## 2 Problem Statement and Formulation
* The value of \(KF\) is associated with the interval of analysis corresponding to \(KF\) frames indexed by \(t=f+(k-1)F=1,2,\ldots,KF\), and represents the time over which the network is in operation. In a real network, the parameter \(KF\) is not specified, in this case being determined by the network lifetime. The value of \(KF\), though, can be determined by the network administrator so that the optimization ceases at a desired moment while the network is fully operative. * The number of sensor nodes is denoted by \(N\), and they are indexed by \(n=1,2,\ldots,N\). A single sink node is assumed. * The energies in the sensor nodes' batteries associated with the \(k\)-th optimization event, \(k=1,2,\ldots,K\), are given by the residual energy matrix \(\mathbf{S}(k)=[\mathbf{s}_{1}(k),\mathbf{s}_{2}(k),\ldots,\mathbf{s}_{F+1}(k )]\in\mathbb{R}^{N\times(F+1)}\), with \(\mathbf{s}_{1}(1)\) being the vector with the initial energies stored in the batteries. After the action of the \(k\)-th optimization event, the residual energy values in the vector \(\mathbf{s}_{F+1}(k)\) become the initial energies for the next optimization event, that is \(\mathbf{s}_{1}(k+1)=\mathbf{s}_{F+1}(k)\). Figure 1: Topology adopted for the MWSN. The mobile sensor nodes inside the coverage area are those that can have direct wireless links to the mobile sink node. Others communicate with the sink node by means of multiple hops. * The activity level matrix is defined by \(\textbf{X}(k)=[\textbf{x}_{1}(k),\textbf{x}_{2}(k),\ldots,\textbf{x}_{F}(k)]\in \mathbb{R}^{N\times F}\), with \(\textbf{x}_{f}(k)\succeq 0\) and \(\textbf{1}^{\mathsf{T}}\textbf{x}_{f}(k)=1\), where \(\succeq\) represents component-wise inequality, **1** is the all-one \(N\)-dimensional vector, and the superscript T denotes transposition. This matrix contains the activity levels assigned by the \(k\)-th optimization event to all sensor nodes in the \(f\)-th frame within the optimization span. If, for instance, an element \(X_{n,f}(k)\) of the activity level matrix is 0.1, this means that the \(n\)-th sensor node can occupy up to 10% of the slots during the \(f\)-th frame pertaining to the \(k\)-th group of \(F\) frames. The amount of energy consumed by the sensor nodes is proportional to the assigned activity levels and depends on the channel conditions between the sensor nodes and the sink node. These channels conditions, in turn, depend on the relative position of each sensor node with respect to the sink node, which is assumed to be unknown. This is important in practice, since channel estimation is a computationally costly process that contributes with the increase of the energy consumption and of the system complexity. * If the maximum activity level of one is assigned to the sensor nodes, the energies that they are expected to spend during the transmissions of \(F\) frames are represented by the maximum consumption matrix \(\textbf{B}(k)=[\textbf{b}_{1}(k),\textbf{b}_{2}(k),\ldots,\textbf{b}_{F}(k)] \in\mathbb{R}^{N\times F}\), with \(\textbf{B}(1)\) denoting the expected maximum consumption during the first \(F\) frames. Then, during the \(k\)-th block of \(F\) frames, the energies that are expected to be consumed by the sensor nodes are given by \(\textbf{B}(k)\circ\textbf{X}(k)\), where the symbol \(\circ\) denotes the Hadamard product (element-wise multiplication). The elements of \(\textbf{B}(k)\) in practice will depend on the sensor nodes' characteristics and on the distances and channels between the sensor nodes and the sink node. * It is assumed that the estimated consumption during the first \(F\) frames is constant, yielding \(\textbf{b}_{1}(1)=\textbf{b}_{2}(1)=\cdots=\textbf{b}_{F}(1)\). This means that transmissions during the first frame are assumed to be done before the sensor nodes start moving. These first transmissions do not need to carry useful sensory information, being used to correctly update the energy consumption for the next optimization event.