1. Stochastic Quasi-Gradient Methods Roger J-B Wets University of California, Davis February 15, 2005

2. Stochastic optimization Formulation Properties: S

3. Subgradients of convex fcns

4. Minimization algorithms Step type 1

5. Minimization algorithms Step type 2 proj

6. “repeated” projections Convex program: quadratic objective function quadratic program if S is a polyhedral set Many applications:  projection is a simple/efficient non-negative, convex, bounded away from 0

7. SQG Iterates basic strategy:

8. SQG: Stochastic Optimization. sqg: justification:

9. SQG: Stochastic Optimization. value estimate: justification:

10. A (simple) location problem Pop. Size of 12 districts: 11  #  26. Probabilistic choice of shopping district: shortage cost: 4, holding cost: 0.5 (excess) decision: location of facilities (shopping malls)

11. “preferences” table 0 1 3 4 6 7 8 … 2 0 1 1 3 5 5 … 7 1 0 1 2 6 5 4 4 …

12. Formulation from objective: probability of sample determined by customer behavior

13. Objective Value: iterates Estimate of the objective per iterate

14. Objective Value (2): iterates Estimate of the objective per iterate Facilities: 18.57 15.90 19.13 16.35 27.25 20.75 21.88 17.81 19.11 17.52 18.62 19.60 Distr.Pop: 14 11 14 13 26 23 22 11 14 12 18 10

15. Objective Value (3): iterates Facilities: 24 22 23 20 26 22 23 22 22 20 22 25 : 271 Distr.Pop: 19 16 19 16 27 21 22 18 19 18 19 20 : 234

16. a.s. Convergence For now presumed optimal sol’n at iteration   projection implies:

17. a.s Convergence taking condition expectation w.r.t. F assumption(a.): with

18. a.s Convergence Hence Assumption(b.): where with

19. a.s. Convergence recursively from (a)

20. a.s. Convergence Thus assumption (c.) and there exists a subsequence such that

21. Review of assumptions (a.) (b.) (c.)

22. “stumbling” blocks Projection Step size: adaptive, adjust (increase, decrease) based on the variance of the stochastic quasi-gradient Stopping criterion: like for step-size, but more generally comparison of the values of the objective:

23. A short history Stochastic approximation methods Robbins & Monro, Kiefer & Wolfowitz (‘50) SQG: Theory Shor, Poljak, Ermoliev, Fabian (‘60), Kushner(‘70),Pflug, Ruszczynski (‘80), Implementation: Gaivoronski, Gupal, Norkin (‘80 … 2005)