In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on the Euclidean space and the nonnegative orthant. For the unconstrained minimization problem, assumming that the function is bounded from below and lower semicontinuous we prove that iterations fxkg given by 0 2 b@(f(:)+(k=2)jj:xk1jj2)(xk) are … Read more
This paper extends the full convergence of the steepest descent algorithm with a generalized Armijo search and a proximal regularization to solve quasiconvex minimization problems defined on complete Riemannian manifolds. Previous convergence results are obtained as particular cases of our approach and some examples in non Euclidian spaces are given. CitationJ. Math. Anal. Appl. 341 … Read more
In this paper we extend in a simple way the transformation of Charnes and Cooper to the case where the functional ratio to be considered are of similar polynomial CitationUniversidad de San Luis Ejercito de Los Andes 950 San Luis(5700) ArgentinaArticleDownload View PDF
This paper generalizes the proximal point method using Bregman distances to solve convex and quasiconvex optimization problems on noncompact Hadamard manifolds. We will proved that the sequence generated by our method is well defined and converges to an optimal solution of the problem. Also, we obtain the same convergence properties for the classical proximal method, … Read more
The minimization of an objective function over a constraint set can often be simplified if the “active manifold” of the constraints set can be correctly identified. In this work we present a simple subproblem, which can be used inside of any (convergent) optimization algorithm, that will identify the active manifold of a “prox-regular partly smooth” … Read more
Fundamental insights into the properties of a function come from the study of its Moreau envelopes and Proximal point mappings. In this paper we examine the stability of these two objects under several types of perturbations. In the simplest case, we consider tilt-perturbations, i.e. perturbations which correspond to adding a linear term to the objective … Read more
The generalized mean function has been widely used in convex analysis and mathematical programming. This paper studies a further generalization of such a function. A necessary and sufficient condition is obtained for the convexity of a generalized function. Additional sufficient conditions that can be easily checked are derived for the purpose of identifying some classes … Read more
The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with … Read more
We show that the block-structured distance to non-surjectivity of a set-valued sublinear mapping equals the reciprocal of a suitable block-structured norm of its inverse. This gives a natural generalization of the classical Eckart and Young identity for square invertible matrices. CitationMathematical Programming 103 (2005) pp. 561–573.
The problem of finding singularities of monotone vectors fields on Hadamard manifolds will be considered and solved by extending the well-known proximal point algorithm. For monotone vector fields the algorithm will generate a well defined sequence, and for monotone vector fields with singularities it will converge to a singularity. It will be also shown how … Read more