The answer to the question Is Neural Network Convex depends on the model. The simplest convex neural network contains one layer. Later layers can be highly recursive functions of their predecessors.
This means that if a network has multiple local minima, each of them has the same value. However, this is not the case if a neural network uses multiple layers. The matrix containing all the second partial derivatives of the model is convex.
Another approach is to use constrained deep learning. This approach can be used to train convex models based on convex optimization. In constrained deep learning, data is used to train models that approximate constraints and dynamics of a system.
Constrained model predictive control is a convex optimization problem, and constrained neural networks are effective in enhancing control performance. A simulation case study shows that convex neural networks outperform the normal L2-regularized version in improving control performance.