papers/earle-genetic-marketing/sections/03-problem-formulation.tex

\section{Problem Formulation}


\subsection{Campaign Genome}

We represent a campaign as a genome $G = (C, T, P, B)$ where:

\begin{align}
C &= (\text{headline}, \text{body}, \text{image}, \text{cta}) \\
T &= (\text{age}, \text{gender}, \text{interests}, \text{behaviors}) \\
P &= (\text{platform}, \text{placement}) \\
B &= (\text{bid\_type}, \text{bid\_amount})
\end{align}

Each component is a gene with a defined allele space:

\begin{itemize}
\item $\text{headline} \in \Sigma^*$: String from character alphabet $\Sigma$.
\item $\text{age} \in [18, 65]^2$: Age range tuple.
\item $\text{interests} \subseteq I$: Subset of interest categories $I$.
\item $\text{cta} \in \{\text{shop\_now}, \text{learn\_more}, \text{sign\_up}, \ldots\}$: Enumerated options.
\end{itemize}

\subsection{Fitness Function}

The fitness function $f: G \rightarrow \mathbb{R}$ evaluates campaign performance. We define a multi-objective fitness:

\begin{equation}
f(G) = w_1 \cdot \text{CVR}(G) - w_2 \cdot \text{CPA}(G) + w_3 \cdot \text{Brand}(G)
\end{equation}

where:
\begin{itemize}
\item $\text{CVR}(G)$: Conversion rate (conversions / impressions).
\item $\text{CPA}(G)$: Cost per acquisition (spend / conversions).
\item $\text{Brand}(G)$: Brand consistency score (semantic similarity to brand guidelines).
\end{itemize}

Weights $w_1, w_2, w_3$ are configurable per campaign objective.

\subsection{Constraints}

Campaigns must satisfy constraints:

\begin{align}
\text{spend}(G) &\leq \text{budget} \\
\text{reach}(T) &\geq \text{min\_audience} \\
\text{policy}(C) &= \text{approved}
\end{align}

The policy constraint ensures compliance with platform advertising policies (no prohibited content, accurate claims).

\subsection{Optimization Objective}

The optimization problem is:

\begin{equation}
\max_{G \in \mathcal{G}} f(G) \quad \text{subject to constraints}
\end{equation}

where $\mathcal{G}$ is the space of valid campaign genomes.