Decomposition Methods

Transcription

1 Decomposition Methods separable problems, complicating variables primal decomposition dual decomposition complicating constraints general decomposition structures Prof. S. Boyd, EE364b, Stanford University

2 Separable problem minimize f 1 (x 1 )+f 2 (x 2 ) subject to x 1 C 1, x 2 C 2 we can solve for x 1 and x 2 separately (in parallel) even if they are solved sequentially, this gives advantage if computational effort is superlinear in problem size called separable or trivially parallelizable generalizes to any objective of form Ψ(f 1,f 2 ) with Ψ nondecreasing (e.g., max) Prof. S. Boyd, EE364b, Stanford University 1

3 Complicating variable consider unconstrained problem, x = (x 1,x 2,y) minimize f(x) = f 1 (x 1,y)+f 2 (x 2,y) y is the complicating variable or coupling variable; when it is fixed the problem is separable in x 1 and x 2 x 1, x 2 are private or local variables; y is a public or interface or boundary variable between the two subproblems Prof. S. Boyd, EE364b, Stanford University 2

4 Primal decomposition fix y and define subproblem 1: minimize x1 f 1 (x 1,y) subproblem 2: minimize x2 f 2 (x 2,y) with optimal values φ 1 (y) and φ 2 (y) original problem is equivalent to master problem minimize y φ 1 (y)+φ 2 (y) with variable y called primal decomposition since master problem manipulates primal (complicating) variables Prof. S. Boyd, EE364b, Stanford University 3

5 if original problem is convex, so is master problem can solve master problem using bisection (if y is scalar) gradient or Newton method (if φ i differentiable) subgradient, cutting-plane, or ellipsoid method each iteration of master problem requires solving the two subproblems (in parallel) if master algorithm converges fast enough and subproblems are sufficiently easier to solve than original problem, we get savings Prof. S. Boyd, EE364b, Stanford University 4

6 Primal decomposition algorithm (using subgradient algorithm for master) repeat 1. Solve the subproblems (in parallel). Find x 1 that minimizes f 1 (x 1,y), and a subgradient g 1 φ 1 (y). Find x 2 that minimizes f 2 (x 2,y), and a subgradient g 2 φ 2 (y). 2. Update complicating variable. y := y α k (g 1 +g 2 ). step length α k can be chosen in any of the standard ways Prof. S. Boyd, EE364b, Stanford University 5

7 x 1,x 2 R 20, y R Example f i are PWL (max of 100 affine functions each); f φ 1 (y) φ 2 (y) φ 1 (y) + φ 2 (y) y Prof. S. Boyd, EE364b, Stanford University 6

8 primal decomposition, using bisection on y f (k) f k Prof. S. Boyd, EE364b, Stanford University 7

9 Dual decomposition Step 1: introduce new variables y 1, y 2 minimize f(x) = f 1 (x 1,y 1 )+f 2 (x 2,y 2 ) subject to y 1 = y 2 y 1, y 2 are local versions of complicating variable y y 1 = y 2 is consensus constraint Prof. S. Boyd, EE364b, Stanford University 8

10 Step 2: form dual problem L(x 1,y 1,x 2,y 2 ) = f 1 (x 1,y 1 )+f 2 (x 2,y 2 )+ν T (y 1 y 2 ) separable; can minimize over (x 1,y 1 ) and (x 2,y 2 ) separately g 1 (ν) = inf x 1,y 1 ( f1 (x 1,y 1 )+ν T y 1 ) = f 1 (0, ν) g 2 (ν) = inf x 2,y 2 ( f2 (x 2,y 2 ) ν T y 2 ) = f 2 (0,ν) dual problem is: maximize g(ν) = g 1 (ν)+g 2 (ν) computing g i (ν) are the dual subproblems can be done in parallel a subgradient of g is y 2 y 1 (from solutions of subproblems) Prof. S. Boyd, EE364b, Stanford University 9

11 Dual decomposition algorithm (using subgradient algorithm for master) repeat 1. Solve the dual subproblems (in parallel). Find x 1, y 1 that minimize f 1 (x 1,y 1 )+ν T y 1. Find x 2, y 2 that minimize f 2 (x 2,y 2 ) ν T y Update dual variables (prices). ν := ν α k (y 2 y 1 ). step length α k can be chosen in standard ways at each step we have a lower bound g(ν) on p iterates are generally infeasible, i.e., y 1 y 2 Prof. S. Boyd, EE364b, Stanford University 10

12 Finding feasible iterates reasonable guess of feasible point from (x 1,y 1 ), (x 2,y 2 ): (x 1,ȳ), (x 2,ȳ), ȳ = (y 1 +y 2 )/2 projection onto feasible set y 1 = y 2 gives upper bound p f 1 (x 1,ȳ)+f 2 (x 2,ȳ) a better feasible point: replace y 1, y 2 with ȳ and solve primal subproblems minimize x1 f 1 (x 1,ȳ), minimize x2 f 2 (x 2,ȳ) gives (better) upper bound p φ 1 (ȳ)+φ 2 (ȳ) Prof. S. Boyd, EE364b, Stanford University 11

13 (Same) example g 1 (ν) g 2 (ν) g 1 (ν) + g 2 (ν) ν Prof. S. Boyd, EE364b, Stanford University 12

14 dual decomposition convergence (using bisection on ν) better bound worse bound g(ν) k Prof. S. Boyd, EE364b, Stanford University 13

15 Interpretation y 1 is resources consumed by first unit, y 2 is resources generated by second unit y 1 = y 2 is consistency condition: supply equals demand ν is a set of resource prices master algorithm adjusts prices at each step, rather than allocating resources directly (primal decomposition) Prof. S. Boyd, EE364b, Stanford University 14

16 Recovering the primal solution from the dual iterates in dual decomposition: ν (k), (x (k) 1,y(k) 1 ), (x(k) 2,y(k) 2 ) x (k) 1,y(k) 1 is minimizer of f 1 (x 1,y 1 )+ν (k)t y 1 found in subproblem 1 x (k) 2,y(k) 2 is minimizer of f 2 (x 2,y 2 ) ν (k)t y 2 found in subproblem 2 ν (k) ν (i.e., we have price convergence) subtlety: we need not have y (k) 1 y (k) 2 0 the hammer: if f i strictly convex, we have y (k) 1 y (k) 2 0 can fix allocation (i.e., compute φ i ), or add regularization terms ǫ y i 2 Prof. S. Boyd, EE364b, Stanford University 15

17 Decomposition with constraints can also have complicating constraints, as in f i, h i, C i convex minimize f 1 (x 1 )+f 2 (x 2 ) subject to x 1 C 1, x 2 C 2 h 1 (x 1 )+h 2 (x 2 ) 0 h 1 (x 1 )+h 2 (x 2 ) 0 is a set of p complicating or coupling constraints, involving both x 1 and x 2 can interpret coupling constraints as limits on resources shared between two subproblems Prof. S. Boyd, EE364b, Stanford University 16

18 fix t R p and define subproblem 1: subproblem 2: Primal decomposition minimize f 1 (x 1 ) subject to x 1 C 1, h 1 (x 1 ) t minimize f 2 (x 2 ) subject to x 2 C 2, h 2 (x 2 ) t t is the quantity of resources allocated to first subproblem ( t is allocated to second subproblem) master problem: minimize φ 1 (t)+φ 2 (t) (optimal values of subproblems) over t subproblems can be solved separately (in parallel) when t is fixed Prof. S. Boyd, EE364b, Stanford University 17

19 Primal decomposition algorithm repeat 1. Solve the subproblems (in parallel). Solve subproblem 1, finding x 1 and λ 1. Solve subproblem 2, finding x 2 and λ Update resource allocation. t := t α k (λ 2 λ 1 ). λ i is an optimal Lagrange multiplier associated with resource constraint in subproblem i λ 2 λ 1 (φ 1 +φ 2 )(t) α k is an appropriate step size all iterates are feasible (when subproblems are feasible) Prof. S. Boyd, EE364b, Stanford University 18

20 Example x 1,x 2 R 20, t R 2 ; f i are quadratic, h i are affine, C i are polyhedra defined by 100 inequalities; p 1.33; α k = 0.5/k f (k) p k Prof. S. Boyd, EE364b, Stanford University 19

21 resource allocation t to first subsystem (second subsystem gets t) k Prof. S. Boyd, EE364b, Stanford University 20

22 form (separable) partial Lagrangian Dual decomposition L(x 1,x 2,λ) = f 1 (x 1 )+f 2 (x 2 )+λ T (h 1 (x 1 )+h 2 (x 2 )) = ( f 1 (x 1 )+λ T h 1 (x 1 ) ) + ( f 2 (x 2 )+λ T h 2 (x 2 ) ) fix dual variable λ and define subproblem 1: subproblem 2: minimize f 1 (x 1 )+λ T h 1 (x 1 ) subject to x 1 C 1 minimize f 2 (x 2 )+λ T h 2 (x 2 ) subject to x 2 C 2 with optimal values g 1 (λ), g 2 (λ) Prof. S. Boyd, EE364b, Stanford University 21

23 h i ( x i ) ( g i )(λ), where x i is any solution to subproblem i h 1 ( x 1 ) h 2 ( x 2 ) ( g)(λ) the master algorithm updates λ using this subgradient Prof. S. Boyd, EE364b, Stanford University 22

24 Dual decomposition algorithm (using projected subgradient method) repeat 1. Solve the subproblems (in parallel). Solve subproblem 1, finding an optimal x 1. Solve subproblem 2, finding an optimal x Update dual variables (prices). λ := (λ+α k (h 1 ( x 1 )+h 2 ( x 2 ))) +. α k is an appropriate step size iterates need not be feasible can again construct feasible primal variables using projection Prof. S. Boyd, EE364b, Stanford University 23

25 Interpretation λ gives prices of resources subproblems are solved separately, taking income/expense from resource usage into account master algorithm adjusts prices prices on over-subscribed resources are increased; prices on undersubscribed resources are reduced, but never made negative Prof. S. Boyd, EE364b, Stanford University 24

26 (Same) example subgradient method for master; resource prices λ k Prof. S. Boyd, EE364b, Stanford University 25

27 dual decomposition convergence; ˆf is objective of projected feasible allocation g ˆf k Prof. S. Boyd, EE364b, Stanford University 26

28 p g(λ) ˆf g(λ) k Prof. S. Boyd, EE364b, Stanford University 27

29 General decomposition structures multiple subsystems (variable and/or constraint) coupling constraints between subsets of subsystems represent as hypergraph with subsystems as vertices, coupling as hyperedges or nets without loss of generality, can assume all coupling is via consistency constraints Prof. S. Boyd, EE364b, Stanford University 28

30 Simple example subsystems, with private variables x 1, x 2, x 3, and public variables y 1, (y 2,y 3 ), and y 4 2 (simple) edges minimize f 1 (x 1,y 1 )+f 2 (x 2,y 2,y 3 )+f 3 (x 3,y 4 ) subject to (x 1,y 1 ) C 1, (x 2,y 2,y 3 ) C 2, (x 3,y 4 ) C 3 y 1 = y 2, y 3 = y 4 Prof. S. Boyd, EE364b, Stanford University 29

31 A more complex example c c 2 c 3 c Prof. S. Boyd, EE364b, Stanford University 30

32 General form K minimize i=1 f i(x i,y i ) subject to (x i,y i ) C i, i = 1,...,K y i = E i z, i = 1,...,K private variables x i, public variables y i net (hyperedge) variables z R N ; z i is common value of public variables in net i matrices E i give netlist or hypergraph row k is e p, where kth entry of y i is in net p Prof. S. Boyd, EE364b, Stanford University 31

33 φ i (y i ) is optimal value of subproblem Primal decomposition minimize f i (x i,y i ) subject to (x i,y i ) C i repeat 1. Distribute net variables to subsystems. y i := E i z, i = 1,...,K. 2. Optimize subsystems (separately). Solve subproblems to find optimal x i, g i φ i (y i ), i = 1,...,K. 3. Collect and sum subgradients for each net. g := K i=1 ET i g i. 4. Update net variables. z := z α k g. Prof. S. Boyd, EE364b, Stanford University 32

34 g i (ν i ) is optimal value of subproblem Dual decomposition minimize f i (x i,y i )+ν T i y i subject to (x i,y i ) C i given initial price vector ν that satisfies E T ν = 0 (e.g., ν = 0). repeat 1. Optimize subsystems (separately). Solve subproblems to obtain x i, y i. 2. Compute average value of public variables over each net. ẑ := (E T E) 1 E T y. 3. Update prices on public variables. ν := ν +α k (y Eẑ). Prof. S. Boyd, EE364b, Stanford University 33

35 A more complex example subsystems: quadratic plus PWL objective with 10 private variables; 9 public variables and 4 nets; p 11.1; α = g(ν) f(ˆx,ŷ) f(x,ŷ) k Prof. S. Boyd, EE364b, Stanford University 34

36 consistency constraint residual y Eẑ versus iteration number y Eẑ k Prof. S. Boyd, EE364b, Stanford University 35