Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming

TL;DR

This paper formulates the problem of setting contextual reserve prices in auctions as a mixed-integer programming problem, demonstrating its computational complexity, proposing an exact MIP model, analyzing its LP relaxation, and showing superior empirical performance.

Contribution

It introduces a novel MIP formulation for reserve price optimization that exactly models the nonconvex revenue function, and analyzes the limitations of its LP relaxation.

Findings

MIP formulation is exact and ideal for single impressions.

LP relaxation can significantly overestimate the optimal revenue.

Empirical results show superior performance over existing algorithms.

Abstract

We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the \emph{Exponential Time Hypothesis} fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem. Finally, we present computational results, showcasing that the MIP formulation, along with its LP relaxation, are able to achieve superior in- and out-of-sample performance, as compared to state-of-the-art algorithms on both real and synthetic datasets. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems.