## 3 Problem Statement
### Multicriteria Optimization of ANN Structure  
The problem of multicriteria unconditional optimization of the ANN structure can be formally expressed as follows [44; 45]:  
\[\left\{E(C,W,af)\rightarrow\min_{C,W,af}\quad CD(C,af)\rightarrow\min_{C,af}\right\} \tag{1}\]  
where \(E(\cdot)\) quantifies the overall root-mean-square error during ANN training, while \(CD(\cdot)\) measures the network's computational workload. The variables \(C\in\mathbb{R}^{N_{n}\times N_{n}}\), \(W\in\mathbb{R}^{N_{n}\times N_{n}}\), and \(\mathbf{af}\in\mathbb{R}^{N_{n}}\) represent the connectivity matrix, the weight matrix for each connection, and the activation-function vector for all neurons, respectively. The symbol \(N_{n}\) indicates the total count of neurons in the network.  
The total RMS error of ANN training is calculated by the following equation:  
\[E(C,W,af)=\frac{1}{m}\left(\sum_{j=1}^{m}\sqrt{\frac{1}{n}\sum_{k=1}^{m}\left(OUT _{j}^{k}(C,W,af)-y_{kj}\right)^{2}}\right) \tag{2}\]  
where \(OUT_{j}^{k}(C,W,af)\) is the actual output of the \(k\)-th neuron in the ANN with structure \((C,W,af)\) when the \(j\)-th image is presented as input, and \(y_{kj}\) is the desired output for the \(k\)-th neuron. Here, \(n\) represents the number of neurons at the output of the network, and \(m\) is the size of the training sample.  
To estimate the computational complexity of the ANN, the following expression is used:  
\[CD(C,af,P)=\sum_{i=1}^{N_{link}(C)}T_{link}(C,P)N_{link}(C)+\sum_{i=1}^{N_{n} }T_{iact}(af,P)N_{n} \tag{3}\]  
where \(N_{link}(\cdot)\) is the number of ANN links, \(T_{link}(\cdot)\) is the processing time for the \(i\)-th ANN link, \(T_{iact}(\cdot)\) is the time to compute the activation function on the \(i\)-th neuron, \(N_{n}\) is the number of neurons in the ANN, and \(P\) is the performance of the computing system.  
The processing time for a single connection within the ANN is given by the following equation:  
\[T_{i}^{link}(C,P)=T\left(\sum_{j=1}^{n+1}w_{ij}\right)-T\left(\sum_{j=1}^{n}w _{ij}\right) \tag{4}\]  
where \(T(x)\) represents the computation time of \(x\), and \(w_{ij}\) is the weight of the \(i\)-th input of the \(j\)-th neuron.  
The processing time remains constant across all links and depends on the software and hardware configuration of the ANN, leading to the following reformulation of Equation (3):  
\[CD(C,af,P)=N_{link}(C)\cdot T_{link}(P)+\sum_{i=1}^{N_{n}}T_{iact}(af,P) \tag{5}\]  
where \(T_{link}(P)\) represents the processing time required for a single ANN link.  
The time \(T_{link}(P)\) depends on the computational power of a specific system. To standardize the computational complexity of the ANN, dimensionless values independent of time are used:  
\[\frac{CD(C,af,P)}{T_{link}(P)}=N_{link}(C)\cdot T_{link}(P)+\sum_{i=1}^{N_{n }}T_{iact}(af,P)\cdot T_{link}(P) \tag{6}\]  
Next, \(CD(C,af)=\frac{CD(C,af,P)}{T_{link}(P)}\) is defined, and a hardware-independent coefficient for the relative complexity of calculating the activation function on the \(i\)-th neuron is introduced as \(K_{i}(af)=\frac{T_{iact}(af,P)}{T_{link}(P)}\). As a result, the criterion for assessing the computational complexity of the ANN model reaches its final form:  
\[CD(C,af)=N_{link}(C)+\sum_{i=1}^{N_{n}}K_{i}(af) \tag{7}\]With the derived expressions for calculating the tuning error (2) and computational complexity (7), the problem of multicriteria unconditional optimization of the ANN structure can be formulated as follows:  
\[\sum_{k=1}^{n}\sum_{j=1}^{m}\sqrt{\left(OUT_{j}^{k}(C,W,af)-y_{kj}\right)^{2}} \rightarrow\min_{C,W,af} \tag{8}\]  
\[N_{link}(C)+\sum_{i=1}^{N_{k}}K_{i}(af)\rightarrow\min \tag{9}\]