MIT 6 079 - Convex optimization examples (25 pages)

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Convex optimization examples



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Convex optimization examples

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Pages:
25
School:
Massachusetts Institute of Technology
Course:
6 079 - Introduction to Convex Optimization

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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 st smin smax in each of T time periods energy consumed in period t is st total energy is E T t 1 st n jobs job i available at t Ai must nish by deadline t Di job i requires total work Wi 0 ti 0 is fraction of processor e ort allocated to job i in period t 1T t 1 Di t Ai tist Wi choose speeds st and allocations ti to minimize total energy E 2 Minimum energy processor speed scheduling work with variables Sti tist st n Sti i 1 Di t Ai Sti Wi solve convex problem T minimize E t 1 st max subject to smin s s t 1 T t n st i 1 Sti t 1 T Di i 1 n t Ai Sti Wi a convex problem when is convex can recover t as ti 1 s t Sti 3 Example T 16 periods n 12 jobs smin 1 smax 6 st s2t jobs shown as bars over Ai Di with area Wi 40 12 30 10 25 8 job i t st 35 20 6 15 4 10 2 5 0 0 1 2 3 4 st 5 6 7 0 0 2 4 6 8 10 12 14 16 18 t 4 Optimal and uniform schedules uniform schedule Sti Wi Di Ai 1 gives E unif 204 3 optimal schedule Sti gives E 167 1 uniform 6 5 5 4 4 st st optimal 6 3 3 2 2 1 1 0 0 2 4 6 8 10 t 12 14 16 18 0 0 2 4 6 8 10 12 14 16 18 t 5 Minimum time optimal control linear dynamical system xt 1 Axt But t 0 1 K x0 xinit inputs constraints umin ut umax t 0 1 K minimum time to reach state xdes f u0 uK min T xt xdes for T t K 1 6 state transfer time f is quasiconvex function of u0 uK f u0 u1 uK T if and only if for all t T K 1 xt Atxinit At 1Bu0 But 1 xdes i e sublevel sets are a ne minimum time optimal control problem minimize f u0 u1 uK subject to umin ut umax t 0 K with variables u0 uK a quasiconvex problem can be solved via bisection 7 Minimum time control example u1 u2 force ut 1 moves object modeled as 3 masses 2 vibration modes force ut 2 used for active vibration suppression goal move



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