Related papers: Applications of Convex Analysis within Mathematics
This note deals with certain properties of convex functions. We provide results on the convexity of the set of minima of these functions, the behaviour of their subgradient set under restriction, and optimization of these functions over an…
The construction of highly incoherent frames, sequences of vectors placed on the unit hyper sphere of a finite dimensional Hilbert space with low correlation between them, has proven very difficult. Algorithms proposed in the past have…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial…
The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperK\"ahler with Torsion). The…
The main result (roughly) is that if (H_i) converges weakly to H and if also f(H_i) converges weakly to f(H), for a single strictly convex continuous function f, then (H_i) must converge strongly to H. One application is that if f(pr(H)) =…
Operator monotone functions, introduced by Lowner in 1934, are an important class of real-valued functions. They arise naturally in matrix and operator theory and have various applications in other branches of mathematics and related…
Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This…
A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently, this method may be very useful for explicit description of solutions of optimal…
We provide a concise analysis about what is known regarding when the closure of the domain of a maximally monotone operator on an arbitrary real Banach space is convex. In doing so, we also provide an affirmative answer to a problem posed…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
Operator learning has been highly successful for continuous mappings between infinite-dimensional spaces, such as PDE solution operators. However, many operators of interest-including differential operators-are discontinuous or set-valued,…
We present several matrix and operator inequalities of Hermite-Hadamard type. We first establish a majorization version for monotone convex functions on matrices. We then utilize the Mond-Pecaric method to get an operator version for convex…
We propose a method for solving constrained fixed point problems involving compositions of Lipschitz pseudo contractive and firmly nonexpansive operators in Hilbert spaces. Each iteration of the method uses separate evaluations of these…
Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this…
In this article, we prove an inner product inequality for Hilbert space operators. This inequality, then, is utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining…
Monotone inclusions have a wide range of applications, including minimization, saddle-point, and equilibria problems. We introduce new stochastic algorithms, with or without variance reduction, to estimate a root of the expectation of…
This works aims at understanding further convergence properties of first order local search methods with complex geometries. We focus on the composite optimization model which unifies within a simple formalism many problems of this type. We…
Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…
This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and their noisy images under an unknown self-adjoint linear…