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Table 1 List of symbols

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