Related papers: Convex Analysis and Duality
Nonlinear contraction theory is a comparatively recent dynamic control system design tool based on an exact differential analysis of convergence, in essence converting a nonlinear stability problem into a linear time-varying stability…
In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the…
We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.
In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined…
Some of the most important results in prediction theory and time series analysis when finitely many values are removed from or added to its infinite past have been obtained using difficult and diverse techniques ranging from duality in…
In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from…
We study the filtering of the perspective of a regular operator map of several variables through a completely positive linear map. By this method we are able to extend known operator inequalities of two variables to several variables; with…
We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…
In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second…
Solutions to network optimization problems have greatly benefited from developments in nonlinear analysis, and, in particular, from developments in convex optimization. A key concept that has made convex and nonconvex analysis an important…
Motivated by the Lyapunov convexity theorem in infinite dimensions, we extend the convexity of the integral of a decomposable set to separable Banach spaces under the strengthened notion of nonatomicity of measure spaces, called…
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…
In this work, optimality conditions and classical results from duality theory are derived for continuous-time linear optimization problems with inequality constraints. The optimality conditions are given in the Karush-Kuhn-Tucker form. Weak…
General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can…
We address the issue of binary classification in Banach spaces in presence of uncertainty. We show that a number of results from classical support vector machines theory can be appropriately generalised to their robust counterpart in Banach…
A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz…
Methods for measuring convexity defects of compacts in R^n abound. However, none of the those measures seems to take into account continuity. Continuity in convexity measure is essential for optimization, stability analysis, global…
We introduce a framework based on Rockafellar's perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such…
Functions that are not differentiable in the classical sense have become a central tool in modern mathematical models for imaging, inverse problems, machine learning, and optimal control of differential equations. These models are…
We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have…