No abstract

No abstract

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.

The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.

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.