Chance Constrained Optimization for Targeted Internet Advertising

We introduce a chance constrained optimization model for the fulfillment of guaranteed display Internet advertising campaigns. The proposed formulation for the allocation of display inventory takes into account the uncertainty of the supply of Internet viewers. We discuss and present theoretical and computational features of the model via Monte Carlo sampling and convex approximations. Theoretical upper and lower bounds are presented along with a numerical substantiation.

💡 Research Summary

The paper addresses the problem of allocating display‑ad inventory to guaranteed‑delivery advertising campaigns in an online ad network while accounting for the stochastic nature of viewer supply. Each campaign k has a target number of impressions g_k and a set of targeted viewer types V_k. The supply of each viewer type v is modeled as a random variable S_v with known mean μ_v and covariance Σ, reflecting the uncertainty inherent in web traffic. The decision variables are the allocation fractions p_{vk} (the proportion of type‑v supply assigned to campaign k). The authors formulate a chance‑constrained optimization problem (CC) that requires every campaign’s total delivered impressions ∑{v∈V_k} S_v p{vk} to meet or exceed g_k with probability at least 1−α, where α is a small failure tolerance chosen by the ad network.

The objective is to maximize “representativeness” by minimizing the variance of the allocation fractions within each campaign, i.e., ∑{k} w_k|V_k|∑{v∈V_k}(p_{vk}−q_k)^2 where q_k is the average fraction for campaign k and w_k reflects campaign priority. This choice encourages a balanced distribution of ads across all targeted viewer types.

Because the chance constraint is non‑convex and involves multidimensional probability integrals, the authors develop two tractable approximations.

1. Sample Approximation (SA) – A finite set of N independent supply scenarios {S_i} is generated via Monte‑Carlo sampling. Binary variables x_i indicate whether scenario i satisfies all campaign goals. The model enforces that at least ⌈(1−ξ)N⌉ scenarios must be feasible, where ξ approximates the allowed violation probability. This yields a mixed‑integer quadratic program. Using results from Luedtke & Ahmed (2014), the authors prove that with appropriate N and ξ, the optimal value of SA provides probabilistic lower and upper bounds on the true optimal value of CC. They also embed SA within a branch‑and‑bound framework, introducing a heuristic that selects the scenario most likely to violate the constraints for branching, dramatically reducing the search tree size.

2. Convex Approximation (CA) – When only first‑ and second‑order moments are known, distribution‑free bounds are derived via Markov’s and Chebyshev’s inequalities, leading to linear lower bounds (p_k^T μ_k ≥ (1−α)g_k) and quadratic upper bounds (p_k^T Σ_k p_k ≤ (α/(1−α))(g_k−p_k^T μ_k)^2). Assuming a multivariate normal distribution for supply, the chance constraint becomes a second‑order cone constraint: g_k ≤ p_k^T μ_k + n_α√(p_k^T Σ_k p_k), where n_α is the α‑quantile of the standard normal distribution. The resulting convex program retains the original variance‑minimization objective and can be solved efficiently with a primal‑dual interior‑point algorithm that exploits Jordan algebra for the cone KKT conditions. An initial feasible point is constructed by proportionally allocating based on expected supply and then refined using a Big‑M technique to satisfy the second‑stage constraints.

Computational Study – Experiments on both real ad‑network logs and synthetic data compare SA and CA under various α, ξ, and N settings. The convex approximation achieves objective values virtually identical to the sample‑approximation while being 10–30 times faster, solving instances with hundreds of campaigns and thousands of viewer types in minutes. The theoretical bounds derived for SA hold empirically, confirming that the sampled model reliably brackets the true optimal solution.

Contributions – (i) Formalization of guaranteed‑delivery ad allocation as a chance‑constrained program, (ii) Development of rigorous sample‑based lower/upper bound theory and a practical branch‑and‑bound heuristic, (iii) Introduction of distribution‑free and normal‑distribution convex relaxations that are solvable at scale, and (iv) Demonstration that the convex model preserves the desired representativeness while meeting probabilistic delivery guarantees.

The work provides a mathematically sound yet computationally feasible framework for ad networks to honor advertisers’ guaranteed impression contracts under uncertain traffic, while maintaining a balanced exposure across targeted audiences. Future extensions could incorporate multi‑stage recourse (spot‑market purchases) and dynamic time‑window decisions for real‑time ad serving.