Quasi-Newton methods are widely used in practise for convex loss minimization problems. These methods exhibit good empirical performance on a wide variety of tasks and enjoy super-linear convergence to the optimal solution. For largescale learning problems, stochastic Quasi-Newton methods have been recently proposed. However, these typically only achieve sub-linear convergence rates and have no...

Abstract. We consider the problem of minimizing the sum of a convex function and of p > 1 fractions subject to convex constraints. The numerators of the fractions are positive convex functions, and the denominators are positive concave functions. Thus, each fraction is quasi-convex. We give a brief discussion of the problem and prove that in spite of its special structure, the problem is NPcomp...

In the frame of fractional calculus, term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus objective this review paper present Hermite–Hadamard (H-H)-type inequalities involving a variety classes convexities pertaining integral operators. Included various are classical convex functions, m-convex r-convex (α,m)-convex (α,m)-geometric...

Recently, Al-Baali (2014) has extended the damped-technique in the modified BFGS method of Powell (1978) for Lagrange constrained optimization functions to the Broyden family of quasi-Newton methods for unconstrained optimization. Appropriate choices for the damped-parameter, which maintain the global and superlinear convergence property of these methods on convex functions and correct the Hess...

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.

A variant of Jessen’s inequality for superquadratic functions is proved. This is a refinement of a variant of Jessen’s inequality of Mercer’s type for convex functions. The result is used to refine some comparison inequalities of Mercer’s type between functional power means and between functional quasi-arithmetic means.

Jensen–Steffensen type inequalities for P -convex functions and functions with nondecreasing increments are presented. The obtained results are used to prove a generalization of Čebyšev’s inequality and several variants of Hölder’s inequality with weights satisfying the conditions as in the Jensen–Steffensen inequality. A few well-known inequalities for quasi-arithmetic means are generalized.

Quasi-homogeneity of copulas is introduced and studied. Quasi-homogeneous copulas are characterized by the convexity and strict monotonicity of their diagonal sections. As a by-product, a new construction method for copulas when only their diagonal section is known is given.

Correspondence: lhorvath@almos. vein.hu Department of Mathematics, University of Pannonia, 8200 Veszprém, Egyetem u. 10., Hungary Abstract In this paper, a new parameter-dependent refinement of the discrete Jensen’s inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of the proofs requires an interesting convergence theorem w...