  We present a new feasible proximal gradient method for constrained
optimization where both the objective and constraint functions are given by the
summation of a smooth, possibly nonconvex function and a convex simple
function. The algorithm converts the original problem into a sequence of convex
subproblems. Formulating those subproblems requires the evaluation of at most
one gradient value of the original objective and constraint functions. Either
exact or approximate subproblem solutions can be computed efficiently in many
cases. An important feature of the algorithm is the constraint level parameter.
By carefully increasing this level for each subproblem, we provide a simple
solution to overcome the challenge of bounding the Lagrangian multipliers and
show that the algorithm follows a strictly feasible solution path till
convergence to the stationary point. We develop a simple, proximal gradient
descent type analysis, showing that the complexity bound of this new algorithm
is comparable to gradient descent for the unconstrained setting, which is new
in the literature. Exploiting this new design and analysis technique, we extend
our algorithms to some more challenging constrained optimization problems where
1) the objective is a stochastic or finite-sum function, and 2) structured
nonsmooth functions replace smooth components of both objective and constraint
functions. Complexity results for these problems also seem to be new in the
literature. Finally, our method can also be applied to convex
function-constrained problems where we show complexities similar to the
proximal gradient method.
