An Inertial Proximal-Gradient Penalization Scheme for Constrained Convex Optimization Problems.

We propose a proximal-gradient algorithm with penalization terms and inertial and memory effects for minimizing the sum of a proper, convex, and lower semicontinuous and a convex differentiable function subject to the set of minimizers of another convex differentiable function. We show that, under suitable choices for the step sizes and the penalization parameters, the generated iterates weakly converge to an optimal solution of the addressed bilevel optimization problem, while the objective function values converge to its optimal objective value.

Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data.

We consider the problem of minimizing a smooth convex objective function subject to the set of minima of another differentiable convex function. In order to solve this problem, we propose an algorithm which combines the gradient method with a penalization technique. Moreover, we insert in our algorithm an inertial term, which is able to take advantage of the history of the iterates. We show weak convergence of the generated sequence of iterates to an optimal solution of the optimization problem, provided a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. We also prove convergence for the objective function values to the optimal objective value. The convergence analysis carried out in this paper relies on the celebrated Opial Lemma and generalized Fejér monotonicity techniques. We illustrate the functionality of the method via a numerical experiment addressing image classification via support vector machines.