  We propose a new unified framework for describing and designing
gradient-based convex optimization methods from a numerical analysis
perspective. There the key is the new concept of weak discrete gradients (weak
DGs), which is a generalization of DGs standard in numerical analysis. Via weak
DG, we consider abstract optimization methods, and prove unified convergence
rate estimates that hold independent of the choice of weak DGs except for some
constants in the final estimate. With some choices of weak DGs, we can
reproduce many popular existing methods, such as the steepest descent and
Nesterov's accelerated gradient method, and also some recent variants from
numerical analysis community. By considering new weak DGs, we can easily
explore new theoretically-guaranteed optimization methods; we show some
examples. We believe this work is the first attempt to fully integrate research
branches in optimization and numerical analysis areas, so far independently
developed.
