ChatGPT May 12 Version says:
The cost function you have specified is a pairwise loss function that penalizes the difference between the output of the ANN for any two input instances. It is similar to the contrastive loss function used in Siamese Networks. Siamese Networks are a type of neural network architecture that consists of two or more identical subnetworks that share the same weights and are used to compare two inputs.
Therefore, a Siamese Network is a suitable choice for minimizing the cost function and satisfying the given constraint. In particular, you can use a Siamese Network with a shared encoder that maps each input instance to a fixed-length vector representation, followed by a distance metric that computes the pairwise distance between the encoded representations. The distance metric can be implemented using a fully connected layer or a custom loss function.
To satisfy the given constraint, you can add a constraint layer to the network that enforces a minimum output value for each input instance. The constraint layer can be implemented using a fully connected layer with a fixed bias term or a custom activation function that saturates at the minimum output value.
Overall, the architecture of the Siamese Network depends on the specifics of the problem, including the dimensionality of the input space and the desired level of accuracy. You may need to experiment with different architectures and hyperparameters to achieve the desired performance.