  In this paper we investigate possible approaches to study general
time-inconsistent optimization problems without assuming the existence of
optimal strategy. This leads immediately to the need to refine the concept of
time-consistency as well as any method that is based on Pontryagin's Maximum
Principle. The fundamental obstacle is the dilemma of having to invoke the {\it
Dynamic Programming Principle} (DPP) in a time-inconsistent setting, which is
contradictory in nature. The main contribution of this work is the introduction
of the idea of the "dynamic utility" under which the original time inconsistent
problem (under the fixed utility) becomes a time consistent one. As a benchmark
model, we shall consider a stochastic controlled problem with multidimensional
backward SDE dynamics, which covers many existing time-inconsistent problems in
the literature as special cases, and we argue that the time inconsistency is
essentially equivalent to the lack of {\it comparison principle}. We shall
propose three approaches aiming at reviving the DPP in this setting: the
duality approach, the dynamic utility approach, and the master equation
approach. Unlike the game approach in many existing works in continuous time
models, all our approaches produce the same value as the original static
problem.
