4 Numerical studies
4.2 Implementation and results
r11 r˜12 ˜r14 k˜1 Slightly stochastic [490,510] [240,260] [5.8,6.2] [1290,1310]
Heavily stochastic [300,700] [50,450] [2,10] [300,2300]
4.2 Implementation and results
For these first illustrative examples, the algorithm has been implemented in MATLAB utilizing the optimization toolbox. However, a more sophisticated and efficient imple-mentation is under way which will allow for a speed-up and fine-tuning of the algorithm and, most importantly, for a comparison with other approaches and SMPEC test problems.
In order to decrease the computation time we employed two types of iteration steps.
In a normal step only one observation of the random data is utilized for the calculations.
At regular intervals, a control step is performed with a sufficiently large sample of ob-servations. Furthermore at such a step the step sizes are adjusted, either automatically or interactively. In the first case the step sizes are calculated according to a rule satisfying the conditions (23). An interactive step size adjustment allows the user to revise the step size according to his observations of the progress of the iteration. For more details on the step size strategy and for reasons for adopting such an interactive approach we refer to the discussion in [Gai88]. Note that with this strategy the step sizes can reach arbitrarily small values but do they not approach zero. Therefore only the vicinity of the optimal solution is reached and the interactive step size selection may be utilized as an indicator for a good automatic step size strategy.
The step sizes were determined according to the rules αkx = 0.1
1 +s, αku = 0.1
1 +sγ (36)
Alternatively, we tested the step size rule αkx = 0.25
2s , αku = 0.25
1.9s, k∈[2s−1,2s] (37) Herek denotes the number of the current iteration step whereassis the iteration step at which the previous control step was performed. We used a regular review interval of 10 steps such thats = dk/10e, the greatest integer which is smaller than or equal tok/10.
The convergence test performed in the iteration evaluates the behavior of the estimation Fkover the previous three iteration steps. A stationary pointxswas identified as optimal when it was within a vicinity of 0.02 % of the actual optimum.
Generally, the iterates show a behavior typical for SQG methods: after a period with heavy oscillations the vicinity of the optimal solution is reached quite fast. From that point on the approximation improves only slowly, small oscillations in the vicinity of the optimum persist. More specific, four different sections can be distinguished for the tested problem. At first the iterates oscillate heavily between two clusters relatively far away from the optimum, possibly some periods with stable objective function values exist. In the second section, the oscillations shift slowly toward a further cluster in the vicinity of the optimum. A short section of consolidation follows. The variance of the oscillations decreases rapidly and the iterates begin to cluster in the vicinity of the optimal solution.
Finally, the iterates oscillate in the vicinity of the optimum. One reason for these oscilla-tions is the step size strategy. Especially due to the behavior of the iterates in the first two periods the estimationFˆof the objective function converges only very slowly. Typically the algorithm terminates because a predefined number of iteration steps was reached. The optimality conditions (11) are often not satisfied and the existence of feasible ascent di-rections is analyzed. This indicates that the convergence and optimality tests are too strict taken into account the stochasticity of the data or the sample size chosen for the test was too small. As a consequence, so-called approximate stationary points could not be iden-tified. Here, a reasonable relaxation of the tests seems appropriate in order to recognize points in a close vicinity of an optimum.
Two sets of experiments were performed. The first set analyzed the segmentICs = {2,5,7} which is found, for example, with the initial points y10 = (0,3.6,0) or y10 = (5.2,6.2,0). On this segment, the deterministic problem has a local optimum aty∗1D =
(2.53,4.87,0)with the recourse decisiony2D∗ = 0, the follower’s responsezD∗ = (3.87,0,0) and the objective function valueFD∗ = 2233.31. Here, the step size strategy (36) proved quite efficient. In the case of slightly stochastic data the periods with heavy oscillations were small and a vicinity of 2 % of the optimal solution was reached after approximately 100 iteration steps and a precision of 0.5 % after further 20 steps. However, even after additional 150 steps the approximation did not increase significantly, the iterates where in a vicinity of about 0.3 % of the optimum. A similar behavior of the iterates can be observed in the case of more random data. Table 1 compares iterates obtained on this segment using highly stochastic and deterministic data.
Table 1: Heavily stochastic and deterministic data,y01 = (0,3.6,0)
heavily stochastic data deterministic data
k y1 Fˆ Fs y1 Fˆ Fs
1 (0.00, 3.60, 0) 650.00 650.00 (0.00, 3.60, 0) 650.00 650.00 2 (5.20, 6.20, 0) -325.00 -1300.00 (5.20, 6.20, 0) -325.00 -1300.00 3 (5.20, 6.20, 0) -650.00 -1300.00 (5.20, 6.20, 0) -650.00 -1300.00 4 (-0.00, 3.60, 0) -325.00 650.00 (-0.00, 3.60, 0) -325.00 650.00 5 (-0.00, 3.60, 0) -260.63 -3.16 (-0.00, 3.60, 0) -130.00 650.00 6 (-0.00, 3.60, 0) -108.86 650.00 (-0.00, 3.60, 0) 0.00 650.00 7 (5.20, 6.20, 0) -279.02 -1300.00 (5.20, 6.20, 0) -185.71 -1300.00 8 (5.20, 6.20, 0) -406.65 -1300.00 (5.20, 6.20, 0) -325.00 -1300.00 9 (5.20, 6.20, 0) -505.91 -1300.00 (5.20, 6.20, 0) -433.33 -1300.00 10 (-0.00, 3.60, 0) -409.86 454.59 (-0.00, 3.60, 0) -325.00 650.00 100 (3.65, 5.42, 0) 560.52 1408.30 (2.72, 4.96, 0) 898.32 2159.00 250 (2.99, 5.09, 0) 1439.17 2008.21 (2.53, 4.87, 0) 1699.36 2233.31 500 (2.83, 5.01, 0) 1815.00 2104.29 (2.53, 4.87, 0) 1966.33 2233.31 1000 (2.58, 4.89, 0) 2007.75 2216.37 (2.53, 4.87, 0) 2099.82 2233.31 1500 (2.87, 5.03, 0) 2077.38 2081.52 (2.53, 4.87, 0) 2144.32 2233.31 2000 (2.36, 4.78, 0) 2114.53 2278.31 (2.53, 4.87, 0) 2166.56 2233.31
However, the second set of experiments shows that the good performance of rule (36) can not be generalized. Choosing the initial point y10 = (0.83,0,210), the iteration is conducted on the segmentICs = {2,7}. The local optimum of the deterministic problem is y1D∗ = (2.53,0.87,0) with the recourse decision y2D∗ = 0, the follower’s response zD∗ = (1.87,1000,0)and the objective valueFD∗ = 1333.29. Here, strategy (36) shows a weaker performance. After a few large initial oscillations a long period of about 300 steps follows with quite stable iterates. During the next 100 steps the vicinity of the optimum is approached with only a few oscillations. Finally, the iterates oscillate in the vicinity of the optimum. For the case of low stochasticity, Figure 1 depicts a typical behavior of the iterates for this strategy on both segments.
With the second step size rule (37), the algorithm performs slightly better for the initial pointy10 = (0.83,0,210), but the performance is worse for the initial pointy10 = (0,3.6,0) (see Figure 2). This observation underlines that a step size strategy which performs equally well for all problems can hardly be found. Rather, at first the algorithm may be run tentatively in interactive mode in order to obtain a conjecture for a good automatic strategy. Such an automatic strategy can for example be chosen from a toolbox containing several alternatives.
A more sophisticated and efficient implementation will allow for a speed-up and
fine-Figure 1: Estimated values of objective function, step size rule (36)
Figure 2: Estimated values of objective function; step size rule (37)
tuning of the proposed algorithm. For example, the test on existence of descent directions is now performed on all index sets. A first improvement may exclude the index sets of already visited segments from this search. Furthermore, a mechanism for identifying neighbor segments through their index sets will restrict the search to only a few segments at each iteration. The tests on stationarity and optimality may be improved using, for example, results by Bayraksan and Morton [BM06] on the quality of solutions of stochas-tic programming problems. Finally, a parallelization of the method may contribute to a considerable reduction of computation time.
5 Conclusions
We studied several formulations of stochastic programming problems with recourse and bilevel structure where connecting upper-level constraints are present. A strong stationar-ity concept has been stated which, under some assumptions on the considered stochastic programming problems, establishes sufficient conditions for optimality. An algorithm for the solution of the presented problem type has been developed utilizing a two-step solution process. This is due to the reformulation of the problems to MPEC-type one-level problems and, hence, the possible nonconvexity of the inducible region caused by
the connecting upper-level constraints. We proved that, under certain conditions on the involved functions, a point in the vicinity of a local optimal solution of the originally studied problems is attained with probability 1. Tentative numerical experiments testify to a reasonable numerical efficiency of the proposed approach.
Future research may include more complex multiperiod problems. For example, the follower’s second-stage problem may represent a recourse problem instead of the two-stage relationship studied now or it may contain inequality constraints. Another conceiv-able extension takes into account that the leader’s perception of the follower’s decision process may be imperfect. This means that the leader will obtain certainty about the actually implemented response only at the end of the first stage. Such a consideration of the uncertainty about the lower-level decision process is especially important for the analysis of agency problems. Finally, also extensions to problems involving multiple fol-lowers appear natural. Then, also the character of the interactions between these decision makers, such as Nash game or further Stackelberg game relations, and their effect on the model properties must be taken into account. Another field for further research may be concerned with enhancing and fine-tuning the proposed solution approach, following the lines of the discussion in Section 4.
References
[AGW06] J.A. Audestad, A.A. Gaivoronski, and A.S. Werner. Extending the stochastic programming framework for the modeling of several decision makers: pric-ing and competition in the telecommunication sector. Annals of Operations Research, 142(1):19–39, 2006.
[BGL04a] S.I. Birbil, G. G¨urkan, and O. Listes. Simulation-based solution of stochastic mathematical programs with complementarity constraints: sample-path anal-ysis. Research report ers-2004-016-lis, Erasmus University, 2004.
[BGL04b] S.I. Birbil, G. G¨urkan, and O. Listes. Solving stochastic mathematical pro-grams with complementarity constraints using simulation. In R.I. Ingalls, M.D. Rossetti, J.S Smith, and B.A. Peters, editors, Proceedings of the 2004 Winter Simulation Conference, pages 550–557, 2004.
[BL97] J.R. Birge and F. Louveaux. Introduction to Stochastic Programming.
Springer, New York, 1997.
[BM06] G. Bayraksan and D.P. Morton. Assessing solution quality in stochastic pro-grams. Mathematical Programming, 108:495–514, 2006.
[CF95] Y. Chen and M. Florian. The nonlinear bilevel programming problem: for-mulations, regularity and optimality conditions. Optimization, 32:193–209, 1995.
[CPS92] R.W. Cottle, J.-S. Pang, and R.E. Stone. The linear complementarity problem.
Academic Press, Boston, 1992.
[Dem02] S. Dempe. Foundations of Bilevel Programming. Kluwer Academic Publish-ers, 2002.
[EP04] A. Evgrafov and M. Patriksson. On the existence of solutions to stochastic mathematical programs with equilibrium constraints.Journal of Optimization Theory and Applications, 121(1):65–76, 2004.
[Erm88] Y. Ermoliev. Stochastic quasigradient methods. In Y. Ermoliev and R.J.-B.
Wets, editors,Numerical Techniques for Stochastic Optimization, pages 141–
186. Springer, 1988.
[EW88a] Y. Ermoliev and R.J.-B. Wets, editors. Numerical Techniques for Stochastic Optimization. Springer Verlag, 1988.
[EW88b] Y. Ermoliev and R.J.-B. Wets. Stochastic programming, an introduction. In Y. Ermoliev and R.J.-B. Wets, editors, Numerical Techniques for Stochastic Optimization, pages 1–32. Springer, 1988.
[FL04] R. Fletcher and S. Leyffer. Solving mathematical programs with complemen-tarity constraints as nonlinear programs.Optimization Methods and Software, 19(1):15–40, 2004.
[Fle05] M.L. Flegel. Constraint Qualifications and Stationarity Concepts for Math-ematical Programs with Equilibrium Constraints. PhD thesis, Institute of Applied Mathematics and Statistics, University of W¨urzburg, 2005.
[FLRS02] R. Fletcher, S. Leyffer, D. Ralph, and S. Scholtes. Local convergence of SQP methods for mathematical programs with equilibrium constraints. Numerical analysis report na/209, Department of Mathematics, University of Dundee, Dundee, UK, 2002.
[Gai88] A.A. Gaivoronski. Stochastic quasigradient methods and their implementa-tion. In Y. Ermoliev and R.J.-B. Wets, editors, Numerical Techniques for Stochastic Optimization, pages 313–351. Springer, 1988.
[Gai04] A.A. Gaivoronski. SQG: stochastic programming software environment. In S.W. Wallace and W.T. Ziemba, editors,Applications of Stochastic Program-ming, MPS-SIAM Series in Optimization, pages 637–670. 2004.
[GR02] C. Greengard and A. Ruszczynski, editors. Decision Making under Uncer-tainty. Energy and Power, volume 128 in The IMA Volumes on Mathematics and its Applications. Springer, 2002.
[Jit84] K. Jittorntrum. Solution point differentiability without strict complementarity in nonlinear programming. Mathematical Programming Study, 21:127–138, 1984.
[JR03] H. Jiang and D. Ralph. Extension of quasi-Newton methods to mathematical programs with complementarity constraints.Computational Optimization and Applications, 25:123–150, 2003.
[KO04] M. Koˇcvara and J.V. Outrata. Optimization problems with equilibrium con-straints and their numerical solution. Mathematical Programming, Ser. B, 101:119–149, 2004.
[LCFar] G.-H. Lin, X. Chen, and M. Fukushima. Solving stochastic mathematical pro-grams with equilibrium constraints via approximation and smoothing implicit programming with penalization. Mathematical Programming, To appear.
[LF06] G.-H. Lin and M. Fukushima. New reformulations for stochastic nonlinear complementarity problems. Optimization Methods and Software, 21(4):551–
564, 2006.
[LPR96] Z.Q. Luo, J.-S. Pang, and D. Ralph.Mathematical Programs with Equilibrium Constraints. Cambridge University Press, 1996.
[LS04] X. Liu and J. Sun. Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints. Mathematical Pro-gramming, Ser. B, 101:231–261, 2004.
[Mir99] J. Mirrlees. The theory of moral hazard and unobservable behaviour: Part I.
The Review of Economic Studies, 66(1):3–21, 1999.
[NV77] E.A. Nurminski and P.I. Verchenko. Convergence of algorithms for finding saddle points. Kibernetika, 3:112–116, 1977.
[PW99] M. Patriksson and L. Wynter. Stochastic mathematical programs with equi-librium constraints. Operations Research Letters, 25:159–167, 1999.
[Sha06] A. Shapiro. Stochastic programming with equilibrium constraints. Journal of Optimization Theory and Applications, 128(1):223–243, 2006.
[SR03] A. Shapiro and A. Ruszczynski. Stochastic Programming. Handbooks in Op-erations Research and Management Science, volume 10. Elsevier, 2003.
[SS00] H. Scheel and S. Scholtes. Mathematical programs with equilibrium con-straints: stationarity, optimality, and sensitivity. Mathematics of Operations Research, 25(1):1–22, 2000.
[SX05] A. Shapiro and H. Xu. Stochastic mathematical programs with equilibrium constraints, modeling and sample average approxima-tion. Preprint, School of Industrial and Systems Engineering, Geor-gia Institute of Technology, Atlanta, Georgia. E-print available at:
http://www.optimization-online.org, 2005.
[Wet89] R.J.-B. Wets. Stochastic optimization. In G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd, editors,Optimization. North-Holland, 1989.
[WG05] A.S. Werner and A.A. Gaivoronski. Utilization of stochastic programming methods in the analysis of agency problems. In Bilevel stochastic program-ming problems: Analysis and application to telecommunications. (Phd The-sis), pages 99–131, 2005.
[WW07] Q. Wang and A.S. Werner. Telecom service resale under uncertainty: Profit and consumer surplus implications. In preparation, 2007.
[Wyn01] L. Wynter. Stochastic bilevel programs. In C. Floudas and P. Pardalos, editors, Encyclopedia of Optimization. Kluwer, 2001.
[Xu06] H. Xu. An implicit programming approach for a class of stochastic mathe-matical programs with complementarity constraints. SIAM Journal on Opti-mization, 16(3):670–696, 2006.
[Ye05] J.J. Ye. Necessary and sufficient optimality conditions for mathematical pro-grams with equilibrium constraints. Journal of Mathematical Analysis and Applications, 307:350–369, 2005.
[ZL01] J.Z. Zhang and G. Liu. A new extreme point algorithm and its application in PSQP algorithms for solving mathematical programs with linear complemen-tarity constraints. Journal of Global Optimization, 14:345–361, 2001.