Convex optimization examples multi-period processor speed scheduling - PowerPoint PPT Presentation

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 • energy consumed in period t is φ ( s t ) ; total energy is E = � T t =1 φ ( s t ) • n jobs – job i available at t = A i ; must finish by deadline t = D i – job i requires total work W i ≥ 0 • θ ti ≥ 0 is fraction of processor effort allocated to job i in period t D i � 1 T θ t = 1 , θ ti s t ≥ W i t = A i • choose speeds s t and allocations θ ti to minimize total energy E 2

Minimum energy processor speed scheduling • work with variables S ti = θ ti s t D i n � � s t = S ti , S ti ≥ W i i =1 t = A i • solve convex problem E = � T minimize t =1 φ ( s t ) s min ≤ s t ≤ s max , subject to t = 1 , . . . , T s t = � n i =1 S ti , t = 1 , . . . , T � D i t = A i S ti ≥ W i , i = 1 , . . . , n • a convex problem when φ is convex • can recover θ ⋆ t as θ ⋆ ti = (1 /s ⋆ t ) S ⋆ ti 3

Example • T = 16 periods, n = 12 jobs • s min = 1 , s max = 6 , φ ( s t ) = s 2 t • jobs shown as bars over [ A i , D i ] with area ∝ W i 40 12 35 10 30 25 8 φ ( s t ) job i 20 6 15 4 10 2 5 0 0 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 16 18 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 E ⋆ = 167 . 1 • optimal schedule: S ⋆ optimal uniform 6 6 5 5 4 4 s t s t 3 3 2 2 1 1 0 0 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 t t 5

Minimum-time optimal control • linear dynamical system: x 0 = x init x t +1 = Ax t + Bu t , t = 0 , 1 , . . . , K, • inputs constraints: u min � u t � u max , t = 0 , 1 , . . . , K • minimum time to reach state x des : f ( u 0 , . . . , u K ) = min { T | x t = x des for T ≤ t ≤ K + 1 } 6

state transfer time f is quasiconvex function of ( u 0 , . . . , u K ) : f ( u 0 , u 1 , . . . , u K ) ≤ T if and only if for all t = T, . . . , K + 1 x t = A t x init + A t − 1 Bu 0 + · · · + Bu t − 1 = x des i.e. , sublevel sets are affine minimum-time optimal control problem: minimize f ( u 0 , u 1 , . . . , u K ) subject to u min � u t � u max , t = 0 , . . . , K with variables u 0 , . . . , u K a quasiconvex problem; can be solved via bisection 7

Minimum-time control example u 1 u 2 • force ( u t ) 1 moves object modeled as 3 masses (2 vibration modes) • force ( u t ) 2 used for active vibration suppression • goal: move object to commanded position as quickly as possible, with | ( u t ) 1 | ≤ 1 , | ( u t ) 2 | ≤ 0 . 1 , t = 0 , . . . , K 8

Ignoring vibration modes • treat object as single mass; apply only u 1 • analytical (‘bang-bang’) solution 3 1 0.5 ( u t ) 1 2.5 0 −0.5 2 −1 ( x t ) 3 −2 0 2 4 6 8 10 12 14 16 18 20 t 1.5 0.1 1 ( u t ) 2 0.05 0 0.5 −0.05 −0.1 0 −2 0 2 4 6 8 10 12 14 16 18 20 −2 0 2 4 6 8 10 12 14 16 18 20 t t 9

With vibration modes • no analytical solution • a quasiconvex problem; solved using bisection 3 1 ( u t ) 1 0.5 2.5 0 −0.5 2 −1 ( x t ) 3 −2 0 2 4 6 8 10 12 14 16 18 20 t 1.5 0.1 1 0.05 ( u t ) 2 0 0.5 −0.05 −0.1 0 −2 0 2 4 6 8 10 12 14 16 18 20 −2 0 2 4 6 8 10 12 14 16 18 20 t t 10

Grasp force optimization • choose K grasping forces on object – resist external wrench – respect friction cone constraints – minimize maximum grasp force • convex problem (second-order cone program): max i � f ( i ) � 2 minimize max contact force i Q ( i ) f ( i ) = f ext subject to � force equillibrium i p ( i ) × ( Q ( i ) f ( i ) ) = τ ext � torque equillibrium � 1 / 2 � µ i f ( i ) f ( i )2 + f ( i )2 ≥ friction cone constraints 3 1 2 variables f ( i ) ∈ R 3 , i = 1 , . . . , K (contact forces) 11

Example 12

Optimal broadcast transmitter power allocation • m transmitters, mn receivers all at same frequency • transmitter i wants to transmit to n receivers labeled ( i, j ) , j = 1 , . . . , n • A ijk is path gain from transmitter k to receiver ( i, j ) • N ij is (self) noise power of receiver ( i, j ) • variables: transmitter powers p k , k = 1 , . . . , m transmitter k receiver ( i, j ) transmitter i 13

at receiver ( i, j ) : • signal power: S ij = A iji p i • noise plus interference power: � I ij = A ijk p k + N ij k � = i • signal to interference/noise ratio (SINR): S ij /I ij problem: choose p i to maximize smallest SINR: A iji p i maximize min � k � = i A ijk p k + N ij i,j subject to 0 ≤ p i ≤ p max . . . a (generalized) linear fractional program 14

Phased-array antenna beamforming ( x i , y i ) θ • omnidirectional antenna elements at positions ( x 1 , y 1 ) , . . . , ( x n , y n ) • unit plane wave incident from angle θ induces in i th element a signal e j ( x i cos θ + y i sin θ − ωt ) ( j = √− 1 , frequency ω , wavelength 2 π ) 15

• demodulate to get output e j ( x i cos θ + y i sin θ ) ∈ C • linearly combine with complex weights w i : n � w i e j ( x i cos θ + y i sin θ ) y ( θ ) = i =1 • y ( θ ) is (complex) antenna array gain pattern • | y ( θ ) | gives sensitivity of array as function of incident angle θ • depends on design variables Re w , Im w (called antenna array weights or shading coefficients ) design problem: choose w to achieve desired gain pattern 16

Sidelobe level minimization make | y ( θ ) | small for | θ − θ tar | > α ( θ tar : target direction; 2 α : beamwidth) via least-squares (discretize angles) i | y ( θ i ) | 2 � minimize subject to y ( θ tar ) = 1 (sum is over angles outside beam) least-squares problem with two (real) linear equality constraints 17

50 ◦ | y ( θ ) | ❅ θ tar = 30 ◦ ❅ ❅ ❘ 10 ◦ sidelobe level ❅ ❅ ❘ 18

minimize sidelobe level (discretize angles) minimize max i | y ( θ i ) | subject to y ( θ tar ) = 1 (max over angles outside beam) can be cast as SOCP minimize t subject to | y ( θ i ) | ≤ t y ( θ tar ) = 1 19

50 ◦ | y ( θ ) | ❅ θ tar = 30 ◦ ❅ ❅ ❘ 10 ◦ sidelobe level ❅ ❅ ❘ 20

Extensions convex (& quasiconvex) extensions: • y ( θ 0 ) = 0 (null in direction θ 0 ) • w is real (amplitude only shading) • | w i | ≤ 1 (attenuation only shading) i =1 | w i | 2 (thermal noise power in y ) • minimize σ 2 � n • minimize beamwidth given a maximum sidelobe level nonconvex extension: • maximize number of zero weights 21

Optimal receiver location • N transmitter frequencies 1 , . . . , N • transmitters at locations a i , b i ∈ R 2 use frequency i • transmitters at a 1 , a 2 , . . . , a N are the wanted ones • transmitters at b 1 , b 2 , . . . , b N are interfering • receiver at position x ∈ R 2 q b 3 a 3 ❛ a 2 q b 2 ❛ x a 1 ❛ q b 1 22

• (signal) receiver power from a i : � x − a i � − α ( α ≈ 2 . 1 ) 2 • (interfering) receiver power from b i : � x − b i � − α ( α ≈ 2 . 1 ) 2 • worst signal to interference ratio, over all frequencies, is � x − a i � − α 2 S / I = min � x − b i � − α i 2 • what receiver location x maximizes S / I? 23

S/I is quasiconcave on { x | S/I ≥ 1 } , i.e. , on { x | � x − a i � 2 ≤ � x − b i � 2 , i = 1 , . . . , N } q b 3 a 3 ❛ q b 2 a 2 ❛ a 1 ❛ q b 1 can use bisection; every iteration is a convex quadratic feasibility problem 24