Catálogo de publicaciones - libros

We investigate the behavior at infinity of a special dissipative system, which consists of two steepest descent equations coupled by a non-autonomous conservative repulsion. The repulsion term is parametrized in time by an asymptotically vanishing factor. We show that under a simple slow parametrization assumption the limit points, if any, must satisfy an optimality condition involving the repulsion potential. Under some additional restrictive conditions, requiring in particular the equilibrium set to be one-dimensional, we obtain an asymptotic convergence result. Finally, some open problems are listed.

Let (, ) be the solution set mapping of a linear parametric optimization problem with parameters in the right hand side and in the objective function. Then, given a point we search for parameter values and as well as for an optimal solution ∈ (, ) such that ‖ − ‖ is minimal. This problem is formulated as a bilevel programming problem. Focus in the paper is on optimality conditions for this problem. We show that, under mild assumptions, these conditions can be checked in polynomial time.

The present paper studies the following constrained vector optimization problem: min (), () ∈ −, () = 0, where : ℝ → ℝ : ℝ → ℝ are functions, : ℝ → ℝ is function, and ⊂ ℝ and ⊂ ℝ are closed convex cones with nonempty interiors. Two type of solutions are important for the consideration, namely -minimizers (weakly efficient points) and -minimizers (isolated minimizers). In terms of the second-order Dini directional derivative second-order necessary conditions a point to be a -minimizer and second-order sufficient conditions to be an -minimizes of order two are formulated and proved. The effectiveness of the obtained conditions is shown on examples.

In this work, a notion of cone-subconvexlikeness of set-valued maps on linear spaces is given and several characterizations are obtained. An alternative theorem is also established for this kind of set-valued maps. Using the notion of vector closure introduced recently by Adán and Novo, we also provide, in this framework, an adaptation of the proper efficiency in the sense of Benson for set-valued maps. The previous results are then applied to obtain different optimality conditions for this Benson-vectorial proper efficiency by using scalarization and multiplier rules.

We discuss some ideas for improvement, extension and application of proximal point methods and the auxiliary problem principle to variational inequalities in Hilbert spaces. These methods are closely related and will be joined in a general framework, which admits a consecutive approximation of the problem data including applications of finite element techniques and the ε-enlargement of monotone operators. With the use of a ”reserve of monotonicity” of the operator in the variational inequality, the concepts of weak- and elliptic proximal regularization are developed. Considering Bregman-function-based proximal methods, we analyze their convergence under a relaxed error tolerance criterion in the subproblems. Moreover, the case of variational inequalities with non-paramonotone operators is investigated, and an extension of the auxiliary problem principle with the use of Bregman functions is studied. To emphasize the basic ideas, we renounce all the proofs and sometimes precise descriptions of the convergence results and approximation techniques. Those can be found in the referred papers.

We consider a general system of equilibrium type problems which can be viewed as an extension of Lagrangean primal-dual equilibrium problems. We propose to solve the system by an inexact proximal point method, which converges to a solution under monotonicity assumptions. In order to make the method implementable, we suggest to make use of a dual descent algorithm and utilize gap functions for ensuring satisfactory accuracy of certain auxiliary problems. Some examples of applications are also given.

The paper considers a general multistage stochastic decision problem which contains Markovian decision processes and multistage stochastic programming problems as special cases. The objective functions, the constraint sets and the probability measures are approximated. Making use of the Bellman Principle, (semi) convergence statements for the optimal value functions and the optimal decisions at each stage are derived. The considerations rely on stability assertions for parametric programming problems which are extended and adapted to the multistage case. Furthermore, new sufficient conditions for the convergence of objective functions which are integrals with respect to decision-dependent probability measures are presented. The paper generalizes results by Langen(1981) with respect to the convergence notions, the integrability conditions and the continuity assumptions.

In this paper one generalizes various types of constrained extremism, keeping the Lagrange or Kuhn-Tucker multipliers rule. The context which supports this development is the nonholonomic optimization theory which requires a holonomic or nonholonomic objective function subject to nonholonomic or holonomic constraints. We refined such a problem using two new ideas: the replacement of the point or velocity constraints by a curve selector, and the geometrical interpretation of the Lagrange and Kuhn-Tucker parameters. The classical optimization theory is recovered as a particular case of extremism constrained by a curve selector.

We consider the interior penalty methods based on the logarithmic and inverse barriers. Under the Mangasarian-Fromovitz constraint qualification and appropriate growth conditions on the objective function, we derive computable estimates for the distance from the subproblem solution to the solution of the original problem. Some of those estimates are shown to be sharp.

In this paper, the problem to control a finite string to the zero state in finite time from a given initial state by controlling the state at the two boundary points is considered. The corresponding optimal control problem where the objective function is the -norm of the controls is solved in the sense that the controls that are successful and minimize at the same time the objective function are determined as functions of the initial state.