Convergence of Nonconvex Douglas-Rachford Splitting and Nonconvex ADMM

Transcription

1 Convergence of Nonconvex Douglas-Rachford Splitting and Nonconvex ADMM Shuvomoy Das Gupta Thales Canada 56th Allerton Conference on Communication, Control and Computing,

2 What is this talk about? this talk is about ADMM and Douglas-Rachford splitting for nonconvex problems the alternating direction method of multipliers (ADMM) originally designed to solve convex optimization problem Douglas-Rachford splitting algorithm ADMM is its special case in a convex setup both guaranteed to converge for convex problems 2

3 Motivation nonconvex ADMM (NC-ADMM) has become a popular heuristic to tackle nonconvex problems recently, NC-ADMM heuristic has been applied to [Erseghe, 2014] optimal power flow problem, [Takapoui et al., 2017] mixed integer quadratic optimization, [Iyer et al., 2014] submodular minimization with nonconvex constraints... [Diamond et al., 2018] Python package NCVX (extension of CVXPY) implements ADMM heuristic (NC-ADMM) often produces lower objective values compared with exact solvers within a time limit nonconvex Douglas-Rachford splitting (NC-DRS): analogous nonconvex heuristic based on Douglas-Rachford splitting not much has been done to improve the theoretical understanding of such heuristics

4 Summary of the results NC-DRS attacks the original problem directly optimal solutions can be characterized via the NC-DRS operator if deviation from a convex setup is bounded it will converge or oscillate in a compact connected set NC-ADMM works on a modified dual problem, not the original nonconvex problem not equivalent to NC-DRS, but there is a relationship between them likely to produce a lower objective value 4

5 Problem in consideration minimize a convex cost function with nonconvex constraint set minimize x subject to f (x) x C (OPT) f is closed, proper, and convex C is compact, but not necessarily convex

6 Reformulation through indicator function indicator function of set C: δ C (x) = { 0, if x C, if x / C δ C is closed and proper, but not necessarily convex we can write ( minimizex f (x) subject to x C ) = minimize x f(x) + δ C (x)

7 Proximal operator of f and projection onto C both NC-DRS and NC-ADMM have same subroutines: first prox γf, then Π C and finally proximal operator of f evaluated at point x with parameter γ > 0: ( 1 prox γf (x) = argmin y f(y) + y x 2) 2γ single-valued, continuous projection onto C: prox γδc (x) = Π C (x) = argmin y C ( y x 2 ) there can be multiple projections one such projection is denoted by Π C ( )

8 NC-ADMM and NC-DRS: NC-ADMM: x n+1 = prox γf (y n z n ) y n+1 = Π C (x n+1 + z n ) z n+1 = z n y n+1 + x n+1 NC-DRS: x n+1 = prox γf (z n ) y n+1 = Π C (2x n+1 z n ) z n+1 = z n + y n+1 x n+1 both have same subroutines, but different inputs 8

9 Pretend C is convex f is closed, proper, convex pretend C is convex x n,y n converge to an optimal solution for any initial condition but C is not necessarily convex in our setup convergence conditions are messy 9

10 Why are convergence conditions messy? the convergence conditions are messy because: subdifferential operator of δ C is monotone, but not maximally monotone Π C : is expansive i.e., not nonexpansive the underlying reflection operator is expansive Little bit of review...

11 Monotone and maximally monotone operators T is monotone if for every (x,u),(y,v) grat x y u v 0 T is maximally monotone if gra T is not properly contained by any other monotone operator s graph monotone, but not maximally monotone maximally monotone 11

12 Subdifferential operator for a nonconvex function g: closed, proper, but not necessarily convex g: subdifferential of g is monotone, but not maximally monotone g (x) = {u R n ( y R n ) g (y) g (x) + u y x } no subgradient at

13 Why are convergence conditions messy? our problem: minimize x f(x) + δ C (x) f: maximally monotone δ C : monotone, but not maximally monotone Π C is expansive (not nonexpansive) What is a nonexpansive operator?

14 What is a nonexpansive operator? T : single-valued operator on R n T is nonexpansive on R n if for every x,y we have T (x) T (y) x y prox γf is nonexpansive ) (2prox γf I n is nonexpansive 14

15 Operators that are expansive T is a single-valued operator on R n T is expansive if there exist x,y such that T (x) T (y) > x y ΠC is expansive 15

16 Characterization of minimizers: NC-DRS operator and its reflection T : NC-DRS operator T = Π C ( 2prox γf I n ) + I n prox γf. R: reflection operator of T NC-DRS in compact form: R = 2 T I n z n+1 = T z n = 1 2 ( R + In ) z n ΠC : expansive R: expansive root of all convergence issues 16

17 Characterization of minimizers argmin(f + δ C ) is the set of minimizers of min. x f(x) + δ C (x) prox γf ( fix T ) argmin(f + δc ) underlying assumptions: 1. zer ( f + δ C ) is nonempty 2. fix T is nonempty 3. fix {( 2Π C I n )( 2proxγf I n )} is nonempty

18 Convergence of NC-DRS: setup ε R xy : expansiveness of R at x,y ε R xy = { R(x) R(y) x y, if x y < R(x) R(y) 0, else σ R xy = ε R xy R(x) R(y) + x y

19 Convergence of NC-DRS: conditions (z n ) n N : sequence of vectors generated for some chosen initial point z 0 if the following holds: there exists a z fix T, such that n=0 (σ R znz) 2 is bounded then... above, and z 0 z 2 is finite z 0 z 2 + n=0 B(z;r): compact ball with center z and radius r define r := ( ) 2 σ R zn z

20 Convergence of NC-DRS then one of the following will happen: 1. convergence to a point: the sequence (z n ) n N converges to a point z B(z;r) 2. cluster points form a continuum: the set of cluster points of (z n ) n N forms a nonempty compact connected set in B(z; r) if situation 1 occurs and lim n (σ R znz ) 2 = 0, then x n = prox γf (z n 1 ) converges to an optimal solution

21 Some comments on convergence for convergence total deviation of R from being a nonexpansive operator over the sequence {(z n,z)} n N is bounded depends on the initial point in our case C is not necessarily convex sanity check: pretend C is convex total deviation of R from being a nonexpansive operator over the sequence {(z n,z)} n N is zero our convergence proof coincides with known convergence results for convex setup

22 Constructing NC-ADMM original problem: minimize x C f(x) take dual and apply NC-DRS to the dual resultant algorithm is relaxed NC-ADMM x n+1 = prox γf (y n z n ) y n+1 = Π convc (z n + x n+1 ) z n+1 = z n y n+1 + x n+1 relaxed NC-ADMM solves minimize x convc f(x)

23 Constructing NC-ADMM (continued) relaxed NC-ADMM x n+1 = prox γf (y n z n ) y n+1 = Π convc (z n + x n+1 ) z n+1 = z n y n+1 + x n+1 solves minimize x convc f(x) replace Π convc with Π C to solve minimize x C f(x) resultant algorithm is NC-ADMM: x n+1 = prox γf (y n z n ) y n+1 = Π C (x n+1 + z n ) z n+1 = z n y n+1 + x n+1

24 Constructing NC-ADMM

25 Pretend C is convex pretend C is convex strong duality holds Π convc = Π C Moreau s decomposition can be applied to δ C : prox δc + prox δ = I C n then: no information loss in constructing NC-ADMM from NC-DRS NC-ADMM and NC-DRS are equivalent to each other

26 However... C is nonconvex NC-ADMM and NC-DRS are not equivalent reasons for the loss of equivalency: 1. strict duality gap 2. Π conv C Π C 3. Moreau s decomposition does not hold NC-ADMM in NCVX-CVXPY works on a modified dual, hence produces a lower objective value relationship between minimizers of the original problem and the NC-ADMM operator breaks down

27 What if? is it possible to establish convergence NC-ADMM ignoring NC-DRS? our convergence analysis is established for nonempty, compact, but not necessarily convex constraint sets it is equally applicable to the smaller subclass of problems with convex constraint sets in this smaller subclass of convex problems NC-ADMM and NC-DRS have the same convergence properties 27

28 Summary NC-DRS and NC-ADMM are very similar looking, but very different heuristics the theoretical rationale to use NC-DRS could be stronger than NC-ADMM... 28

29 Limitations strong assumptions in minimizer characterization step size is 1, does not consider adaptive step sizes no numerical experiments in the paper

30 End of talk if you are working on nonconvex problems using ADMM/DRS, please talk to me! Thank you! Questions?

31 Diamond, S, Takapoui, R, & Boyd, S A general system for heuristic minimization of convex functions over non-convex sets. Optimization Methods and Software, 33(1), Erseghe, Tomaso Distributed optimal power flow using ADMM. IEEE transactions on power systems, 29(5), Iyer, Rishabh, Jegelka, Stefanie, & Bilmes, Jeff Monotone Closure of Relaxed Constraints in Submodular Optimization: Connections Between Minimization and Maximization. Pages of: Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence. UAI 14. Arlington, Virginia, United States: AUAI Press. 30

32 Takapoui, Reza, Moehle, Nicholas, Boyd, Stephen, & Bemporad, Alberto A simple effective heuristic for embedded mixed-integer quadratic programming. International Journal of Control,