Convexity Theory Research Articles

Cognitive radio inspired RSMA-MEC under hardware impairments: Performance analysis and optimization

In this paper, we investigate a cognitive radio (CR) inspired rate splitting multiple access (RSMA) aided mobile edge computing (MEC) network, where hardware impairments (HIs) in all transceivers are considered. Given that allowing the secondary user (SU) to share the resource with the primary user (PU) for data offloading does not degrade the PU’s offloading performance, we derive closed-form expressions for the rate-splitting parameter and power allocation coefficient at the SU to maximize its offloading rate. Using the derived parameters, we then derive the successful computation probability (SCP), which is defined as the probability that both the PU and SU can successfully compute their task bits within a given latency budget, into the closed form. To further enhance the performance of the CR inspired RSMA-MEC network with HIs, a SCP maximization-based problem is formulated to jointly optimize the task offloading ratio and task offloading time of both the SU and PU. Leveraging the convex theory, we obtain the optimal solutions in the closed form. Simulation results confirm the following two insights. First, the presence of HIs leads to a decreasing on the SCP and the SCP approaches to a constant which is less than 1 with the increase of the transmit power. Second, with the optimal task offloading ratio and time, the CR inspired RSMA-MEC achieves the highest SCP compared to the existing schemes.

On the Geodetic and Strong Geodetic Parameters of Certain Hypercubic Networks

Networks derived from the hypercube are noted for their efficiency in simulating any bounded-degree communication network and hence they play a significant role in parallel machines. For a graph [Formula: see text], a geodetic cover is a subset [Formula: see text] of [Formula: see text], satisfying the condition that every vertex of [Formula: see text] other than the vertices in [Formula: see text] are covered using geodesics connecting any two vertices in [Formula: see text]. The geodetic number of [Formula: see text] is the least order of such covers in [Formula: see text]. This seminal distance-based parameter is well employed in the areas of location theory, convexity theory and game theory. The strong variant and its edge version are recent variations of the problem that were introduced in due of their applications in social networks. The most established and strong hypercube derivative networks are studied in this paper with respect to certain geodetic parameters. Notably, among the networks that we have considered, the butterfly and the Benes networks belong to the class of bipartite graphs for which the geodetic number problem and its strong version are NP-complete. We have determined the geodetic parameters that include the geodetic number, the strong geodetic number and their corresponding edge variants of the networks taken into consideration. In the process, we have identified a new class of graphs, as semi-extreme edge geodesic graphs and that the enhanced butterfly networks belong to this class. The study of such geodetic parameters in the structural models of parallel computation could be useful for the design of efficient algorithms for the structures considered.

Hermite-Hadamart Integral Inequality for Differentiable m- Convex Functions

Convexity in connection with integral inequalities is an interesting research domain in recent years. The convexity theory plays a fundamental role in the development of various branches of applied sciences since it includes the theory of convex functions that possesses two important attributes viz. a boundary point is where the maximum value is reached and any local minimum value is a global one. Convexities and inequalities are connected which has a basic character in many branches of pure and applied disciplines. The most important inequality related to convex function is the Hermite-Hadamard integral inequality. The extensions, enhancements and generalizations of this inequality has motivated the researchers in recent years. This paper is an extension of some inequalities connected with difference of the left-hand part as well as the right-hand part from the integral mean in Hermite- Hadamard’s inequality for the case of m- convex functions.

Results on E-quasiconvex Functions for a linear map E

Abstract It is well known that convexity theory and its generalization play an important role and have many applications in mathematical programming. In the previous article [2], the authors studied E-convex and semi-E-convex functions. In the present article, we deal with a class of E-quasiconvex functions and give a weak condition for a lower semi-continuous function on ℝ n to be an E-quasiconvex function.

Strongly Geodesic Log-Preinvex Functions

In this article, we delve into the intriguing concept of strongly geodesic log-preinvex functions in Riemannian manifolds. We present essential preliminaries and fundamental results that shed light on this specialized area of study. By examining the properties and implications of these functions, we aim to contribute to the growing body of knowledge in convexity theory within the context of Riemannian manifolds.

Distributed GNE Seeking Strategy for Second-Integrator Multiplayer Systems Over Directed Topologies.

This article studies the generalized Nash equilibrium (GNE) seeking problem of second-integrator multiplayer systems. In particular, each player is endowed with an individual payoff function with respect to collective decision variables, and simultaneously, a coupling inequality constraint and a set constraint are imposed to each player. The players communicate with their local neighbors over a directed topology. To begin with, a distributed-observer-based seeking strategy is synthesized by leveraging a proper composite variable. It is first demonstrated using nonsmooth analysis that the established distributed observer enables each player to accurately estimate the decision variables of others in terms of a strongly connected topology condition. Upon this basis, all the decision variables are then shown to converge to the expected GNE asymptotically borrowing from convex theory. In addition, three extension results are also given under the built GNE seeking framework. First, under the postulation that the velocity information is unavailable, a velocity-free distributed GNE seeking strategy is synthesized for second-integrator systems by implementing a proper auxiliary dynamics. Second, we consider nonlinear Euler-Lagrange systems with unknown inertia parameters and synthesize an improved distributed GNE seeking strategy resorting to an adaptation technique. Third, we focus on integrator chain systems and synthesize a modified distributed GNE seeking strategy using a new composite variable based on a proper coordinate transformation. For three extension cases, we all show in detail the achievement of the GNE seeking objective. Finally, a practical example is simulated to confirm the built GNE seeking results.

Fractional Hermite–Hadamard-Type Inequalities for Differentiable Preinvex Mappings and Applications to Modified Bessel and q-Digamma Functions

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher’s interest even more.

The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (BK,ρ)−Invexity

Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the K−directional derivative, that is, uniform (BK,ρ)−invex, strictly uniform (BK,ρ)−invex, uniform (BK,ρ)−pseudoinvex, strictly uniform (BK,ρ)−pseudoinvex, uniform (BK,ρ)−quasiinvex and weakly uniform (BK,ρ)−quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions.

AutoBar: Automatic Barrier Coverage Formation for Danger Keep Out Applications in Smart City.

Barrier coverage is a fundamental application in wireless sensor networks, which are widely used for smart cities. In applications, the sensors form a barrier for the intruders and protect an area through intrusion detection. In this paper, we study a new branch of barrier coverage, namely warning barrier coverage (WBC). Different from the classic barrier coverage, WBC has the inverse protect direction, which moves the sensors surrounding a dangerous region and protects any unexpected visitors by warning them away from the dangers. WBC holds a promising prospect in many danger keep out applications for smart cities. For example, a WBC can enclose the debris area in the sea and alarm any approaching ships in order to avoid their damaging propellers. One special feature of WBC is that the target region is usually dangerous and its boundary is previously unknown. Hence, the scattered mobile nodes need to detect the boundary and form the barrier coverage themselves. It is challenging to form these distributed sensor nodes into a barrier because a node can sense only the local information and there is no global information of the unknown region or other nodes. To this end, in response to the newly proposed issue of the formation of barrier cover, we propose a novel solution AutoBar for mobile sensor nodes to automatically form a WBC for smart cities. Notably, this is the first work to trigger the coverage problem of the alarm barrier, wherein the regional information is not pre-known. To pursue the high coverage quality, we theoretically derive the optimal distribution pattern of sensor nodes using convex theory. Based on the analysis, we design a fully distributed algorithm that enables nodes to collaboratively move toward the optimal distribution pattern. In addition, AutoBar is able to reorganize the barrier even if any node is broken. To validate the feasibility of AutoBar, we develop the prototype of the specialized mobile node, which consists of two kinds of sensors: one for boundary detection and another for visitor detection. Based on the prototype, we conduct extensive real trace-driven simulations in various smart city scenarios. Performance results demonstrate that AutoBar outperforms the existing barrier coverage strategies in terms of coverage quality, formation duration, and communication overhead.

I.V-CR-γ-Convex Functions and Their Application in Fractional Hermite–Hadamard Inequalities

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR-γ-convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen’s inequalities using CR-γ-convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard’s, Hermite-Hadamard-Fejer, and Pachpatte’s type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations.

Fertilitopes

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation pi is a particular set mathcal V(pi ) of integer compositions that appears in a formula for the fertility of pi , which is defined to be |s^{-1}(pi )|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that mathcal V(pi ) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of pi to be the convex hull of mathcal V(pi ), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of pi directly from pi using Bousquet-Mélou’s notion of the canonical tree of pi . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that sum _{sigma in s^{-1}(pi )}x^{textrm{des}(sigma )+1} is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

Velocity-free distributed coordinated optimal control for second-order multi-agent systems

This paper solves the coordinated optimal consensus problem for second-order multi-agent systems without velocity information from the distributed perspective. Particularly, all the agents are driven towards a consensus at an optimal solution inside the intersection of multiple feasible sets that minimizes a global target function aggregated by a group of convex local ones. Each local target function is exclusively assigned to a respective agent. Also, each feasible set constructed by inequality and set constraints privately belongs to a respective agent. Resorting to a novel auxiliary dynamics, a velocity-free distributed coordinated optimal control strategy is proposed without relying on global topology information. Using convex theory under the Lyapunov framework, it is demonstrated that the concerned coordinated optimal control objective is fulfilled by implementing the proposed distributed control strategy despite no velocity information. Examples are finally given to confirm the control effectiveness.

Collision and obstacle avoidance for agents based on velocity programming in unknown environments under kinematical constraints

AbstractThis paper conducts research on obstacle and collision avoidance under kinematical constraints in unknown environments, while communication and simultaneous localization and mapping (SLAM) are unavailable to agents. Then a strategy based on mixed‐integer programming is proposed, in which velocity constraints are established with a modified Barrier function for obstacles that can be completely detected. As for obstacles that cannot be completely detected, a feasible set is built for the velocity programming based on the convex theory, and the contradicted constraints are addressed with the logic metric method. Besides, the actuator saturation is avoided by converting kinematical constraints into the restrictions on the magnitude, the restrictions on the direction, and the negative correlations between the components of the velocity respectively. Given that invalid nominal velocity leads to collisions, a virtual goal is estimated for nominal velocity improvement. In addition, the local extremum brought by empty programming space due to multiple constraints is repaired by fixing the velocity constraints. The feasibility of the proposed strategy is analyzed, and numerical simulations are provided to verify the effectiveness of the proposed strategy.

Novel Mean-Type Inequalities via Generalized Riemann-Type Fractional Integral for Composite Convex Functions: Some Special Examples

This study deals with a novel class of mean-type inequalities by employing fractional calculus and convexity theory. The high correlation between symmetry and convexity increases its significance. In this paper, we first establish an identity that is crucial in investigating fractional mean inequalities. Then, we establish the main results involving the error estimation of the Hermite–Hadamard inequality for composite convex functions via a generalized Riemann-type fractional integral. Such results are verified by choosing certain composite functions. These results give well-known examples in special cases. The main consequences can generalize many known inequalities that exist in other studies.

Ostrowski Type Inequalities via Some Exponentially s-Preinvex Functions on Time Scales with Applications

Integral inequalities concerned with convexity have many applications in several fields of mathematics in which symmetry plays an important role. In the theory of convexity, there exist strong connections between convexity and symmetry. If we are working on one of the concepts, then it can be applied to the other of them. In this paper, we establish some novel generalizations of Ostrowski type inequalities for exponentially s-preinvex and s-preinvex functions on time scale by using Hölder inequality and Montgomery Identity. We also obtain applications to some special means. These results are motivated by the symmetric results obtained in the recent article by Abbasi and Anwar in 2022 on Ostrowski type inequalities for exponentially s-convex functions and s-convex functions on time scale. Moreover, we discuss several special cases of the results obtained in this paper.

Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation

<abstract><p>There are many benefits derived from the speculation regarding convexity in the fields of applied and pure science. According to their definitions, convexity and integral inequality are linked concepts. The construction and refinement of classical inequalities for various classes of convex and nonconvex functions have been extensively studied. In convex theory, Godunova-Levin functions play an important role, because they make it easier to deduce inequalities when compared to convex functions. Based on Bhunia and Samanta's total order relation, harmonically cr-$ h $-Godunova-Levin function is defined in this paper. Utilizing center order (CR) relationship, various types of inequalities can be introduced. (CR)-order relation enables us to derive some Hermite-Hadamard ($ \mathcal{H.H} $) inequality along with a Jensen-type inequality for harmonically $ h $-Godunova-Levin interval-valued functions (GL-$ \mathcal{IVFS} $). Many well-known and new convex functions are unified by this kind of convexity. For further verification of the accuracy of our findings, we provide some numerical examples.</p></abstract>

Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes

<abstract><p>It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.</p></abstract>

Some new estimates of well known inequalities for $ (h_1, h_2) $-Godunova-Levin functions by means of center-radius order relation

<abstract><p>In this manuscript, we aim to establish a connection between the concept of inequalities and the novel Center-Radius order relation. The idea of a Center-Radius (CR)-order interval-valued Godunova-Levin (GL) function is introduced by referring to a total order relation between two intervals. Consequently, convexity and nonconvexity contribute to different kinds of inequalities. In spite of this, convex theory turns to Godunova-Levin functions because they are more efficient at determining inequality terms than other convexity classes. Our application of these new definitions has led to many classical and novel special cases that illustrate the key findings of the paper. Using total order relations between two intervals, this study introduces CR-$ (h_1, h_2) $-Goduova-Levin functions. It is clear from their properties and widespread usage that the Center-Radius order relation is an ideal tool for studying inequalities. This paper discusses various inequalities based on the Center-Radius order relation. With the CR-order relation, we can first derive Hermite-Hadamard ($ \mathcal{H.H} $) inequalities and then develop Jensen-type inequality for interval-valued functions ($ \mathcal{IVFS} $) of type $ (h_1, h_2) $-Godunova-Levin function. Furthermore, the study includes examples to support its conclusions.</p></abstract>

New Class Up and Down Pre-Invex Fuzzy Number Valued Mappings and Related Inequalities via Fuzzy Riemann Integrals

Numerous applications of the theory of convex and nonconvex mapping exist in the fields of applied mathematics and engineering. In this paper, we have defined a new class of nonconvex functions which is known as up and down pre-invex (pre-incave) fuzzy number valued mappings (F-N-V∙Ms). The well-known fuzzy Hermite–Hadamard (

Optimal Selection and Integration of Batteries and Renewable Generators in DC Distribution Systems through a Mixed-Integer Convex Formulation

The problem concerning the optimal placement and sizing of renewable energy resources and battery energy storage systems in electrical DC distribution networks is addressed in this research by proposing a new mathematical formulation. The exact mixed-integer nonlinear programming (MINLP) model is transformed into a mixed-integer convex model using McCormick envelopes regarding the product between two positive variables. Convex theory allows ensuring that the global optimum is found due to the linear equivalent structure of the solution space and the quadratic structure of the objective function when all the binary variables are defined. Numerical results in the 21-bus system demonstrate the effectiveness and robustness of the proposed solution methodology when compared to the solution reached by solving the exact MINLP model. Numerical results showed that the simultaneous allocation of batteries and renewable energy resources allows for the best improvements in the daily operating costs, i.e., about 53.29% with respect to the benchmark case of the 21-bus grid, followed by the scenario where the renewable energy resources are reallocated while considering a fixed location for the batteries, with an improvement of 43.33%. In addition, the main result is that the difference between the exact modeling and the proposed formulation regarding the final objective function was less than 3.90% for all the simulation cases, which demonstrated the effectiveness of the proposed approach for operating distributed energy resources in monopolar DC networks.