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Stochastic variational inequalities: single-stage to multistage

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Abstract

Variational inequality modeling, analysis and computations are important for many applications, but much of the subject has been developed in a deterministic setting with no uncertainty in a problem’s data. In recent years research has proceeded on a track to incorporate stochasticity in one way or another. However, the main focus has been on rather limited ideas of what a stochastic variational inequality might be. Because variational inequalities are especially tuned to capturing conditions for optimality and equilibrium, stochastic variational inequalities ought to provide such service for problems of optimization and equilibrium in a stochastic setting. Therefore they ought to be able to deal with multistage decision processes involving actions that respond to increasing levels of information. Critical for that, as discovered in stochastic programming, is introducing nonanticipativity as an explicit constraint on responses along with an associated “multiplier” element which captures the “price of information” and provides a means of decomposition as a tool in algorithmic developments. That idea is extended here to a framework which supports multistage optimization and equilibrium models while also clarifying the single-stage picture.

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Notes

  1. Of course a full probability setting is important to have available in the long run, but we plan to deal with that in a follow-up article.

  2. Generalizations beyond (1.1) are available. The set-valued normal cone mapping \(N_C\) is the subdifferential mapping \(\partial \delta _C\) associated with the indicator \(\delta _C\), which is a closed proper convex function. A natural step therefore is to replace \(N_C\) in (1.1) by the subdifferential mapping \(\partial f\) for any closed proper convex function f on \(\mathbb {R}^n\) (which goes back to the earliest days of the subject) or perhaps any set-valued mapping T that, like such \(\partial f\), is maximal monotone. It is possible also to take F to be set-valued, built out of other subdifferential mappings, say. Despite the genuine interest in such generalizations and their eventual importance in applications, the fundamental version in (1.1) will be our touchstone here.

  3. Such a variational inequality representation of the KKT conditions was the “generalized equation” which inspired Robinson in his pioneering work [17]. In [7] the term “generalized equation” is applied also to the case of (1.1) in which \(N_C\) is replaced by a more general set-valued mapping.

  4. For other L it can characterize first-order optimality in an amazingly large class of problems which can even be “nonsmooth”; see [20], [25, 11.46–11.47].

  5. The equilibrium in (1.9) is a true Nash equilibrium when \(f_i(x_i,x_{-i})\) is convex in \(x_i\), so that first-order optimality in its local sense coincides with global optimality. However, equilibrium is an apt term even without the convexity, since it’s hardly reasonable to burden agents with mastering global minimization in a context where the actions of competitors render perceptions local at best anyway.

  6. An analogy with overdetermined systems of linear equations can be considered, where a single solution x ought to exist but the equations disagree slightly because of measurement errors in the eoefficients, rendering their simultaneous solution impossible. One might think similarly of collection of linear programming problems, say, which are identical except for “coefficient noise” and look for an approximate common solution.

  7. Note that, in this kind of notation, \({\mathcal {F}}(x(\cdot ))(\xi )\) could be used for \(F(x(\xi ),\xi )\), but \({\mathcal {F}}(x(\xi ))\) wouldn’t make any sense, since \({\mathcal {F}}\) acts on elements of \({\mathcal {L}}_n\), not on vectors in \(\mathbb {R}^n\).

  8. An immediate criterion, as indicated in the background discussion, is the boundedness of \({\mathcal {C}}\), which corresponds to the boundedness of the sets \(C(\xi )\). Later, in the multistage development, broader criteria will be presented.

  9. Get the second condition from the first by taking \(w(\xi )=z(\xi )-E[z(\xi )]\); get the first from the second by taking expectations.

  10. In a two-stage precedent of sorts for going beyond a pure single-stage model is present in in [3], where the “approximate” x in the ERM approach is treated as a first-stage decision compared to a \(\xi \)-dependent hindsight solution. This can still be viewed as an error minimization model rather than solving an actual variational inequality.

  11. Here we terminate with an observation, but we could instead terminate with a decision as in the two-stage preview. That alternative will be taken up later.

  12. This way of treating information serves us here as the being the simplest for purpose at hand. It fits as a special case of a more sophisticated treatment of finitely many scenarios that was laid out in [24]. This information structure could also be rendered in the form of a “scenario tree” with transition probabilities.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    R. Tyrrell Rockafellar

  2. Department of Mathematics, University of California-Davis, Davis, CA, 95615, USA

    Roger J-B Wets

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  1. R. Tyrrell Rockafellar
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  2. Roger J-B Wets
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Correspondence to R. Tyrrell Rockafellar.

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This material is based in part on work supported by the U.S. Army Research Office under Grant W911NF-12-1-0273.

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Rockafellar, R.T., Wets, R.JB. Stochastic variational inequalities: single-stage to multistage. Math. Program. 165, 331–360 (2017). https://doi.org/10.1007/s10107-016-0995-5

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