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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