Fractional Action-Like Variational Problem and Its Noether Symmetries for a Nonholonomic System

For an in-depth study on the symmetric properties for nonholonomic non-conservative mechanical systems,the fractional action-like Noether symmetries and conserved quantities for nonholonomic mechanical systems are studied,based on the fractional action-like approach for dynamics modeling proposed by El-Nabulsi.Firstly,the fractional action-like variational problem is established,and the fractional action-like Lagrange equations of holonomic system and the fractional action-like differential equations of motion with multiplier for nonholonomic system are given;secondly,according to the invariance of fractional action-like Hamilton action under infinitesimal transformations of group,the definitions and criteria of fractional action-like Noether symmetric transformations and quasi-symmetric transformations are put forward;finally,the fractional action-like Noether theorems for both holonomic system and nonholonomic system are established,and the relationship between the fractional action-like Noether symmetry and the conserved quantity is given.

On the Variational Problems of the Functionals with Derivatives of Higher Orders and Undetermined Boundary

According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.

A VARIATIONAL PROBLEM ARISING IN REGISTRATION OF DIFFUSION TENSOR IMAGES

The existence of a global minimizer for a variational problem arising in registration of diffusion tensor images is proved, which ensures that there is a regular spatial transformation for the registration of diffusion tensor images.

Minimizing sequences of variational problems with small parameters

A class of variational problems with small parameters is studied. Their zeroth-order asymptotic solutions are constructed. It is shown that the zeroth-order asymptotic solution is just the minimizing sequence of variational problems as the small parameter approaches to zero.

ON THE SINGULAR VARIATIONAL PROBLEMS

The authors deal with the singular variational problemS(α,b,λ0)as well asS= S(α,b,λ1,λ2)where Nm/N-m+m(b-a),α,β(?)1,E= D1α,m(RN). The aim of this paper is to show the existence of minimizer for 5(α, b,λ0) and S(α,b,λ1,λ2).

ON THE REGULARITY OF THE MINIMUM SOLUTION OF THE RESTRAINED VARIATIONAL PROBLEMS

This paper is concerned with the regularity of minimum solution u of the following functional L(u) = integral(Omega) a alpha(beta)(x)g(ij)(u)D alpha u(i)D(beta)upsilon(i)dx on the restraint E = {u is an element of W-0(1,2) (Omega, R(N))\parallel to u parallel to L(D) = 1}. Under appropriate conditions, the bounded minimum solution u of the above functional is proved to be nothing but Holder continuous.

Necessary Optimality Conditions for Multi-Objective Semi-Infinite Variational Problem

In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.

LOCAL AND GLOBAL STABILITY OF INFINITE-HORIZON VARIATIONAL PROBLEM

This paper presents an existence theorem of the optimal solution for the infinite-horizon variational problem with the inclusion constraints on the state variables and velocity variables, give the estimation of the optimal solution. Under some conditions, the authors obtain the local stability and global stability of the stationary solution.

An Iterative Method for Split Variational Inclusion Problem and Split Fixed Point Problem for Averaged Mappings

In this paper, we use resolvent operator technology to construct a viscosity approximate algorithm to approximate a common solution of split variational inclusion problem and split fixed point problem for an averaged mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and split fixed point problem for averaged mappings which is also the unique solution of the variational inequality problem. The results presented here improve and extend the corresponding results in this area.

The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.

Optimality and Duality for Multiobjective Semi-infinite Variational Problem Using Higher-Order B-type I Functions

The notion of higher-order B-type I functional is introduced in this paper.This notion is utilized to study optimality and duality for multiobjective semi-infinite variational problem in which the index set of inequality constraints is an infinite set.The concept of efficiency is used as a tool for optimization.Mond–Weir type of dual is proposed for which weak,strong,and strict converse duality theorems are proved to relate efficient solutions of primal and dual problems.

Existence and uniqueness for variational problem from progressive lens design

We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.

An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions.To this end,wefirst reformulate the original problem into a minimax problem corresponding to a feasible augmented La-grangian,which can be solved by the augmented Lagrangian method in an infinite dimensional setting.Based on this,by expressing the primal and dual variables with two individual deep neural network functions,we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimiza-tion method together with a projection technique.Compared to the traditional penalty method,the new method admits two main advantages:i)the choice of the penalty parameter isflexible and robust,and ii)the numerical solution is more accurate in the same magnitude of computational cost.As typical applications,we apply the new ap-proach to solve elliptic problems and(nonlinear)eigenvalue problems with essential boundary conditions,and numerical experiments are presented to show the effective-ness of the new method.

A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM,FIXED POINT PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS,AND QUASI-VARIATIONAL INCLUSION PROBLEM

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Wakahashi and Toyoda [9], Takahashi and Takahashi [I0] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [II], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

A Smoothing Newton Method for the Box Constrained Variational Inequality Problems

The box constrained variational inequality problem can be reformulated as a nonsmooth equation by using median operator.In this paper,we present a smoothing Newton method for solving the box constrained variational inequality problem based on a new smoothing approximation function.The proposed algorithm is proved to be well defined and convergent globally under weaker conditions.

STRONGLY CONVERGENT ITERATIVE METHODS FOR SPLIT EQUALITY VARIATIONAL INCLUSION PROBLEMS IN BANACH SPACES

The purpose of this paper is to introduce and study the split equality variational inclusion problems in the setting of Banach spaces. For solving this kind of problems, some new iterative algorithms are proposed. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm are proved. As applications, we shall utilize the results presented in the paper to study the split equality feasibility prob- lems in Banach spaces and the split equality equilibrium problem in Banach spaces. The results presented in the paper are new.

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

RELAXED INERTIAL METHODS FOR SOLVING SPLIT VARIATIONAL INEQUALITY PROBLEMS WITHOUT PRODUCT SPACE FORMULATION

Many methods have been proposed in the literature for solving the split variational inequality problem.Most of these methods either require that this problem is transformed into an equivalent variational inequality problem in a product space,or that the underlying operators are co-coercive.However,it has been discovered that such product space transformation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting nature of the split variational inequality problem.On the other hand,the co-coercive assumption of the underlying operators would preclude the potential applications of these methods.To avoid these setbacks,we propose two new relaxed inertial methods for solving the split variational inequality problem without any product space transformation,and for which the underlying operators are freed from the restrictive co-coercive assumption.The methods proposed,involve projections onto half-spaces only,and originate from an explicit discretization of a dynamical system,which combines both the inertial and relaxation techniques in order to achieve high convergence speed.Moreover,the sequence generated by these methods is shown to converge strongly to a minimum-norm solution of the problem in real Hilbert spaces.Furthermore,numerical implementations and comparisons are given to support our theoretical findings.

A CLASS OF VARIATIONAL DIFFERENCE SCHEMES FOR A SINGULAR PERTURBATION PROBLEM

In this paper, a singularly perturbed boundary value problem for second order self-adjoint ordinary differential equation is discussed. A class of variational difference schemes is constructed by the finite element method. Uniform convergence about small parameter is proved under a weaker smooth condition with respect to the coefficients of the equation. The schemes studied in refs. [1], [3], [4] and [51 belong to the cllass.

A HYBRID METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEMS

By using Fukushima's differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993.In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix Δ F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty,polyhedral,closed and convex is proposed.Armijo type line search and trust region strategies as well as Fukushima's differentiable merit function are incorporated into the method.It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al.,the method reduces to the basic Newton method and hence the rate of convergence is quadratic.Computational experiences show the efficiency of the proposed method.