  Lattices with minimal normalized second moments are designed using a new
numerical optimization algorithm. Starting from a random lower-triangular
generator matrix and applying stochastic gradient descent, all elements are
updated towards the negative gradient, which makes it the most efficient
algorithm proposed so far for this purpose. A graphical illustration of the
theta series, called theta image, is introduced and shown to be a powerful tool
for converting numerical lattice representations into their underlying exact
forms. As a proof of concept, optimized lattices are designed in dimensions up
to 16. In all dimensions, the algorithm converges to either the previously best
known lattice or a better one. The dual of the 15-dimensional laminated lattice
is conjectured to be optimal in its dimension and its exact normalized second
moment is computed.
