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arXiv · 2110.02765

Quadratic Regularization of Bilevel Pricing Problems and Application to Electricity Retail Markets

Quentin Jacquet·Wim van Ackooij·Clémence Alasseur·Stéphane Gaubert
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Abstract

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.

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Quentin Jacquet, Wim van Ackooij, Clémence Alasseur, Stéphane Gaubert. 2021-10-06. Quadratic Regularization of Bilevel Pricing Problems and Application to Electricity Retail Markets. https://arxiv.org/abs/2110.02765

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