In this manuscript, we introduce concepts of (m1,m2)-logarithmically convex (AG-convex) functions and establish some Hermite-Hadamard type inequalities of these classes of functions.

Constrained optimization problems have a wide range of applications in science, economics, and engineering. In this paper, a neural network model is proposed to solve a class of nonsmooth constrained optimization problems with a nonsmooth convex objective function subject to nonlinear inequality and affine equality constraints. It is a one-layer non-penalty recurrent neural network based on the...

In this paper we study the differentially private Empirical Risk Minimization (ERM) problem in different settings. For smooth (strongly) convex loss function with or without (non)-smooth regularization, we give algorithms that achieve either optimal or near optimal utility bounds with less gradient complexity compared with previous work. For ERM with smooth convex loss function in high-dimensio...

Convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is (possibly) multi–dimensional. Such regression problems arise in operations research, economics, and other disciplines in which imposing a convexity constraint on the regression function is natural. This paper studies a least squares estimator that ...

[1] The channel network (S) is a nonconvex set, while its basin [C(S)] is convex. We remove open-end points of the channel connectivity network iteratively to generate a traveltime sequence of networks (Sn). The convex hulls of these traveltime networks provide an interesting topological quantity, which has not been noted thus far. We compute lengths of shrinking traveltime networks L(Sn) and a...

This paper studies a new multi-objective fuzzy optimization prob- lem. The objective function of this study has dierent levels. Therefore, a suitable optimized solution for this problem would be an optimized solution with preemptive priority. Since, the feasible domain is non-convex; the tra- ditional methods cannot be applied. We study this problem and determine some special structures related...

A convex non-convex variational model is proposed for multiphase image segmen-tation. We consider a specially designed non-convex regularization term which adapts spatially tothe image structures for better controlling of the segmentation and easy handling of the intensityinhomogeneities. The nonlinear optimization problem is efficiently solved by an Alternating Direc-tions Meth...

in this paper, we first introduce the notion of $c$-affine functions for $c> 0$.then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. moreover, a hyers–-ulam stability result for strongly convex functions is shown.

Continuing the work of Hiriart-Urruty and Phelps, we discuss (in both locally convex spaces and Banach spaces) the formulas for the conjugates and subdifferentials of the precomposition of a convex function by a continuous linear mapping and the marginal function of a convex function by a continuous linear mapping. We exhibit a certain (incomplete) duality between the operations of precompositi...