Exact relaxations of non-convex variational problems

Here, we solve non-convex, variational problems given in the form

Shifted Legendre direct method for variational problems

The shifted Legendre polynomial series is employed to solve variational problems. The solution is carried out by using an operational matrix for integrating the shifted Legendre polynomial vector. Variational problems are reduced to solving algebraic equations. Two illustrative examples are given, and the computational results obtained by Legendre series direct method are compared with the exact solutions.     

Laguerre series direct method for variational problems

A direct method for solving variational problems via Laguerre series is presented. First, an operational matrix for the integration of Laguerre polynomials is introduced. The variational problems are reduced to the solution of algebraic equations. An illustrative example is given.     

On characterizing the set of possible effective tensors of composites: The variational method and the translation method

A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T₁. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.     

A solution method for linear variational relation problems

In this paper, we consider a particular class of variational relation problem namely linear variational relation problem wherein the sets are defined by linear inequalities. The purpose is to study the existence of the solution set and its nature for this class of problem. Using these results, we provide algorithms to obtain the solutions of the problem based on which we present some numerical illustrations.     

Semidefinite relaxations of ordering problems

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element $u$ is before $v$ in the ordering. In the second case, the profit depends on whether $u$ is before $v$ and $r$ is before $s$ . The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.     

解变分不等式问题的混合方法

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 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.     

Modified subgradient extragradient method for variational inequality problems

In this paper, we introduce an algorithm as combination between the subgradient extragradient method and inertial method for solving variational inequality problems in Hilbert spaces. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. The proposed algorithm can be considered as an improvement of the previously known inertial extragradient method over each computational step. The performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.     

Combination of the variational iteration method and numerical algorithms for nonlinear problems

A very simple and efficient local variational iteration method (LVIM), or variational iteration method with local property, for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate.     

Solution of optimal control problems by a variational method

An algorithm for numerically solving optimal control problems by methods applied to ill-posed problems is discussed. The stable algorithms for solving such problems on compact sets developed by Academician A.N. Tikhonov in the twentieth century can be applied to problems of optimal control. The special feature of optimal control problems is the discontinuity of a control function. This difficulty is overcome by introducing a moving computational grid. The step size of the grid is determined by solving the speed problem.     

Optimal control in coefficients for weak variational problems in Hilbert space

In this paper we give sufficient conditions for the existence of solutions of a problem of parametric optimization. We use continuity with respect to a functional parameter of weak solutions of a variational problem in a Hilbert space.We consider a problem of optimization with the control in coefficients of linear parabolic equation as an example. Using results of Spagnolo we characterize the closure of the reachable set. Finally, we construct an example of an optimization problem with the control in coefficients of a parabolic equation which does not have an optimal solution.     

Restricted variational principle method for problems of low-density aerodynamics

Rosen's restricted variational principle representation of the Boltzmann equation is applied to the problem of determining the transitional-regime, low-density, hypersonic flow over slender, conical vehicles with diffusely reflecting surfaces. If the trial distribution function is suitably chosen, the Euler-Lagrange equations associated with Rosen's functional result in a semilinear hyperbolic system amenable to solution by classical characteristics methods. Sample calculations are given and compared with the low-density cone flow experiments of Hickman to assess the accuracy of the method presented. It is believed that the present method constitutes the first solution method for boundary value problems in the low-density transitional flow regime.This work was supported by the United States Air Force, Contract No. AF04(695)-1001.     

A QP-free constrained Newton-type method for variational inequality problems

Received January 5, 1997 / Revised version received November 19, 1997 Published online November 24, 1998     

Convergence of a non-interior smoothing method for variational inequality problems

The variational inequality problem can be reformulated as a system of equations. One can solve the reformulated equations to obtain a solution of the original problem. In this paper, based on a symmetric perturbed min function, we propose a new smoothing function, which has some nice properties. By which we propose a new non-interior smoothing algorithm for solving the variational inequality problem, which is based on both the non-interior continuation method and the smoothing Newton method. The proposed algorithm only needs to solve at most one system of equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results are reported.     

An extragradient method for relaxed cocoercive variational inequality and equilibrium problems

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansivemappings, the set of an equilibrium problem and the set of solutions of the variational inequality problem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao, Takahashi and many others.