  This paper firstly proposes a convex bilevel optimization paradigm to
formulate and optimize popular learning and vision problems in real-world
scenarios. Different from conventional approaches, which directly design their
iteration schemes based on given problem formulation, we introduce a
task-oriented energy as our latent constraint which integrates richer task
information. By explicitly re-characterizing the feasibility, we establish an
efficient and flexible algorithmic framework to tackle convex models with both
shrunken solution space and powerful auxiliary (based on domain knowledge and
data distribution of the task). In theory, we present the convergence analysis
of our latent feasibility re-characterization based numerical strategy. We also
analyze the stability of the theoretical convergence under computational error
perturbation. Extensive numerical experiments are conducted to verify our
theoretical findings and evaluate the practical performance of our method on
different applications.
