From: Evaluation of linear relaxations in Ad Network optimization for online marketing
Symbol | Definition |
---|---|
\(\mathcal {C}\) | Campaign set |
\(k \in \mathcal {C}\) | A campaign |
B _{ k } | Budget of campaign k in number ofclicks |
S _{ k } | Starting time of the campaign |
L _{ k } | Lifetime of the campaign |
P _{req} | Probability that a request is receivedby the Ad Network |
\(\mathcal {G}\) | Set of possible user profiles |
\(P_{\mathcal {G}}: \mathcal {G} \rightarrow [0,1]\) | Probability that a user belongs to auser profile i |
\(i \in \mathcal {G}\) | A user profile |
CTR(i,k) | Click-through rate indicates theprobability that user i clicks on ad k |
c c _{ k } | Revenue generated by a click oncampaign k |
\(\mathcal {S}\) | Set of fully observable MDP states |
\(s \in \mathcal {S}\) | A state of the MDP model |
\(\mathcal {A}\) | Set of possible actions |
\(a \in \mathcal {A}\) | An action of the MDP model |
\(\mathcal {A}(s,t)\) | Set of valid actions at instant t in state\(s \in \mathcal {S}\) |
\(\mathcal {D}\) | Set of decision epochs |
\(t \in \mathcal {D}\) | A decision epoch |
\(\mathcal {T}:\mathcal {S}\times \mathcal {A}\times \mathcal {S}\times \mathcal {D}\rightarrow [0,1]\) | Transition function |
\(\mathcal {T}(s, a, s', t)\) | Transition from state s to s ^{′} whenexecutes action a in time t |
Reward function | |
\(\pi :\mathcal {S}\times \mathcal {D} \rightarrow \mathcal {A}\) | Non-stationary deterministic policy |
V ^{π}(s,t) | Value function of policy π |
V ^{∗}(s,t) | Value function of optimal policy π ^{∗} |
\(G \in \mathcal {G} \cup \{0\}\) | User profile that is generating a request(0 stands for no request to attend to) |
\(\mathcal {J}_{j}\) | Set of intervals defined by thecampaign time constraints |
\(\mathbb {T}_{j}\) | Length of the interval j |
x _{ j,i,k } | Variable that indicates how many adsfrom campaign k should be displayedto users with user profile i at theinterval j |
η _{ k } | Average cost per impression ofcampaign k |
ctr_{ k } | CTR(1,k) for one-profile scenario |
R | Expected revenue when the AdNetwork applies the solution of theLP relaxation |
γ | Variable that artificially increases thebudget, \(B_{k}^{'} = \gamma B_{k}\), γ>1 |