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Register a new userProbabilistic feasibility guarantees for solution sets to uncertain variational inequalities
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Added by
Filippo Fabiani
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We develop a data-driven approach to the computation of a-posteriori feasibility certificates to the solution sets of variational inequalities affected by uncertainty. Specifically, we focus on instances of variational inequalities with a deterministic mapping and an uncertain feasibility set, and represent uncertainty by means of scenarios. Building upon recent advances in the scenario approach literature, we quantify the robustness properties of the entire set of solutions of a variational inequality, with feasibility set constructed using the scenario approach, against a new unseen realization of the uncertainty. Our results extend existing results that typically impose an assumption that the solution set is a singleton and require certain non-degeneracy properties, and thereby offer probabilistic feasibility guarantees to any feasible solution. We show that assessing the violation probability of an entire set of solutions, rather than of a singleton, requires enumeration of the support constraints that shape this set. Additionally, we propose a general procedure to enumerate the support constraints that does not require a closed form description of the solution set, which is unlikely to be available. We show that robust game theory problems can be modelling via uncertain variational inequalities, and illustrate our theoretical results through extensive numerical simulations on a case study involving an electric vehicle charging coordination problem.
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In this paper a distribution-free methodology is presented for providing robustness guarantees for Nash equilibria (NE) of multi-agent games. Leveraging recent a posteriori developments of the so called scenario approach (Campi et al., 2018), we provide probabilistic guarantees for feasibility problems with polytopic constraints. This result is then used in the context of multi-agent games, allowing to provide robustness certificates for constraint violation of any NE of a given game. Our guarantees can be used alongside any NE seeking algorithm that returns some equilibrium solution. Finally, by exploiting the structure of our problem, we circumvent the need of employing computationally prohibitive algorithms to find an irreducible support subsample, a concept at the core of the scenario approach. Our theoretical results are accompanied by simulation studies that investigate the robustness of the solutions of two different problems, namely, a 2-dimensional feasibility problem and an electric vehicle (EV) charging control problem.
Variational inequalities are modelling tools used to capture a variety of decision-making problems arising in mathematical optimization, operations research, game theory. The scenario approach is a set of techniques developed to tackle stochastic optimization problems, take decisions based on historical data, and quantify their risk. The overarching goal of this manuscript is to bridge these two areas of research, and thus broaden the class of problems amenable to be studied under the lens of the scenario approach. First and foremost, we provide out-of-samples feasibility guarantees for the solution of variational and quasi variational inequality problems. Second, we apply these results to two classes of uncertain games. In the first class, the uncertainty enters in the constraint sets, while in the second class the uncertainty enters in the cost functions. Finally, we exemplify the quality and relevance of our bounds through numerical simulations on a demand-response model.
We investigate the probabilistic feasibility of randomized solutions to two distinct classes of uncertain multi-agent optimization programs. We first assume that only the constraints of the program are affected by uncertainty, while the cost function is arbitrary. Leveraging recent a posteriori developments of the scenario approach, we provide probabilistic guarantees for all feasible solutions of the program under study. This result is particularly useful in cases where numerical difficulties related to the convergence of the solution-seeking algorithm hinder the exact quantification of the optimal solution. Furthermore, it can be applied to cases where the agents incentives lead to a suboptimal solution, e.g., under a non-cooperative setting. We then focus on optimization programs where the cost function admits an aggregate representation and depends on uncertainty while constraints are deterministic. By exploiting the structure of the program under study and leveraging the so called support rank notion, we provide agent-independent robustness certificates for the optimal solution, i.e., the constructed bound on the probability of constraint violation does not depend on the number of agents, but only on the dimension of the agents decision. This substantially reduces the number of samples required to achieve a certain level of probabilistic robustness as the number of agents increases. All robustness certificates provided in this paper are distribution-free and can be used alongside any optimization algorithm. Our theoretical results are accompanied by a numerical case study involving a charging control problem of a fleet of electric vehicles.
We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=Ccap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $mathcal H_1$ and $mathcal H_2$, respectively, and the operator $A$ acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto $S$ by another metric projection onto a half-space which contains $S$. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set $S$. For the split part we make use of the Landweber transform.
The regularity theory for variational inequalities over polyhedral sets developed in a series of papers by Robinson, Ralph and Dontchev-Rockafellar in the 90s has long become classics of variational analysis. But in the available proofs of almost all main results, fairly nontrivial as they are, techniques of variational analysis do not play a significant part. In the paper we develop a new approach that allows to obtain some generalizations of the the mentioned results without invoking anything beyond elementary geometry of convex polyhedra and some basic facts of the theory of metric regularity.
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