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.

我们考虑电力零售商解决的利润最大化问题，该零售商旨在设计合同菜单。这是单位需求无嫉妒定价问题的延伸：客户的目标是根据预订账单和多个价格系数（属性）选择一份最大化其效用的合同。基本方法假设客户具有确定性效用；然后，每个客户的反应对价格高度敏感，因为它集中于最好的报价。第二种经典方法是考虑使用逻辑模型在客户的选择中添加概率行为。为了规避前者的内在不稳定性和后者的解决困难，我们引入了客户响应的二次正则化模型，这导致了互补约束下的二次规划（QPCC）。这可以增强确定性模型，同时保持强大的几何结构。特别是，我们表明客户的响应受到多面体复合体的控制，其中每个多面体单元确定一组有效选择的合同。此外，确定性模型被恢复为正则化模型的极限情况。我们利用这些几何特性来开发一种旋转启发式方法，并将其与双层编程中的隐式或非线性方法进行比较，显示该方法的有效性。在整篇论文中，电力零售商问题是我们的指导方针，我们对此应用案例进行了数值研究。