System Of Inequalities Research Articles

Liouville-type theorems for a system of elliptic inequalities on weighted graphs

Let (V,E) be an infinite, connected, locally finite weighted graph. We are concerned with the system of elliptic inequalities \begin{cases}-\Delta u\ge v^{p}\quad\text{in }V,\\ -\Delta v\ge u^{q}\quad\text{in }V,\end{cases} where p are q are real numbers and \Delta is the standard graph Laplacian. For p\leq 0 or q\leq 0 or p,q>0 and pq\leq 1 , we show that the system has no positive solution. Moreover, we also establish some non-existence results of positive solutions of the system in the case where p,q>0 and pq>1 under some assumptions on the volume growth of graph and weight. We also construct an explicit example to show the existence result in the super-critical case. Our result is, in particular, a natural extension of some results in [Gu, Huang, and Sun, Calc. Var. Partial Differential Equations 62 (2023), no. 2, article no. 42] to the system of inequalities.

A Novel Zeroing Neural Network for the Effective Solution of Supply Chain Inventory Balance Problems

The issue of inventory balance in supply chain management represents a classic problem within the realms of management and logistics. It can be modeled using a mixture of equality and inequality constraints, encompassing specific considerations such as production, transportation, and inventory limitations. A Zeroing Neural Network (ZNN) model for time-varying linear equations and inequality systems is presented in this manuscript. In order to convert these systems into a mixed nonlinear framework, the method entails adding a non-negative slack variable. The ZNN model uses an exponential decay formula to obtain the desired solution and is built on the specification of an indefinite error function. The suggested ZNN model’s convergence is shown by the theoretical results. The results of the simulation confirm how well the ZNN handles inventory balance issues in limited circumstances.

Orbit-finite Linear Programming

An infinite set is orbit-finite if, up to permutations of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

On Symmetrically Stochastic System of Fractional Differential Equations and Variational Inequalities

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard’s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory.

On the existence of solutions of the system of nonlinear quasi-mixed equilibrium problems and their stability of the procedure scheme

In this paper, we provide a simplified proof of the existence of solutions for the system of nonlinear quasi-mixed equilibrium problems studied by Suantai and Petrot (Appl. Math. Lett. 24:308–313, 2011), utilizing Banach’s fixed point theorem. This result remains valid under weaker assumptions through the product space approach. Moreover, we show that the stability analysis of the iterative algorithm proposed in their work (and in (J. Inequal. Appl. 2010:437976, 2010)) contains a flaw, which we address using Ostrowski’s classical result from 1967 (Z. Angew. Math. Mech. 47:77–81, 1967). Finally, we examine and improve the convergence theorem for solving a system of variational inequalities as established by Chang et al. (Appl. Math. Lett. 20:329–334, 2007).

Optimal control for a dynamic contact problem governed by a system of hemivariational inequalities in thermo-electro-viscoelasticity

Optimal control for a dynamic contact problem governed by a system of hemivariational inequalities in thermo-electro-viscoelasticity

The metric projection over a polyhedral set through the relative interiors of its faces

Abstract We first characterize the region of the n-dimensional Euclidean space for which two optimization problems with the square distance function as common objective function, but different constraints, are equivalent. The affine hull of a certain face of a closed convex set $$C\subseteq {\mathbb {R}}^n$$ C ⊆ R n is the constraint associated to one problem and the whole closed convex set C is the constraint associated to the other problem. Such optimization problems are best approximation problems which can be reformulated in terms of the metric projection. Using the language of the metric projection, we characterize the region of $${\mathbb {R}}^n$$ R n which is metrically projected over a face of C in the same way that it is projected over the affine hull of the face itself. The metric projection over such a face is the one associated to the entire closed convex set, and the metric projection over the affine hull of such a face is the one associated to the affine hull. It turns out that this region is the closure of the inverse image, through the metric projection over the entire closed convex set, of the relative interior of the face. We also characterize analytically the closure of the regions of $${\mathbb {R}}^n$$ R n that are projected over the relative interiors of the faces of a polyhedral set, through the metric projection of the polyhedral set itself. We show that these regions are polyhedral convex sets by explicitly characterizing them through systems of linear inequalities.

Stochastic optimal switching and systems of variational inequalities with interconnected obstacles

Stochastic optimal switching and systems of variational inequalities with interconnected obstacles

Liouville-type theorems for systems of elliptic inequalities involving $ p $-Laplace operator on weighted graphs

Liouville-type theorems for systems of elliptic inequalities involving $ p $-Laplace operator on weighted graphs

New results on system of variational inequalities and fixed point theorem

In (Croat. Oper. Res. Rev, 13(1), 131-135, (2022).) A. Benhadid showed by a counterexample that a number of publications in the research on the variational inequalities system contains inaccurate results about applying the Lipschitz continuity concept in relation to both the first and second variables during the proof of the main theorem on the part of authors. In this paper, We suggest to applying fixed point theorem to correct the main results of some publications.

Gradient-tracking-based distributed Nesterov accelerated algorithms for multiple cluster games over time-varying unbalanced digraphs

This paper studies the design of accelerated distributed algorithms for seeking the Nash Equilibrium (NE) of multiple cluster games over time-varying unbalanced digraphs, where players in the same cluster cooperatively minimize the summation of the local cost function and do not concern the interest of other clusters. In this game, the players have limited access to others’ decisions, while they can communicate with others over the inter-cluster and intra-cluster topologies. Accelerated distributed algorithms are proposed based on the decision estimation, Nesterov acceleration, and pseudo gradient estimation to seek the NE of multiple cluster games. We prove that the proposed algorithm linearly converges to the NE using the multistep contraction and linear systems of inequalities. Moreover, three variants of the proposed algorithm are also given for dealing with cases where only partial communication topologies are time-varying and gossip-type. Lastly, the effectiveness of proposed algorithms and the acceleration effect are verified by solving the intrusion–interception confrontation problem of Unmanned Vehicle (UV) swarms in simulations.

Optimal Portfolio and Retirement Decisions with Costly Job Switching Options

In this paper, we consider the utility maximization problem of an agent regarding optimal consumption-investment, job-switching strategy, and the optimal early retirement date. The agent can switch between two jobs or job categories at any time before retirement, but incurs a cost when switching to a position offering higher labor income. The agent's utility maximization involves a combination of stochastic control for consumption and investment, switching control for job-switching, and optimal stopping for early retirement decisions, making it a non-trivial and highly challenging problem. By utilizing the dynamic programming principle, we can derive the nonlinear Hamilton-Jacobi-Bellman (HJB) equation in the form of a system of variational inequalities with obstacle constraints, which arises from the agent's optimization problem. We employ guess and verify methods based on economic intuition to derive the closed-form solution of this HJB equation and demonstrate, through a verification theorem, that this solution aligns with the solution to the agent's utility maximization problem.

Existence of optimal strategies in bimatrix game and applications

This paper delves into interval-valued bimatrix games, where precise payoffs remain elusive, but lower and upper bounds on payoffs can be determined. The study explores several key questions in this context. Firstly, it addresses the issue of the existence of a universally applicable equilibrium across all instances of interval values. The paper establishes a fundamental equivalence by demonstrating that this property hinges on the solvability of a specific system of interval linear inequalities. Secondly, the research endeavors to characterize the comprehensive set of weak and strong equilibrium using a system of interval linear inequalities. The findings in this paper shed light on the complexities and intricacies of interval-valued bimatrix games, offering valuable insights into their equilibrium properties and computational aspects. Through illustrative examples, we underscore the practical utility of these approaches and compare them with previously developed state-of-the-art methods, demonstrating their ability to generate conservative solutions in the face of interval uncertainty. The findings of this research not only offer valuable insights into the equilibrium properties and computational aspects of interval-valued bimatrix games but extend their practical implications. In particular, the paper delves into real-life applications, exemplifying the significance of these findings for crude oil trading decision-making.

Stabilization of associated prime ideals of monomial ideals – Bounding the copersistence index

The sequence (Ass(R/In))n∈N of associated primes of powers of a monomial ideal I in a polynomial ring R eventually stabilizes by a known result by Markus Brodmann. Lê Tuân Hoa gives an upper bound for the index where the stabilization occurs. This bound depends on the generators of the ideal and is obtained by separately bounding the powers of I after which said sequence is non-decreasing and non-increasing, respectively. In this paper, we focus on the latter and call the smallest such number the copersistence index. We take up the proof idea of Lê Tuân Hoa, who exploits a certain system of inequalities whose solution sets store information about the associated primes of powers of I. However, these proofs are entangled with a specific choice for the system of inequalities. In contrast to that, we present a generic ansatz to obtain an upper bound for the copersistence index that is uncoupled from this choice of the system. We establish properties for a system of inequalities to be eligible for this approach to work. We construct two suitable inequality systems to demonstrate how this ansatz yields upper bounds for the copersistence index and compare them with Hoa's. One of the two systems leads to an improvement of the bound by an exponential factor.

Active fault tolerant PD control law for PWRs under external disturbance

NPP is a complex time varying nonlinear system that have to operate under severe constraints while complying with safe operating conditions in order to ensure the power demand and prevent the plant from contingent network instability. Towards this goal, this paper proposes the design of active fault tolerant proportional and derivative (PD) control law for Pressurized Water Reactors (PWRs) under external disturbance. To achieve this, the reactor core’s nonlinear dynamic is transformed into input/output (I/O) second-order system with respect to the power level. Based on the new model, a control strategy is proposed that aims to detect, identify, estimate and compensate for actuator faults. The stability of the system is proven using Lyapunov stability theory, where a quadruple Linear Matrix Inequalities (LMIs) system is derived to provide gains for both the control law and observer. Numerical simulations are conducted to assess the performance of the newly built control strategy and a comparison has been made. It follows that the designed controller not only effectively manages faults and external disturbances but also demonstrate exceptional performance when compared to the model-free controller (MFC) and the L1 adaptive robust controller (L1 ARC).

Bi-attribute utility preference robust optimization: A continuous piecewise linear approximation approach

In this paper, we consider a bi-attribute decision making problem where the decision maker’s (DM’s) objective is to maximize the expected utility of outcomes with two attributes but where the true utility function which captures the DM’s risk preference is ambiguous. To tackle this ambiguity, we propose a maximin bi-attribute utility preference robust optimization (BUPRO) model where the optimal decision is based on the worst-case utility function in an ambiguity set of plausible utility functions constructed using partially available information such as the DM’s specific preference for certain lotteries. Specifically, we consider a BUPRO model with two attributes, where the DM’s risk attitude is bivariate risk-averse and the ambiguity set is defined by a linear system of inequalities represented by the Lebesgue–Stieltjes integrals of the DM’s utility functions. To solve the inner infinite-dimensional minimization problem, we propose a continuous piecewise linear approximation approach to approximate the DM’s unknown true utility. Unlike the univariate case, we partition the domain of the utility function into a set of small non-overlapping rectangles and then divide each rectangle into two triangles by either the main diagonal (Type-1) or the counter diagonal (Type-2). The inner minimization problem based on the piecewise linear utility function can be reformulated as a mixed-integer linear program and the outer maximization problem can be solved efficiently by the derivative-free method. In the case that all the small triangles are partitioned either in Type-1 or in Type-2, the inner minimization can be formulated as a finite dimensional linear program and the overall maximin as a single mixed-integer program. To quantify the approximation errors, we derive, under some mild conditions, the error bound for the difference between the BUPRO model and the approximate BUPRO model in terms of the ambiguity set, the optimal value and the optimal solutions. Finally, we carry out some numerical tests to examine the performance of the proposed models and computational schemes. The results demonstrate the efficiency of the computational schemes and highlight the stability of the BUPRO model against data perturbations.

COMPARATOR REGRESSION ANALYSIS IN CONDITIONS OF FUZZY INITIAL DATA

Abstract. The canonical task of regression analysis consists in studying the problem of estimation and mathematical description of the influence caused by one or more independent variables (predictors, explanatory variables) on the dependent (explained) variable. The subject of comparator identification problem is a specific modification of the standard regression analysis problem in the case when, for some reason, it is not possible to measure values of the dependent, explained variable. However, the problem of reconstructing a regression polynomial can be solved if it is possible to qualitatively compare the results of experiments and rank them, for example, in the order of increasing. Taking into account the results of ranking the values of the explained variable in different experiments, a corresponding system of inequalities is formed. The purpose of the research consists in developing a method for mathematical processing of the resulting system of inequalities for analytical assessment of the level of influence of explanatory variables on the explained variable. The method for solving the problem is based on transforming a system of inequalities into a system of linear algebraic equations. A significant drawback of the known approach to solving this problem consists in the fact that the resulting system of linear algebraic equations is an underdetermined one (the number of unknowns in this system exceeds the number of equations). At the same time, the system has an infinite number of solutions, among which there may be an uncontrollable number of unacceptable ones. The exhaustive search method for finding acceptable solutions is futile. In this regard, the problem of developing a correct method of comparator identification remains relevant. Another difficulty that accompanies the actual procedure of comparator identification when solving practical problems consists in the fact that measurements of values of the controlled parameters are not accurate. The uncertainty that arises in the conditions of a small sample of initial data can be removed using the tools of fuzzy mathematics. It is assumed that, based on preliminary research, a mathematical description of the corresponding membership function can be obtained for each of the controlled parameters. As a result, effective approaches to solving emerging problems have been proposed, based on the use of developed optimization procedures in the conditions of a small sample of fuzzy initial data.

A low-complexity solution for LMIs of hardly constrained nonlinear MIMO tracking control systems

This research is dedicated to the discrete-time tracking control of nonlinear multi-input multi-output (MIMO) Newtonian mechanics systems with an equal number of inputs and outputs, in the presence of input saturation and high-order dynamics of the actuation system. Firstly, the continuous-time model of the plant and actuation system has been transformed into the discrete-time domain using Adams–Bashforth and Adams–Moulton discretization techniques. In the next step, considering saturation boundaries, the modified position and velocity references are derived using the discrete-time reaching law-based control approach. These modified references are obtained through an optimization algorithm that calculates the closest achievable position and velocity trajectories with respect to the desired reference trajectory. Modifying desired trajectories in MIMO systems results in the formation of a system of linear matrix inequalities (LMIs). Solving such systems of inequalities requires the use of complex methods, and this difficulty increases with the increase in the system’s degrees of freedom (DOFs). The proposed strategy offers a more straightforward approach to solving these inequalities. Once the modified references are calculated, the corresponding position and velocity control signals can be obtained using the mentioned reaching law strategies. The final step of the proposed algorithm is to combine the resulting position and velocity control signals using a Gaussian-weighted averaging method to form the main control signal. A desirable performance for the control system can be achieved by appropriately adjusting the standard deviations of the Gaussian weighing functions. To evaluate the performance characteristics of the proposed method, it has been applied in the tracking control procedure of a nonlinear 3-DOF mechanical system. The numerical simulation results demonstrate the effectiveness of the method.

Seen but not heard: the voice of women at work and the mediating role of culture

While there is now an extensive body of literature on employee voice behaviour in the global North, research evidence from the global South is limited. This has constrained our understanding of the barriers that female workers face in expressing their views and concerns in developing countries such as Nigeria. This article examines the cultural factors that shape female employee voice behaviour in Nigerian workplaces. Using a meta-synthesis of 52 semi-structured interviews and approximately 200 h of non-participant observation, we identify a high-power distance orientation and patriarchal norms as two cultural factors that contribute to gender imbalance in the workplace, making it difficult for female employees to express their opinions, suggestions, ideas, or complaints about important workplace issues. Our findings highlight a system of patriarchal hegemony and gender inequality that makes voice behaviour difficult for female workers. The findings also show that contextualised religious norms and teachings encourage silence among female employees. We provide valuable insights into the cultural norms that inhibit female employee voice behaviour in the Global South context.

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.