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In this paper we prove two general minimax theorems which gen- eralize famous classical saddle point theorems of M. Sion [6] and J. von Neu- mann. Our theorems also include some results of S. Park [3]-[5]. Results of this type have many applications in the Game theory, because they gives existence of solution of zero sums games. [ABSTRACT FROM AUTHOR]

Divergence is usually used to determine the dissipation of a dynamical system, but some researchers have noticed that it can lead to elusive contradictions. In this article, a criterion, dissipative power, beyond divergence for judging the dissipation of a system is presented, which is based on the knowledge of classical mechanics and a novel dynamic structure by Ao. Moreover, the relationship between the dissipative power and potential function (or called Lyapunov function) is derived, which reveals a very interesting, important, and apparently new feature in dynamical systems: to classify dynamics into dissipative or conservative according to the change of “energy function” or “Hamiltonian,” not according to the change of phase space volume. We start with two simple examples corresponding to two types of attractors in planar dynamical systems: fixed points and limit cycles. In judging the dissipation by divergence, these two systems have both the elusive contradictions pointed by researchers and new ones noticed by us. Then, we analyze and compare these two criteria in these two examples, further consider the planar linear systems with the coefficient matrices being the four types of Jordan’s normal form, and find that the dissipative power works when divergence exhibits contradiction. Moreover, we also consider another nonlinear system to analyze and compare these two criteria. Finally, the obtained relationship between the dissipative power and the Lyapunov function provides a reasonable way to explain why some researchers think that the Lyapunov function does not coexist with the limit cycle. Those results may provide a deeper understanding of the dissipation of dynamical systems.

Differential geometry is a powerful tool to analyze the vapor–liquid critical point on the surface of the thermodynamic equation of state. The existence of usual condition of the critical point (∂p/∂V)T=0 requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume (∂p/∂T)V=0. The distinction between the critical point and other points on the surface leads us to further assume that the critical point is geometrically represented by zero Gaussian curvature. A slight extension of the van der Waals equation of state is to letting the two parameters a and b in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.

We introduce a generalized numerical algorithm to construct the solution landscape, which is a pathway map consisting of all stationary points and their connections. Based on the high-index optimization-based shrinking dimer (HiOSD) method for gradient systems, a generalized high-index saddle dynamics (GHiSD) is proposed to compute any-index saddles of dynamical systems. Linear stability of the index-k saddle point can be proved for the GHiSD system. A combination of the downward search algorithm and the upward search algorithm is applied to systematically construct the solution landscape, which not only provides a powerful and efficient way to compute multiple solutions without tuning initial guesses, but also reveals the relationships between different solutions. Numerical examples, including a three-dimensional example and the phase field model, demonstrate the novel concept of the solution landscape by showing the connected pathway maps. [ABSTRACT FROM AUTHOR]

An analytical solution is obtained for the diffraction problem. In an infinite cylindrical duct, the sound waves are emanating by a ring source. The duct is rigid for z < l and perforated for z > l. The mixed boundary value problem is defined by a Wiener Hopf equation, by using the Fourier transform technique. Then the numerical solution is obtained. The influence of the parameters of the problem on the diffraction phenomenon is displayed graphically. The present study can be used as a model for different applications. Reducing noise in exhaust systems, ventilation systems are some of these applications. [ABSTRACT FROM AUTHOR]

Using the idea of the dimensional splitting method, we present an iteration method for solving three-by-three block saddle point problems which appear in linear programming and finite element discretization of the Maxwell equation. We prove that the method is unconditionally convergent. Then the induced preconditioner is used to accelerate the convergence of the GM RES method for solving the system. Numerical results are presented to compare the performance of the method with some existing ones. [ABSTRACT FROM AUTHOR]

A new notion of linking is introduced to treat minima as minimax points in a unitary way. Critical points are located in conical annuli making possible to obtain multiplicity. For functionals defined on a Cartesian product, the localization of critical points is given on components and the variational properties of the components can differ, part of them being of minimum type, others of mountain pass type. [ABSTRACT FROM AUTHOR]

We propose here an approach in order to estimate parameters of the CIR model with jumps in the case where the distribution of jump amplitude is estimated non-parametrically. Since the knowledge of the exact distribution of the jump amplitude is a challenge, in this paper we choose not to fix this law in advance but to estimate it on the basis of the available observations. The method of estimation we propose here is based on the approximation of the closed form of transition density. Since the CIR does not have an explicit solution, it is approximated by the second order Milstein scheme in order to have a more accurate approximation. The method of estimation is then applied on real data, which are the Federal Funds rate and 3 Month T-Bill rate. These two sets of data are used to estimated parameters of the CIR model. We then compare our results to those obtained from Vasicek and Brennon–Swartz models with jumps. Results indicate that there is no clear winner of models competitions. Apparently depending on the nature and structural components of the data, there is a winner. The challenge here is that, there is a trade off between the sample size, the number of jumps and the efficiency of estimates. More data involves the likelihood to have more jumps and thereby less efficient are estimates. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special class of optimization problems which contains infinitely many inequality constraints and finitely many complementarity constraints known as the semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC). We propose Lagrange type dual model for the SIMPEC and obtain their duality results using convexity assumptions. Further, we discuss the saddle point optimality conditions for the SIMPEC. Some examples are given to illustrate the obtained results.

The paper studies the open-loop saddle point and the open-loop lower and upper values, as well as their relationship for two-person zero-sum stochastic linear-quadratic (LQ) differential games with deterministic coefficients. It derives a necessary condition for the finiteness of the open-loop lower and upper values and a sufficient condition for the existence of an open-loop saddle point. It turns out that under the sufficient condition, a strongly regular solution to the associated Riccati equation uniquely exists, in terms of which a closed-loop representation is further established for the open-loop saddle point. Examples are presented to show that the finiteness of the open-loop lower and upper values does not ensure the existence of an open-loop saddle point in general. But for the classical deterministic LQ game, these two issues are equivalent and both imply the solvability of the Riccati equation, for which an explicit representation of the solution is obtained. [ABSTRACT FROM AUTHOR]

We consider in this work small random perturbations (of multiplicative noise type) of the gradient flow. We prove that under mild conditions, when the potential function is a Morse function with additional strong saddle condition, the perturbed gradient flow converges to the neighborhood of local minimizers in O (ln (ε − 1)) time on the average, where ε is the scale of the random perturbation. Under a change of time scale, this indicates that for the diffusion process that approximates the stochastic gradient method, it takes (up to logarithmic factor) only a linear time of inverse stepsize to evade from all saddle points. This can be regarded as a manifestation of fast convergence of the discrete-time stochastic gradient method, the latter being used heavily in modern statistical machine learning. [ABSTRACT FROM AUTHOR]

In this paper, some properties of the lower (upper) semi-continuity for set-valued maps taking values in an abstract pre-ordered set are showed, which are then applied to study the existence of solutions for abstract set optimization problems. Moreover, some relationships between minimal solutions for abstract set-valued optimization problems with the vector and set criteria are given under mild conditions. As applications, existence results of solutions for set optimization problems are applied to obtain the existence of saddle points for the set-valued map taking values in the pre-ordered set with the set criterion and of solutions for vector optimization problems whose image space is a real vector space not necessarily endowed with a topology. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Localization of a magnetic skyrmion on a nonmagnetic defect in a two-dimensional triangular lattice is investigated within the framework of the generalized Heisenberg model, which includes exchange, anisotropy, Dzyaloshinskii–Moriya interaction, and interaction with an external magnetic field. It is shown that there is a threshold magnetic field, below which there are two locally stable positions of the defect inside the skyrmion. The energy difference between the states with a different localization of defects results in a fine energy structure of skyrmions, depending on the strength of the magnetic field.

The fact that both the Lorenz 1963 and 1969 models suggest finite predictability is well known. However, less well known is the fact that the mechanisms (i.e., sensitivities) within both models, which lead to finite predictability, are different. Additionally, the mathematical and physical relationship between these two models has not been fully documented. New analyses, along with a literature review, are performed here to provide insights regarding similarities and differences for these two models. The models represent different physical systems, one for convection and the other for barotropic vorticity. From the perspective of mathematical complexities, the Lorenz 1963 (L63) model is limited-scale and nonlinear; and the Lorenz 1969 (L69) model is closure-based, physically multiscale, mathematically linear, and numerically ill-conditioned. The former possesses a sensitive dependence of solutions on initial conditions, known as the butterfly effect, and the latter contains numerical sensitivities due to an ill-conditioned matrix with a large condition number (i.e., a large variance of growth rates). Here, we illustrate that the existence of a saddle point at the origin is a common feature that produces instability in both systems. Within the chaotic regime of the L63 nonlinear model, unstable growth is constrained by nonlinearity, as well as dissipation, yielding time varying growth rates along an orbit, and, thus, a dependence of (finite) predictability on initial conditions. Within the L69 linear model, multiple unstable modes at various growth rates appear, and the growth of a specific unstable mode (i.e., the most unstable mode during a finite time interval) is constrained by imposing a saturation assumption, thereby yielding a time varying system growth rate. Both models were interchangeably applied for qualitatively revealing the nature of finite predictability in weather and climate. However, only single type solutions were examined (i.e., chaotic and linearly unstable solutions for the L63 and L69 models, respectively), and the L69 system is ill-conditioned and easily captures numerical instability. Thus, an estimate of the predictability limit using either of the above models, with or without additional assumptions (e.g., saturation), should be interpreted with caution and should not be generalized as an upper limit for atmospheric predictability.

Searching saddle points on the potential energy surface is a challenging problem in the rare event. When there exist multiple saddle points, sampling different initial guesses are needed in local search methods in order to find distinct saddle points. In this paper, we present a novel global optimization-based dimer method (GOD) to efficiently search saddle points by coupling ant colony optimization (ACO) algorithm with optimization-based shrinking dimer (OSD) method. In particular, we apply OSD method as a local search algorithm for saddle points and construct a pheromone function in ACO to update the global population. By applying a two-dimensional example and a benchmark problem of seven-atom island on the (111) surface of an FCC crystal, we demonstrate that GOD shows a significant improvement in computational efficiency compared with OSD method. Our algorithm is the first try to apply the global optimization technique in searching saddle points and it offers a new framework to open up possibilities of adopting other global optimization methods. [ABSTRACT FROM AUTHOR]

We consider an abstract mixed variational problem consisting of two inequalities. The first one is governed by a functional φ, possibly non-differentiable. The second inequality is governed by a nonlinear term depending on a non negative parameter ǫ. We study the existence and the uniqueness of the solution by means of the saddle point theory. In addition to existence and uniqueness results, we deliver convergence results for ǫ → 0. Finally, we illustrate the abstract results by means of two examples arising from contact mechanics.

Kinematic relationships between degrees of freedom, also named multi-point constraints, are frequently used in structural mechanics. In this paper, the Craig variant of the Golub-Kahan bidiagonalization algorithm is used as an iterative method to solve the arising linear system with a saddle point structure. The condition number of the preconditioned operator is shown to be close to unity and independent of the mesh size. This property is proved theoretically and illustrated on a sequence of test problems of increasing complexity, including concrete structures enforced with pretension cables and the coupled finite element model of a reactor containment building. The Golub-Kahan algorithm converges in only a small number of steps for all considered test problems and discretization sizes. Furthermore, it is robust in practical cases that are otherwise considered to be difficult for iterative solvers. [ABSTRACT FROM AUTHOR]

A type of security is proposed that combines the possibilities of simultaneous exercise of infinite call and put options on a common stock by two market players. A zero-sum game is defined in which the players apply threshold exercise strategies. We show that unique best response functions exist for the players, and the game has a solution in pure strategies. A method to search for a saddle point is suggested. [ABSTRACT FROM AUTHOR]

We consider an abstract mixed variational problem consisting of two inequalities. The first one is governed by a functional, possibly non-differentiable. The second inequality is governed by a nonlinear term depending on a non negative parameter: We study the existence and the uniqueness of the solution by means of the saddle point theory. In addition to existence and uniqueness results, we deliver convergence results for ! 0. Finally, we illustrate the abstract results by means of two examples arising from contact mechanics. [ABSTRACT FROM AUTHOR]

A general theory of online power control for discrete-time battery limited energy harvesting communications is developed, which leads to, among other things, an explicit characterization of a maximin optimal policy. This policy only requires the knowledge of the (effective) mean of the energy arrival process and maximizes the minimum asymptotic expected average reward (with the minimization taken over all energy arrival distributions of a given (effective) mean). Moreover, it is universally near optimal and has a strictly better worst-case performance as well as a strictly improved lower multiplicative factor in comparison with the fixed fraction policy proposed by Shaviv and Özgür when the objective is to maximize the throughput over an additive white Gaussian noise channel. The competitiveness of this maximin optimal policy is also demonstrated via numerical examples. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a separate approach as a new strategy to solve the saddle point problem arising from the stochastic Galerkin finite element discretization of Stokes problems. The preconditioner is obtained by replacing the (1,1) and (1,2) blocks in the RHSS preconditioner by others well chosen and the parameter α in (2,2) −block of the RHSS preconditioner by another parameter β. The proposed preconditioner can be used as a preconditioner corresponding to the stationary itearative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergence properties of the GMRHSS iteration method are derived. Meanwhile, we analyzed the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Finally, numerical results show the effectiveness of the proposed preconditioner as compared with other preconditioners. [ABSTRACT FROM AUTHOR]

Molecular biology studies on cancer, using gene expression datasets, have revealed that the datasets have a very small number of samples. Obtaining medical data is difficult and expensive due to privacy constraints. Accuracy of classifiers depends greatly on the quality and quantity of input data. The problem of small sample size or small data size has been addressed by augmentation. Owing to the sensitivity of synthetic data samples for the cancer data classification for gene expression data, this paper is motivated to investigate data augmentation using GAN. GAN is based on the principle of two blocks (generator and discriminator) working in a collaborative yet adversarial way. This paper proposes modified generator GAN (MG-GAN) where the generator is fed with original data and multivariate noise to generate data with Gaussian distribution. As the generated data lie within latent space, we reach saddle point faster. GAN has been widely used in data augmentation for image datasets. As per our understanding, this is the first attempt of using GAN for augmentation on gene expression dataset. The performance merit of proposed MG-GAN was compared with KNN and Basic GAN. As compared to KNN and GAN, MG-GAN improves classification accuracy by 18.8% and 11.9%, respectively. The loss value of the error function for MG-GAN is drastically reduced, from 0.6978 to 0.0082, ensuring sensitivity of the generated data. Improved classification accuracy and reduction in the loss value make our improved MG-GAN method better suited for critical applications with sensitive data. [ABSTRACT FROM AUTHOR]

The theory of calculus of variations is a mathematical tool which is widely used in different scientific areas in particular in physics and chemistry. This theory is strongly related with optimisation. In fact the former seeks to optimise an integral related with some physical magnitude over some space to an extremum by varying a function of the coordinates. On the other hand, reaction paths and potential energy surfaces, in particular their stationary points, are the basis of many chemical theories, in particular reactions rate theories. We present a review where it is gathered together the variational nature of many types of reaction paths: steepest descent, Newton trajectories, artificial force induced reaction (AFIR) paths, gradient extremals, and gentlest ascent dynamics (GAD) curves. The variational basis permits to select the best optimisation technique in order to locate important theoretical objects on a potential energy surface. [ABSTRACT FROM AUTHOR]

The present paper proposes a new preconditioner for solving saddle point problems. The preconditioner is obtained by replacing the (1 , 2) and (2 , 2) blocks in the original saddle point matrix A by another well chosen block. The proposed preconditioner can be used as a preconditioner corresponding to the stationary iterative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergent and semi-convergent analyses of the regularized iteration method are presented. The eigenvalue distribution and the forms of the eigenvectors of the preconditioned matrix are analyzed. Finally, numerical results show the effectiveness of the proposed preconditioner as compared to other preconditioners. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the optimality conditions and saddle point theorems of a class of generalized convex (multi-objective fractional programming) problem ( MFP ) with respect to approximate weakly efficient solutions. Firstly, by using parameter method, the problem ( MFP ) is transformed into a (multi-objective non-fractional programming) problem (MNP) . Secondly, the concepts of type-I function and approximate pseudo-quasi-type-I function are introduced for the objective and constraint functions of problem ( MNP ). Finally, under assumptions of the introduced generalized convexities, the optimality conditions and saddle point theorems to problem ( MNP ) are established. [ABSTRACT FROM AUTHOR]

This paper concerns saddle points of rational functions, under general constraints. Based on optimality conditions, we propose an algorithm for computing saddle points. It uses Lasserre's hierarchy of semidefinite relaxation. The algorithm can get a saddle point if it exists, or it can detect its nonexistence if it does not. Numerical experiments show that the algorithm is efficient for computing saddle points of rational functions. [ABSTRACT FROM AUTHOR]

There are works of the Maeda–Morokuma group, which propose the artificial force induced reaction (AFIR) method (Maeda et al., J. Comput. Chem. 2014, 35, 166 and 2018, 39, 233). We study this important method from a theoretical point of view. The understanding of the proposers does not use the barrier breakdown point of the AFIR parameter, which usually is half of the reaction path between the minimum and the transition state which is searched for. Based on a comparison with the theory of Newton trajectories, we could better understand the method. It allows us to follow along some reaction pathways from minimum to saddle point, or vice versa. We discuss some well‐known two‐dimensional test surfaces where we calculate full AFIR pathways. If one has special AFIR curves at hand, one can also study the behavior of the ansatz. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of constrained vector optimization problems by using image space analysis. A class of vector-valued separation functions and a C-solution notion are proposed for the constrained vector optimization problems, respectively. Moreover, existence of a saddle point for the vector-valued separation function is characterized by the (regular) separation of two suitable subsets of the image space. By employing the separation function, we introduce a class of generalized vector-valued Lagrangian functions without involving any elements of the feasible set of constrained vector optimization problems. The relationships between the type-Ⅰ(Ⅱ) saddle points of the generalized Lagrangian functions and that of the function corresponding to the separation function are also established. Finally, optimality conditions for C-solutions of constrained vector optimization problems are derived by the saddle-point conditions. [ABSTRACT FROM AUTHOR]

The topological structure of field theory often makes inevitable the existence of stable and unstable localized solutions of the field equations. These are minima and saddle points of the energy. Saddle point solutions occurring this way are known as sphalerons, and the most interesting one is in the electroweak theory of coupledW, Z and Higgs bosons. The topological ideas underpinning sphalerons are reviewed here. [ABSTRACT FROM AUTHOR]