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Salkuyeh et al. (D.K. Salkuyeh, H. Aslani, Z.Z. Liang, An alternating positive semi-definite splitting preconditioner for the three-by-three block saddle point problems, Math. Commun. 26 (2021) 177-195) has recently established an alternating positive semi-definite splitting (APSS) method for nonsymmetric block three-by-three nonsingular saddle point problems arising from the Picard iteration method for a class of mixed finite element scheme. In this work, we analyse the semi-convergence of the APSS method for solving a class of nonsymmetric block three-by-three singular saddle point problems. The APSS induced preconditioner is applied to improve the semi-convergence rate of the exible GMRES (FGMRES) method. Numerical results are designated to support the theoretical results. These results show that the served preconditioner is efficient compared with FGMRES without a preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

A parametric version of the ellipsoid method with space scaling by a scalar parameter λ > 0 is considered. For certain values of the parameter λ, it is reduced to the well-known versions of the ellipsoid method, namely, the Shor, Khachiyan, Nemirovsky, and Yudin ones. The properties of two algorithmic implementations of the parameterized ellipsoid method are proved. The first algorithm is based on updating the asymmetric matrix B, and the second one updates the symmetric matrix H = BBT. The applications of the algorithms for solving convex programming problems and finding the saddle point of a convex-concave function are described. [ABSTRACT FROM AUTHOR]

In this study, the behaviour of the triple product ( n i τ E T i ) and height of the saddle point are investigated on the basis of the particle and power balance equations by considering primary reaction and side reactions among all product and reactant species in a spherical torus nuclear fusion reactor. For this purpose, a set of equations are coupled and solved numerically. Ignition capability is investigated and compared for different types of side reactions and other features like, the ratio between particle and energy confinement time and hot ion modes. Finally, the results are compared with each other to improve future research. [ABSTRACT FROM AUTHOR]

The authors prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. They address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

The paper proves the minimax theorem for a specific class of functions that are defined on branching polylines in a linear space, not on convex subsets of a linear space. The existence of a saddle point for such functions does not follow directly from the classical minimax theorem and needs individual consideration based both on convex analysis and graph theory. The paper presents a self-sufficient analysis of the problem. It contains everything that enables a plain understanding of the main result and its proof and avoids using concepts outside the scope of obligatory mathematical education of engineers. The paper is addressed to researchers in applied mechanics, engineering, and other applied sciences as well as to mathematicians who lecture convex analysis and optimization methods to non-mathematicians. [ABSTRACT FROM AUTHOR]

Minimax problems of the form minx maxy Ѱ(x, y) have attracted increased interest largely due to advances in machine learning, in particular generative adversarial networks and adversarial learning. These are typically trained using variants of stochastic gradient descent for the two players. Although convex-concave problems are well understood with many efficient solution methods to choose from, theoretical guarantees outside of this setting are sometimes lacking even for the simplest algorithms. In particular, this is the case for alternating gradient descent ascent, where the two agents take turns updating their strategies. To partially close this gap in the literature we prove a novel global convergence rate for the stochastic version of this method for finding a critical point of Ѱ(.):= maxy Ѱ (.,y) in a setting which is not convex-concave. [ABSTRACT FROM AUTHOR]

In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak-Lojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, we propose Riemannian Hamiltonian methods (RHMs) and present their convergence analyses. We extend RHMs to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHMs in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz's function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems. [ABSTRACT FROM AUTHOR]

In this work, we use the notion of convexificators to discuss optimality conditions for a fractional interval-valued optimization problem. We illustrate the sufficient optimality conditions established in the paper by the example of a nonconvex fractional interval-valued optimization problem with the help of generalized invex functions. Further, we study saddle point criteria of a Lagrange function defined for a fractional interval-valued optimization problem. [ABSTRACT FROM AUTHOR]

We consider a predator‐prey model with dissimilar functional and numerical responses that induce an Allee effect. There is a time lag between consumption and digestion of prey biomass by predator. Hence, a time delay has been incorporated in the numerical response function. The system consists of two interior equilibria. Taking time delay as the bifurcation parameter, four different dynamic behaviors appear, viz., (R1) system undergoes no change in its stability for all time delay, (R2) system undergoes stability change, (R3) system undergoes stability switching, and (R4) system undergoes instability switching. Here, finding four distinct dynamics in a single population model with only one delay is a novelty in this contribution. This variation in dynamics emerges due to asymmetricity in functional and numerical responses. All the relevant theorems in establishing stability are provided, and these are verified numerically. We analytically prove that if an interior equilibrium is a saddle point in absence of time delay, then the equilibrium cannot be stabilized by varying the time delay. It is popularly believed that existence of two distinct pair of purely imaginary roots of the characteristic function leads to stability switching. However, we provide examples where the system remains unstable, stability changes, and instability switching occurs. This is another new and interesting observation in our work. The numerical examples are furnished with phase portraits, time series plots, bifurcation diagrams, and eigenvalues evaluation with delay, for better understanding. Our model with a single delay exhibits variety of dynamics, which were not explored before. [ABSTRACT FROM AUTHOR]

We formulate a variational problem for harmonic waves in a viscoelastic body with short-term memory generated by distributed loads of a given frequency. The conditions of its well-posedness and equivalence to the saddle-point problem for the corresponding Lagrangian are established. We construct an h -adaptive scheme of the finite-element method for the solution of the formulated problem with an a posteriori error estimator determined element by element and a criterion for the local improvement of Delaunay triangulations used for the evaluation of approximations with prescribed accuracy. The efficiency of the proposed methods is illustrated by an example of numerical investigation of the resonance frequencies of a square plate containing a square hole. [ABSTRACT FROM AUTHOR]

Finding the saddle point of a matrix game is a classical problem that arises in various fields, e.g., economics, computer science, and engineering. The standard problem-solving methods consist of formulating the problem as a linear program (LP). However, this approach seems less efficient when many instances need to be solved. In this paper, we propose a Convolutional Neural Network based approach, which is able to predict both the strategy profile (x , y) and the optimal value v of the game. We call this approach Matrix Game-Conventional Neural Network or MG-CNN for short. Thanks to a global pooling technique, MG-CNN can solve matrix games with different shapes. We propose a specialized algorithm to train MG-CNN, which includes both data generation and model training. Our numerical experiments show that MG-CNN outperforms standard LP solvers in terms of computational CPU time and provides a high-quality prediction. [ABSTRACT FROM AUTHOR]

In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a (u , v − u) + b (v − u , λ) + j (v) − j (u) + J (u , v) − J (u , u) ≥ (f , v − u) X , b (u , μ − λ) − ψ (μ) + ψ (λ) ≤ 0 , (v ∈ K ⊆ X , μ ∈ Λ ⊂ Y ), where (X , (⋅ , ⋅) X) and (Y , (⋅ , ⋅) Y) are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ (⋅) = ϵ ψ ¯ (⋅) (ϵ > 0) . Then, we deliver a convergence result for ϵ → 0 , the case ψ ≡ 0 appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones. [ABSTRACT FROM AUTHOR]

Saddle dynamics is a time continuous dynamics to efficiently compute the any-index saddle points and construct the solution landscape. In practice, the saddle dynamics needs to be discretized for numerical computations, while the corresponding numerical analyses are rarely studied in the literature, especially for the high-index cases. In this paper we propose the convergence analysis of discrete high-index saddle dynamics. To be specific, we prove the local linear convergence rates of numerical schemes of high-index saddle dynamics, which indicates that the local curvature in the neighborhood of the saddle point and the accuracy of computing the eigenfunctions are main factors that affect the convergence of discrete saddle dynamics. The proved results serve as compensations for the convergence analysis of high-index saddle dynamics and are substantiated by numerical experiments. [ABSTRACT FROM AUTHOR]

We study the solution landscape and bifurcation diagrams of nematic liquid crystals confined on a rectangle, using a reduced two-dimensional Landau-de Gennes framework in terms of two geometry-dependent variables: half short edge length λ and aspect ratio b. First, we analytically prove that, for any b with a small enough λ or for a large enough b with a fixed domain size, there is a unique stable solution that has two line defects on the opposite short edges. Second, we numerically construct solution landscapes by varying λ and b, and report a novel X state, which emerges from saddle-node bifurcation and serves as the parent state in such a solution landscape. Various new classes are then found among these solution landscapes, including the X class, the S class, and the L class. By tracking the Morse indices of individual solutions, we present bifurcation diagrams for nematic equilibria, thus illustrating the emergence mechanism of critical points and several effects of geometrical anisotropy on confined defect patterns. [ABSTRACT FROM AUTHOR]

This paper aims at solving a stochastic two-player zero-sum Nash game problem studied in Singh and Lisser (2019). The main contribution of our paper is that we model this game problem as a dynamical neural network (DNN for short). In this paper, we show that the saddle point of this game problem is the equilibrium point of the DNN model, and we study the globally asymptotically stable of the DNN model. In our numerical experiments, we present the time-continuous feature of the DNN model and compare it with the state-of-the-art convex solvers, i.e., Splitting conic solver (SCS for short) and Cvxopt. Our numerical results show that our DNN method has two advantages in dealing with this game problem. Firstly, the DNN model can converge to a better optimal point. Secondly, the DNN method can solve all problems, even when the problem size is large. [ABSTRACT FROM AUTHOR]

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence. [ABSTRACT FROM AUTHOR]

In this paper, we study the lower iteration complexity bounds for finding the saddle point of a strongly convex and strongly concave saddle point problem: min x max y F (x , y) . We restrict the classes of algorithms in our investigation to be either pure first-order methods or methods using proximal mappings. For problems with gradient Lipschitz constants ( L x , L y and L xy ) and strong convexity/concavity constants ( μ x and μ y ), the class of pure first-order algorithms with the linear span assumption is shown to have a lower iteration complexity bound of Ω L x μ x + L xy 2 μ x μ y + L y μ y · ln 1 ϵ , where the term L xy 2 μ x μ y explains how the coupling influences the iteration complexity. Under several special parameter regimes, this lower bound has been achieved by corresponding optimal algorithms. However, whether or not the bound under the general parameter regime is optimal remains open. Additionally, for the special case of bilinear coupling problems, given the availability of certain proximal operators, a lower bound of Ω L xy 2 μ x μ y · ln (1 ϵ) is established under the linear span assumption, and optimal algorithms have already been developed in the literature. By exploiting the orthogonal invariance technique, we extend both lower bounds to the general pure first-order algorithm class and the proximal algorithm class without the linear span assumption. As an application, we apply proper scaling to the worst-case instances, and we derive the lower bounds for the general convex concave problems with μ x = μ y = 0 . Several existing results in this case can be deduced from our results as special cases. [ABSTRACT FROM AUTHOR]

The minimum (worst case) value of a long-only portfolio of bonds, over a convex set of yield curves and spreads, can be estimated by its sensitivities to the points on the yield curve. We show that sensitivity based estimates are conservative, i.e., underestimate the worst case value, and that the exact worst case value can be found by solving a tractable convex optimization problem. We then show how to construct a long-only bond portfolio that includes the worst case value in its objective or as a constraint, using convex–concave saddle point optimization. [ABSTRACT FROM AUTHOR]

In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a (u , v − u) + b (v − u , λ) + j (v) − j (u) + J (u , v) − J (u , u) ≥ (f , v − u) X , b (u , μ − λ) − ψ (μ) + ψ (λ) ≤ 0 , (v ∈ K ⊆ X , μ ∈ Λ ⊂ Y ), where (X , (⋅ , ⋅) X) and (Y , (⋅ , ⋅) Y) are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ (⋅) = ϵ ψ ¯ (⋅) (ϵ > 0) . Then, we deliver a convergence result for ϵ → 0 , the case ψ ≡ 0 appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones. [ABSTRACT FROM AUTHOR]

This paper is concerned with one kind of two-person zero-sum linear-quadratic stochastic differential game with jumps. A closed-loop formulation for the saddle points of the game is established in a mixed-strategy-law form, for which the associated Riccati equation and Hamiltonian system are solved. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Radiation of sound waves by a coaxial duct is considered. The coaxial duct has an inner rigid wall which is infinite, while the outer wall is rigid for z < 0 and perforated for z > 0 which makes the problem more interesting. Applying the Fourier transform technique, the mixed boundary value problem is described by a Wiener–Hopf equation and then the solution of the problem is found by means of the Wiener–Hopf (WH) technique. The influence of the parameters of the problem like inner and outer duct radii, frequency and porosity on the sound pressure level is displayed graphically. This study is important because of its applications in noise reduction in modern aircraft jet and turbofan engines. The current problem is also compared with subproblems with and without perforated and very good agreement is observed. [ABSTRACT FROM AUTHOR]

Gradient-descent (GD) based algorithms are an indispensable tool for optimizing modern machine learning models. The paper considers distributed stochastic GD (D-SGD)|a network-based variant of GD. Distributed algorithms play an important role in large-scale machine learning problems as well as the Internet of Things (IoT) and related applications. The paper considers two main issues. First, we study convergence of D-SGD to critical points when the loss function is nonconvex and nonsmooth. We consider a broad range of nonsmooth loss functions including those of practical interest in modern deep learning. It is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. It is well known that classical GD avoids saddle points; however, analogous results have been absent for distributed variants of GD. For this problem, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation. [ABSTRACT FROM AUTHOR]

In this paper, we consider one optimal control problem with a multipoint quality functional described by a system of nonlinear hyperbolic equations with Goursat boundary conditions. Using a modified version of the increment method, various necessary first-order optimality conditions such as the Pontryagin maximum principle and the linearized maximum condition are proved. [ABSTRACT FROM AUTHOR]

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution C± with interior defects, high-index 'fake defect' solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics. [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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Confined nematic liquid crystals can admit multiple stable equilibria. These equilibrium states usually exhibit peculiar defect patterns in order to satisfy the topological constraints imposed by the boundary conditions. In this work, we systematically investigate transition pathways between different stable equilibria of nematic liquid crystals confined in a cylinder with homeotropic boundary condition. To do so, we implement spectral method to minimize the Landau-de Gennes free energy and develop a multiscale string method to accurately compute both minimal energy path and transition state. We first compute the transition pathways under the assumption that system state is invariant along the axial direction of the cylinder. This axial invariant assumption allows us to reduce the simulation domain to a two-dimensional disk. We subsequently remove the axial invariant assumption to search the transition pathways in the three-dimensional cylinder. Numerical simulations demonstrate that there exists a threshold for the cylinder height above which a domino-like transition pathway is more energetically favored than the axial invariant transition pathway. Moreover, we show that the transition pathway from escape radial with ring defects (ERRD) to escape radial has a lower energy barrier than the one from ERRD to planar polar, even though the free energy of the planar polar state is lower than that of the escape radial state. Our approach provides an accurate and efficient method to compute minimal energy path and transition state of confined nematic liquid crystals. [ABSTRACT FROM AUTHOR]

We present a high-index optimization-based shrinking dimer (HiOSD) method to compute index-k saddle points as a generalization of the optimization-based shrinking dimer method for index-1 saddle points [L. Zhang, Q. Du, and Z. Zheng, SIAM J. Sci. Comput., 38 (2016), pp. A528{A544]. We first formulate a minimax problem for an index-k saddle point that is a local maximum on a k-dimensional manifold and a local minimum on its orthogonal complement. The k-dimensional maximal subspace is spanned by the k eigenvectors corresponding to the smallest k eigenvalues of the Hessian, which can be constructed by the simultaneous Rayleigh-quotient minimization technique or the locally optimal block preconditioned conjugate gradient method. Under the minimax framework, we implement the Barzilai{Borwein gradient method to speed up the convergence. We demonstrate the eficiency of the HiOSD method for computing high-index saddle points by applying finite-dimensional examples and semilinear elliptic problems. [ABSTRACT FROM AUTHOR]

• Explicit choice of a 2 degree of freedom potential surface. • Rolling of a ball on this surface and trajectories tracked by a digital camera. • Comparisons with the outcome of numerical simulation. • Identification of unstable equilibria. • Assessment of the factors influencing the accuracy of the approach. This paper details a new method to estimate the location of unstable equilibria, specifically saddle-points, based on transient trajectories from experiments. We describe a system in which saddle-points (not easily observed in a direct sense) influence the behavior of trajectories that pass 'close-by' them. This influence is used to construct a model and thus identify a more accurate estimate of the location using a number of refinements associated with linearization and regression. Both simulations and experiments were conducted to verify the method. The experiment consists of a small ball rolling on a relatively shallow curved surface under the influence of gravity: a potential energy surface in two dimensions. Tracking the motion of the ball with a digital camera provides data that compares closely with the output of numerical simulation. The experimental results suggest that this method can effectively locate the saddle equilibria in a system, and the robustness of the approach is assessed relative to the effect of noise, size of the local neighborhood, etc., in addition to providing information on the local dynamics. Given the relative simplicity of the experiment system used and a priori knowledge of the saddle-points, it is a useful testing environment for system identification in a nonlinear context. [ABSTRACT FROM AUTHOR]