Understanding Deep Learning through Energy Landscapes

A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. A non-convex function "curves up and down" -- it is neither convex nor concave. However note that this function is convex from -pi to 0, and concave from 0 to pi. If the bounds on the variables restrict the domain of the objective and constraints to a region where the functions are convex, then the overall problem is convex. Linear functions are convex, so linear programming problems are convex problems.