A bornological approach to rotundity and smoothness applied to approximation.
A Clarke–Ledyaev Type Inequality for Certain Non–Convex Sets
We consider the question whether the assumption of convexity of the set involved in Clarke-Ledyaev inequality can be relaxed. In the case when the point is outside the convex hull of the set we show that Clarke-Ledyaev type inequality holds if and only if there is certain geometrical relation between the point and the set.
A critical point result for non-differentiable indefinite functionals
In this paper, two deformation lemmas concerning a family of indefinite, non necessarily continuously differentiable functionals are proved. A critical point theorem, which extends the classical result of Benci-Rabinowitz [14, Theorem 5.29] to the above-mentioned setting, is then deduced.
A duality-based approach to elliptic control problems in non-reflexive Banach spaces
Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...
A duality-based approach to elliptic control problems in non-reflexive Banach spaces*
Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...
A generalized differential of real functionals.
A Mean Value Theorem for non Differentiable Mappings in Banach Spaces
We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition....
A minimum effort optimal control problem for elliptic PDEs
This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...
A minimum effort optimal control problem for elliptic PDEs
This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...
A new approach to the constrained controllability problem
We consider the problem of internal regional controllability with output constraints. It consists in steering a hyperbolic system to a final state between two prescribed functions only on a subregion of the evolution system domain. This problem is solved by characterizing the optimal control in terms of a subdifferential associated with the minimized functional.
A new method for nonsmooth convex optimization.
A new subdifferential in quasiconvex analysis.
A note on a class of equilibrium problems with equilibrium constraints
The paper concerns a two-level hierarchical game, where the players on each level behave noncooperatively. In this way one can model eg an oligopolistic market with several large and several small firms. We derive two types of necessary conditions for a solution of this game and discuss briefly the possibilities of its computation.
A note on Banach spaces and sub differentials of convex functions
A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory
The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower...
A note on Minty type vector variational inequalities
The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space are introduced....
A note on Minty type vector variational inequalities
The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space Y are introduced. Under...
A note on singular points of convex functions in Banach spaces
A note on the characterization of the global maxima of a (tangentially) convex function over a convex set.
A polynomial of degree four not satisfying Rolle’s Theorem in the unit ball of
We give an example of a fourth degree polynomial which does not satisfy Rolle’s Theorem in the unit ball of .