Quasiconvex Research Articles

Improved incentive pricing-based quasi-linear utility function of wireless networks

The model of the incentive pricing scheme-based quasi-linear utility function in wireless network was designed. Previous research seldom focusses on user’s satisfaction while using network. Therefore, the model is then attempted to be set up that is derived from the modification of bundling and models of reverse charging and maintain the quality of service to users by utilizing quasi-linear utility function. The pricing schemes then are applied to local data server traffic. The model used is known as mathematical programming problem that can be solved by LINGO 13.0 program as optimization tool to get the optimal solution. The optimal results show that the improved incentive pricing can achieve better solution compared to original reverse charging where the models will be obtained in flat fee, usage-based, and two-part tariff strategies for homogeneous consumers.

Fractional Weighted Ostrowski-Type Inequalities and Their Applications

An important area in the field of applied and pure mathematics is the integral inequality. As it is known, inequalities aim to develop different mathematical methods. Nowadays, we need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition and its properties. Furthermore, there is a strong correlation between convexity and symmetry concepts. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the last few years. In this study, by using a new identity, we establish some new fractional weighted Ostrowski-type inequalities for differentiable quasi-convex functions. Further, further results for functions with a bounded first derivative are given. Finally, in order to illustrate the efficiency of our main results, some applications to special means are obtain. The obtained results generalize and refine certain known results.

On quasimonotone Stampacchia variational inequalities on Hadamard manifolds

In this paper, which is deeply inspired from Aussel and Hadjisavvas [On quasimonotone variational inequalities. J Optim Theory Appl. 2004;121:445–450] and Daniilidis and Hadjisavvas [Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions. J Optim Theory Appl. 1999;102(3):525–536], we study the existence of solutions of the Stampacchia variational inequality for a quasimonotone set-valued vector field on a Hadamard manifold. Moreover, the existence results are obtained under weak assumptions like quasimonotonicity and upper-sign continuity. An application of our results is also given.

A regularity result for minimal configurations of a free interface problem

We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region varOmega subseteq {mathbb {R}}^n. We will deal with the energy functional {mathscr {F}}(v,E):=int _varOmega [F(nabla v)+1_E G(nabla v)+f_E(x,v)],dx+P(E,varOmega ). The bulk energy depends on a function v and its gradient nabla v. It consists in two strongly quasi-convex functions F and G, which have polinomial p-growth and are linked with their p-recession functions by a proximity condition, and a function f_E, whose absolute valuesatisfies a q-growth condition from above. The surface penalization term is proportional to the perimeter of a subset E in varOmega . The term f_E is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with {mathscr {F}}. The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove that u is locally Hölder continuous and A is equivalent to an open set {tilde{A}}. We finally get P(A,varOmega )={mathscr {H}}^{n-1}(partial {tilde{A}}cap varOmega ).

Characterization of Nonsmooth Quasiconvex Functions and their Greenberg–Pierskalla’s Subdifferentials Using Semi-Quasidifferentiability notion

The relationships between Greenberg–Pierskalla’s subdifferential and some variants of it and semi-quasidifferentials of quasiconvex functions are studied in this paper. Also, some characterizations of quasiconvex functions in terms of semi-quasidifferentials and the connection between quasimonotonicity of semi-quasidifferentials and quasiconvexity are investigated.

Characterizing quasiconvexity of the pointwise infimum of a family of arbitrary translations of quasiconvex functions, with applications to sums and quasiconvex optimization

It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasiconvex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property. Three concrete applications in quasiconvex optimization are presented: First, we establish the convexity of the (Benson) proper efficient solution set to a quasiconvex vector optimization problem; second, we derive conditions that allow us to reduce a constrained optimization problem to one with a single inequality constraint, and finally, we show a class of quasiconvex minimization problems having zero duality gap.

NEW MULTI-FUNCTIONAL APPROACH FOR κTH-ORDER DIFFERENTIABILITY GOVERNED BY FRACTIONAL CALCULUS VIA APPROXIMATELY GENERALIZED (ψ, ℏ)-CONVEX FUNCTIONS IN HILBERT SPACE

This work addresses several novel classes of convex function involving arbitrary non-negative function, which is known as approximately generalized [Formula: see text]-convex and approximately [Formula: see text]-quasiconvex function, with respect to Raina’s function, which are elaborated in Hilbert space. To ensure the feasibility of the proposed concept and with the discussion of special cases, it is presented that these classes generate other classes of generalized [Formula: see text]-convex functions such as higher-order strongly (HOS) generalized [Formula: see text]-convex functions and HOS generalized [Formula: see text]-quasiconvex function. The core of the proposed method is a newly developed Simpson’s type of identity in the settings of Riemann–Liouville fractional integral operator. Based on the HOS generalized [Formula: see text]-convex function representation, we established several theorems and related novel consequences. The presented results demonstrate better performance for HOS generalized [Formula: see text]-quasiconvex functions where we can generate several other novel classes for convex functions that exist in the relative literature. Accordingly, the assortment in this study aims at presenting a direction in the related fields.

Asymptotic behavior for a quasi-autonomous gradient system of expansive type governed by a quasiconvex function

We consider the quasi-autonomous first-order gradient system $$\displaylines{ \dot{u}(t)=\nabla\phi(u(t))+f(t),\quad t\in [0,+\infty)\cr u(0)=x_0\in H, }$$ where \(\phi:H\to\mathbb{R}\) is a differentiable quasiconvex function such that \(\nabla\phi\) is Lipschitz continuous. We study the asymptotic behavior of solutions to this system in continuous and discrete time. We show that each solution either approaches infinity in norm or converges weakly to a critical point of \(\phi\). This further concludes that the existence of bounded solutions and implies that \(\phi\) has a nonempty set of critical points. Some strong convergence results, as well as numerical examples, are also given in both continuous and discrete cases.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/15/abstr.html

New quantum integral inequalities for some new classes of generalizedψ-convex functions and their scope in physical systems

AbstractIn the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as theψ-s-convex andψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard ({\mathcal{ {\mathcal H} {\mathcal H} }})-type inequalities regarding our proposed technique are derived via generalizedψ-quasi-convex and generalizedψ-s-convex functions. Considering an identity, several new inequalities connected to the{\mathcal{ {\mathcal H} {\mathcal H} }}type for twice differentiable functions for the aforesaid classes are derived. The consequences elaborated here, being very broad, are figured out to be dedicated to recapturing some known results. Appropriate links of the numerous outcomes apprehended here with those connecting comparatively with classical quasi-convex functions are also specified. Finally, the proposed study also allows the description of a process analogous to the initial and final condition description used by quantum mechanics and special relativity theory.

New Inequalities of Fejer´ and Hermite-Hadamard type Concerning Convex and QuasiConvex Functions With Applications

This research contains new integral inequalities of Fejer and ´ Hermite-Hadamard type involving convex and quasi-convex functions. Applications of the newly established results for special means of positive real numbers are given.

Quasi-Duality Result and Linearization in Multiobjective Quasiconvex Programming

In this paper, multiobjective optimization problems with nondifferentiable quasiconvex functions are considered. We obtain some duality results and a linear representation for the considered problems. Since the well-known strong duality result is not valid for the problems, we present a weaker form of it, named quasi-strong duality result.

Asymptotic behaviour of a dynamical system governed by non-monotone potential and non-potential operators

We consider the following second order equation where is quasi-nonexpansive and Lipschitz continuous on bounded sets and is a continuously differentiable quasiconvex function such that is Lipschitz continuous on bounded sets. We study the asymptotic behaviour of solutions to this equation. Assuming some mild conditions on the operators, we prove weak and strong convergence of solutions to some point in . We also obtain similar results for the asymptotic behaviour of solutions to the discrete version of the above equation. Finally, we apply our results to solving a minimization problem and approximating a common fixed point of two mappings. Our work is motivated by the papers of H. Attouch and P. E. Maingé [Asymptotic behaviour of second-order dissipative evolution equations combining potential with non-potential effects. ESAIM Control Optim Calc Var. 2011;17:836–857.], X. Goudou and J. Munier [The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math Program Ser B. 2009;116:173–191.], and F. Alvarez and H. Attouch [An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping. Well-posedness in optimization and related topics. Set Valued Anal. 2001;9:3–11.], and extends some of their results.

Cebyšev inequalities for co-ordinated \(QC\)-convex and \((s,QC)\)-convex

In this paper, we establish some new Cebyšev type inequalities for functions whose modulus of the mixed derivatives are co-ordinated quasi-convex and \(\alpha \)-quasi-convex and \(s\)-quasi-convex functions.

Best approximation for multivalued mappings in paranormed spaces

In this paper, we give some best approximation results for multivalued quasiconvex mappings in paranormed spaces. As corollaries, we obtain some fixed point results for continuous for single-valued and multi-alued mappings.

Some General Integral Operator Inequalities Associated with φ -Quasiconvex Functions

This paper deals with generalized integral operator inequalities which are established by using φ -quasiconvex functions. Bounds of an integral operator are established which have connections with different kinds of known fractional integral operators. All the results are deducible for quasiconvex functions. Some fractional integral inequalities are deduced.

On global subdifferentials with applications in nonsmooth optimization

The notions of global subdifferentials associated with the global directional derivatives are introduced in the following paper. Most common used properties, a set of calculus rules along with a mean value theorem are presented as well. In addition, a diversity of comparisons with well-known subdifferentials such as Frechet, Dini, Clarke, Michel–Penot, and Mordukhovich subdifferential and convexificator notion are provided. Furthermore, the lower global subdifferential is in fact proved to be an abstract subdifferential. Therefore, the lower global subdifferential satisfies standard properties for subdifferential operators. Finally, two applications in nonconvex nonsmooth optimization are given: necessary and sufficient optimality conditions for a point to be local minima with and without constraints, and a revisited characterization for nonsmooth quasiconvex functions.

Convex and quasiconvex functions in metric graphs

<p style='text-indent:20px;'>We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest viscosity subsolution to a simple differential equation, <inline-formula><tex-math id="M1">\begin{document}$ u'' = 0 $\end{document}</tex-math></inline-formula> on the edges, plus nonlinear transmission conditions at the vertices. We also study the analogous problem for quasiconvex functions and obtain a characterization of the largest quasiconvex function that is below a given datum.</p>

Lower semi-continuity for 𝒜-quasiconvex functionals under convex restrictions

We show weak lower semi-continuity of functionals assuming the new notion of a “convexly constrained” 𝒜-quasiconvex integrand. We assume 𝒜-quasiconvexity only for functions defined on a set K which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent “convexly constrained” 𝒜-free sequences. In a motivating example, the integrand is − det1/d−1 and the convex constraint is positive semi-definiteness of a matrix field.

The q-difference operator associated with the multivalent function bounded by conical sections

In this paper we obtain some inclusion relations of k - starlike functions, k - uniformly convex functions and quasi-convex functions. Furthermore, we obtain coe¢ cient bounds for some subclasses of fractional q-derivative multivalent functions together with generalized Ruscheweyh derivative.

$\mathcal{A}$-Quasiconvexity, Gårding Inequalities, and Applications in PDE Constrained Problems in Dynamics and Statics

A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimizers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$-local minimizers in the classical $\mathcal{A}={curl}$ case.