. In this article, we consider the problem of approximating the solution of bilevel split variational inequality problem in real Hilbert spaces. The underlying operators in the lower level problem are quasimonotone and Lipschitz continuous. The proposed algorithm is a combination of the modified subgradient extragradient and modified Tseng's extragradient methods. Compared with the existing modified subgragadient extragradient methods for solving bilevel split variational inequality problem, our suggested method does not required computation of the projections onto two half-spaces, containing the feasibility sets. The step sizes employed in our algorithm do not need the prior knowledge of the norm of the bounded linear operator and the the Lipschitz constants of the underlying operators. We obtain the strong convergence results of the new method using some mild conditions on the control parameters. The proposed method involves double inertial terms which permits it to accelerate its convergence speed. To show the advantage and potential of our method over some existing methods, we present some numerical experiments. direction.
. This paper establishes sufficient optimality conditions for nonlinear continuous-time optimization problems using the concept of second-order KKT-invexity. Second-order KKT-invexity is a type of generalized convexity that is appropriate to work with second-order stationary solutions, which in turn are solutions that satisfy second-order necessary optimality conditions. It is shown that the second-order necessary optimality conditions become sufficient for problems satisfying this generalized convexity concept. Furthermore, it is shown that in some cases this concept is the most general, in the sense that if the problem is such that every second-order stationary solution is an optimal solution, then the problem necessarily satisfies the generalized convexity concept.
This article is devoted to the study of the exact controllability of a star-shaped network of strings in time-varying domains. The system consists of a single node with three connected arcs. Each arc is governed by the wave equation. Robin boundary control acts on only two endpoints. At the central node, the arcs are coupled in such a way that the energy is non-increasing. Using multiplier technique and careful energy estimate, we first establish observability estimates for the corresponding adjoint system. Then, by applying the Hilbert Uniqueness Method (HUM), we show that the boundary controllability of the original system holds for a sufficiently large time.
In this article, a Leslie-Gower prey-predator model that incorporates cooperation among the predator populations during hunting and a refuge mechanism for the prey are proposed and then analyzed. Additionally, a strong Allee effect in prey growth is included to address both biological and mathematical considerations. Initially, the topological equivalence method is employed to discuss the dynamical behavior of the system in the neighborhood of the origin. Subsequently, the existence and stability of the model's non-negative equilibria are examined. We identify parameter subsets where the system exhibits co dimension-one local bifurcations, specifically saddle-node and Hopf bifurcations, and demonstrate the presence of bistability. The first Lyapunov number is calculated to determine the stability of limit cycles that emerge from the Hopf bifurcation. Using Sotomayor's theorem, we derive the existence of a saddle-node bifurcation in the system. Furthermore, we analyze the influence of hunting cooperation on the model both analytically and numerically, revealing that hunting cooperation not only reduces the density of the prey population, but also destabilizes the system's dynamics. We also investigate the impact of refuge on the model numerically, finding that refuge stabilizes the system's dynamics.
This paper focuses on an accelerated modulus-based matrix splitting iteration method for solving a class of horizontal nonlinear complementarity problems. First, by introducing a secondary matrix splitting based on the principles of stepwise updates and immediate utilization of the latest computational information, the equivalent modulus equation of the horizontal nonlinear complementarity problem is reformulated and thereby the accelerated modulus-based matrix splitting iteration scheme is constructed. Subsequently, under the assumption that the system matrices are H+-matrices, the convergence conditions for the proposed method are established. Finally, the computational efficiency of the new method is demonstrated by numerical experiments.
We study the Stackelberg-Nash null controllability of a coupled system governed by two linear forward stochastic parabolic equations. The system includes one leader control localized in a subset of the domain, two additional leader controls in the diffusion terms, and m follower controls, where m & ecaron; 2. We consider two different scenarios for the followers: first, when the followers minimize a functional involving both components of the system's state, and second, when they minimize a functional involving only the second component of the state. For fixed leader controls, we first establish the existence and uniqueness of the Nash equilibrium in both scenarios and provide its characterization. As a byproduct, the problem is reformulated as a classical null controllability issue for the associated coupled forward-backward stochastic parabolic system. To address this, we derive new Carleman estimates for the adjoint stochastic systems. As far as we know, this problem is among the first to be discussed for stochastic coupled systems.
. This paper provides a rigorous analysis of finite-time multi-cluster flocking for a modified Cucker-Smale model. We modify the interaction law by applying a componentwise sign-power-type map to the velocity-difference term. Since the exponent lies between zero and one, the resulting vector field is non-Lipschitz continuous, and uniqueness of solutions is not guaranteed in general. To study the system from a multi-cluster viewpoint, we partition the agents into several groups and decompose each state variable into a cluster mean and a fluctuation around that mean. This reformulation separates the motion of cluster centers from the deviations of agents within each cluster. Using this structure, we derive basic identities and uniform bounds for the average and deviation variables. We then establish a dissipative structure for the position-velocity deviations under suitable assumptions on the initial data and system parameters. We also prove that distinct cluster centers separate as time evolves. Based on these estimates, we show that the desired multi-cluster flocking structure emerges within a common finite time for all solutions starting from the given initial data. Numerical simulations are provided to support the theoretical results.
This study proposes a new Conjugate Gradient (CG) method that is both theoretically sound and computationally efficient for solving large-scale unconstrained optimization problems. The method integrates the Dai-Liao (DL) parameter with a gradient-difference vector formulation inspired by Powell (1978) and incorporates the Barzilai-Borwein step size to enhance convergence performance. Under standard assumptions, the proposed method is proven to satisfy the sufficient descent condition and to guarantee global convergence. Extensive numerical experiments demonstrate that the new method outperforms classical CG algorithms including the Hestenes-Stiefel (HS) and DL methods in terms of iteration count, function evaluations, and computational time. Furthermore, the method is applied to train a feedforward neural network aimed at predicting the seismic vulnerability of 641 multistory buildings in Zakho City, Iraq. Using a dataset of structural and geotechnical features, the neural network trained with the proposed CG method achieves faster convergence and maintains high prediction accuracy, confirming the method's practical utility in real-world applications.
This paper proposes a generalized SOR-type iterative method for solving the generalized tensor absolute value equation. The convergence of the method is rigorously analyzed under mild assumptions on the underlying tensor splitting and the selection of parameter pairs. In contrast to existing SOR-type iterative method, our analysis establishes a weaker convergence condition, which significantly enlarges the class of problems for which the method is applicable. Furthermore, we provide numerical experiments on a variety of test problems to validate the theoretical findings. The results confirm that the proposed method not only converges reliably under broader conditions but also achieves improved computational efficiency compared with existing approaches.
. In real-world scenarios, decision-making often involves challenges arising from imprecise, uncertain, or incomplete information. Such uncertainties may stem from decision-makers' intuition, subjective judgments, evaluations, assumptions, data collection methods, or behavioral tendencies. While the fuzzy set theory introduced a foundational approach to addressing vagueness, it proved insufficient for more complex cases. The fuzzy rough set is a powerful mathematical tool for handling data characterized by uncertainty, incompleteness, or imprecision. This article aims to propose an effective method for addressing a novel class of n -players continuous differential games within a fuzzy rough environment. In the proposed method, the fuzzy rough n-players games are reformulated into two problems under a fuzzy environment, corresponding to the lower and upper approximations of fuzzy rough numbers. Additionally, by applying the alpha-cut set, four distinct problems are identified: the lower-upper level problem, upper-lower level problem, upperupper level problem, and lower-lower level problem. The study also establishes sufficient and necessary conditions for achieving equilibrium strategies in fuzzy rough continuous differential games. Finally, the suggested method is validated through a numerical example, demonstrating its effectiveness and practical applicability.
. It is well-known that a symmetric matrix with its entries +/- 1 can not be positive definite. But this is not true for symmetric tensors (hypermatrix). In this paper, we mainly discussed the positive (semi-)definiteness criterion of a class of fourth-order, three-dimensional symmetric tensors with entries tijkl is an element of {-1, 0,1}. Through theoretical derivations and detailed classification discussions, the criteria for determining the positive (semi-)definiteness of such a class of tensors were provided based on the relationships and number values of its entries. This has established some unique properties of higher symmetric tensors that are distinct from the ones of matrices.
. In this paper, we propose a lung tumor segmentation model using low-rank and sparse representation. It decomposes the input image into a lowrank component for normal tissues, and a sparse component for tumor regions, with an & ell;1 + & ell;2,1 mixed-norm regularizer to enhance sparsity and robustness. By combining PET and CT information, our approach effectively captures metabolic activity and anatomical structures. The optimization problem is efficiently solved via the linearized alternating direction method of multipliers. Experimental results demonstrate that the proposed strategy significantly improves higher segmentation accuracy compared to CT-only baselines and & ell;1-norm regularization.
. This paper focuses on the open-loop Nash equilibrium for networked control systems (NCSs) with asymmetric information. In this NCSs model, player 1 shares its observations and historical control inputs with player 2, whereas player 2 does not disclose any of its information to player 1. Using the maximum principle, we obtain an explicit analytical Nash equilibrium by decoupling solving the forward and backward stochastic difference equations (FBSDEs), while the optimal gain matrices are given by coupled Riccati equations. The asymmetric information leads to coupled forward and backward Riccati equations, which complicates the calculation. To address it, a forward iteration algorithm is employed to obtain a suboptimal solution for the open-loop Nash equilibrium strategy with asymmetric information. Numerical results illustrate that incorporating the Nash equilibrium into NCSs can improve the system performance.
. In this paper, we study a nondifferentiable bilevel optimization problem, where the functions involved are characterized by their tangential subdifferentials. To address this nonsmooth extremum problem, we derive sufficient efficiency conditions and establish duality results. Thus, the hierarchical structure of the problem is transformed into a single-level formulation using an optimal value reformulation. Under the assumption that the functions are Dini generalized convex and described by tangential subdifferentials, we develop sufficient efficiency conditions for the problem. Additionally, we present weak and strong duality results based on Mond-Weir dual problems, expressed in terms of tangential subdifferentials. The theoretical findings are illustrated through examples of nondifferentiable bilevel optimization models.
This paper presents a semi-analytical approach to solving the optimal control problem associated with thermal propagation in multilayer biological tissues during hyperthermia cancer treatment. The heat source at the tumor site is modeled as a distributed control function, and the problem is analyzed using a five-layer composite structure comprising skin, fat, muscle, tumor, and muscle layers. The approximate controllability of the multilayer bioheat problem is proved through infinite-dimensional systems theory, and the optimal control problem is formulated as a constrained Bolza problem in a one-dimensional framework. A hybrid solution method is proposed, integrating infinite-dimensional systems theory to model the multilayer bioheat problem as a system with inputs and outputs, along with strongly continuous semigroups theory to solve the corresponding abstract differential equation. Novel convolution operators are introduced for each layer, facilitating the computation of hybrid solutions for state and co-state problems. Additionally, a numerical Laplace inversion technique is applied to determine interface functions between layers. The effectiveness of the proposed approach is demonstrated through numerical simulations, providing valuable insights into the controllability of thermal propagation in multilayer tissues. These findings contribute to the optimization of hyperthermia treatment strategies for tumor therapy, improving precision and efficacy in clinical applications.
. In this paper, we study a convex adjustable robust problem (ARP) with general constraints, in which adjustable, non-adjustable, and uncertainty variables directly impact each other in each constraint. We first present reformulations of (ARP) in the forms of semi-infinite and generalized semi-infinite problems. Optimality conditions are also provided under suitable constraint qualifications (CQs) such as Slater CQ, local Farkas-Minkowski CQ, and Mangasarian-Fromovit CQ. The relationship between these CQs are also considered. The Wolfe-type dual problem and Mond-Weir type dual problem are formulated. We present duality results between the primal one and its dual problems. Some examples are provided to illustrate the presented results.
In this paper, we propose a modified relaxed inertial subgradient extragradient method for approximating a common solution of the quasimonotone variational inequality problem and the multivalued demicontractive fixed point problem. We prove the strong convergence result of our proposed method under some mild conditions on the control parameter. Finally, we carry out numerical experiments to show our method's performance and computational advantage over some existing methods in the literature.
. This study investigates an MAP/PH/1 G-queue system with unreliable repairs and two types of repair processes. The negative arrivals cause removal of positive arrivals and deteriorates the servers service rate. Since the repair process is unreliable, there exists a probability of failure while attempting to restore the server to its original efficiency. To address these challenges, two types of repair processes are incorporated. The first, Type-I repair, activates after a threshold number of negative arrivals has entered the system, a mechanism frequently discussed in the existing literature. The second, Type-II repair, is initiated when the server remains idle in any deteriorated state, an aspect often overlooked in related studies. The system is analyzed by modeling it as a quasi-birth-death (QBD) process, and steady-state probabilities are obtained using the matrix analysis method. A comparative performance and economic evaluation of models with and without Type-II repair is conducted. The optimized value of the profit function of both models is determined using Sequential Quadratic Programming (SQP). Numerical experiments illustrate the importance of Type II repair process in reducing the impact of negative arrivals and an unreliable repair process.
