In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.

This paper is devoted to constructing a definite efficient scheme for non-smooth optimization. A separating plane algorithm with additional clipping is proposed. The algorithm is used for solving the unconstrained non-smooth convex optimization problem. The latter problem can be reformulated as the computation of the value of a conjugate function at the origin. The algorithm convergence is prov...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

Abstract Given an exact symplectic map T of a cylinder with generating function H satisfying the so-called negative twist condition, H 12 > 0 , we study locally maximis...

The notions of a quasi-monotone operator and of a cyclically quasi-monotone operator are introduced, and relations between such operators and quasi-convex functions are established.

The semidefinite matrix rank minimization, which has a broad range of applications in system control, statistics, network localization, econometrics and so on, is computationally NPhard in general due to the noncontinuous and non-convex rank function. A natural way to handle this type of problems is to substitute the rank function into some tractable surrogates, most popular ones of which inclu...

Logistic regression and matrix factorization specifically SVD are both very popular in application areas of computational advertising and recommender system. This project basically can be divided into two parts. The first part is to test different optimization methods to solve sparse 2-class logistic regression. This is a convex non-smooth problem. Particularly, I’ll try to answer the question ...

Regularization is an effective strategy for reducing noise in tomographic reconstruction. This paper proposes a spatially weighted non-convex (SWNC) regularization method for digital breast tomosynthesis (DBT) image reconstruction. With a non-convex cost function, this method can suppress noise without blurring microcalcifications (MC) and spiculations of masses. To minimize the non-convex cost...

The equivalence can be formalized as follows: For a particular c in (21), there is a corresponding δ > 0 in the optimization in (A-1). We focus on `1-ARD where f(x) = ‖x‖1. Then the objective is concave in H. One natural way to solve (A-1) iteratively is to use an MM procedure by upper bounding the objective function with its tangent (first-order Taylor expansion) at the current iterate H. This...

The semidefinite matrix rank minimization, which has a broad range of applications in system control, statistics, network localization, econometrics and so on, is computationally NPhard in general due to the noncontinuous and non-convex rank function. A natural way to handle this type of problems is to substitute the rank function into some tractable surrogates, most popular ones of which inclu...