Abstract Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Keyphrases toll pricing problem certain stochastic bilevel optimization problem stackelberg game complementarity constraint stochastic optimization problem equilibrium model sample-path method recent progress sufficient condition certain stochastic mathematical program steady-state function equilibrium constraint stochastic mathematical program almostsure convergence simulation-based method transportation network.
Powered by:. C3: Every vender pays the company for the whole lunches he buys, i. C5: Even if there are any unsold lunches, the venders cannot return them to the company but they can dispose of the unsold lunches with no cost. We suppose that the demands of lunches depend on the price and the weather on that day. Since the weather is uncertain, we may treat it as a random variable. Suppose there are m venders located at different spots. Now there are two cases.
Here-and-now model: Suppose that both the company and the venders have to make decisions on Saturday, without knowing the weather of Sunday. Lower-level wait-and-see model: Suppose that the company makes a decision on Saturday, but the venders can make their decisions on Sunday morning after knowing the weather of that day. Below we report our numerical experience with these two models.
In our implementation, we used the classical constructions method in [25] to approx- imate the continuous distributions by discrete ones. Table 1 Data for the demand i u i0 u i1 vi0 vi1 functions 1 20 12 3 2 Table 5 Values of price x L L here-and-now wait-and-see 3 6. We have also computed the solutions of the two models with various values of L. Moreover, the prices of Sunday set in the lower-level wait-and-see model are higher than the prices of Saturday set in the here-and-now model.
Such results indicate that both models are appropriate for applications in real world. If the venders want to buy lunches with lower price, they have to make their decisions earlier.
In other words, the difference between the two prices represents the value of information about the weather. We may extend the approaches to the lower-level wait-and-see problems.
In consequence, the conclusions given in Sect. Comparing with the results given in the literature, the assumptions employed in Sect. Acknowledgments The authors are grateful to the anonymous referees for their helpful suggestions and comments. Anitescu, M. SIAM J. Bertsekas, D. Academic Press, New York 3. Birbil, S. Birge, J. Springer, New York 5. Chen, X. Chen, Y. Optimization 32, — 7. Cottle, R. Academic, New York 8. Facchinei, F. Springer, New York Fischer, A.
Optimization 24, — By simulating this system and using gradient with deterministic constraints, appeared in Plambeck et estimation techniques or automatic differentiation capabili- al.
Using these together with appropriate methods appeared in Rubinstein and Shapiro under deterministic techniques we can then compute the optimal the name of stochastic counterpart methods. The basic ap- taxation scheme of the leader and the equilibrium activity proach and its variants is also known as sample average levels of the followers. In Sec- stochastic constraints. From the viewpoint well. This aspect feasible point. Following a similar effort in order to account for the uncertain data.
It is well known see e. Suppose we observe some sequences of functions see Luo, Pang, Ralph The following mathematical description of the toll pric- Although useful, continuous convergence itself does ing problem is somewhat standard in the literature and our not guarantee neither the existence of approximating solu- exposition borrows largely from the model given by Dirkse tions nor their convergence.
To guarantee these, one needs and Ferris Consider a transportation network de- to impose an additional regularity condition. The partic- scribed by a set of nodes N , and a set of arcs A. The set of destination nodes is denoted by Scheel and Scholtes A commodity is associated with a destination node refer the reader to Scheel and Scholtes and Birbil, and it is denoted by k.
The cost other regularity concepts. We assume that the monetary costs of toll long enough on any sample-path , the solution set of 13 prices are converted, for instance through simple weighting, becomes nonempty and compact. For each commodity k, there By assuming an additional condition at the solution is an associated demand dik at node i and the demand vector point, similar convergence results to Theorem 1 can be for the same commodity over the network is denoted by also established for the C-stationary and strong stationary d k.
In practice the demand is a function of the minimum concepts mentioned above. For network controllers, one the congestion on the network. Accordingly, d k s.
Adding now the toll prices to this age. The in tollmpec. However, in prac- tollmpec. In the stochastic setting, parameters values. Numerical experiments corporating this type of uncertainty into the mathematical are currently underway. This expected costs. Thus, the controller may decide to relate her extends the range of applicability of sample-path methods.
We also gave an illustrative example of how it can be used Formally, the situation can be modelled using the following to solve a stochastic toll pricing problem. Simulation- Birbil, S. Approximation to optimization problems: sis. Cited By Stochastic mathematical programs with probabilistic complementarity constraints: SAA and distributionally robust approaches.
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