 Multi-period processor speed scheduling processor adjusts its speed s t [ s min , s max ] in each of T time periods T energy consumed in period t is ( s t ) ; total energy is E = ( s t ) t =1 n jobs job i available at t = A i ; must finish by deadline t = D i job i requires total work W i ti is fraction of processor effort allocated to job i in period t D i 1 T t = 1 , ti s t W i t = A i choose speeds s t and allocations ti to minimize total energy E 2 Minimum energy processor speed scheduling work with variables S ti = ti s t n D i s t = S ti , S ti W i i =1 t = A i solve convex problem T minimize E = t =1 ( s t ) subject to s min s n t s max , t = 1 , . . . , T s t = i =1 S ti , t = 1 , . . . , T D i S ti W i , i = 1 , . . . , n t = A i a convex problem when is convex can recover t as = (1 /s t ) S ti ti 3 Example T = 16 periods, n = 12 jobs s min = 1 , s max = 6 , ( s t ) = s 2 t jobs shown as bars over [ A i , D i ] with area W i 40 12 35 10 30 25 0 2 4 6 8 10 12 14 16 18 t ( s t ) 20 15 10 5 0 job i 8 6 4 2 0 1 2 3 4 5 6 7 s t t 4 Optimal and uniform schedules uniform schedule: S ti = W i / ( D i A i + 1) ; gives E unif = 204 . 3 ti ; gives optimal schedule: S E = 167 . 1 optimal uniform 6 6 5 5 4 4 0 2 4 6 8 10 12 14 16 18 s t 3 2 1 0 0 2 4 6 8 10 12 14 16 18 s t 3 2 1 0 t t 5 Minimum-time optimal control linear dynamical system: x t +1 = Ax t + Bu t , t...
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MIT6_079F09_lec10 - Convex optimization examples...

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