SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).

REFERENCE FUNCTIONAL AND CHARACTERISTIC SPACE FOR LAGRANGE AND BERNSTEIN OPERATORS

This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating operator and the De Casteljau one for the Bernstein-Bezier approximating operator. Both schemes fall intothe framework of transformations of the form where the F iare given numbers (forexample, at the initial step they coincide with the values of the function on a given lattice), and the coefficients (x) are linear polynomials valued in x and x is fixed. A general theory for such sequence of transformations can be found in [2] where it is also proved that these tranformations are completely characterized in term of a linear functional, reference functional. This functional is associated with a linear space., characteristic space.The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functional: associated with these operators.

FURTHER STUDY ON A DUAL ALGORITHM

The dual algorithm for minimax problems is further studied in this paper.The resulting theoretical analysis shows that the condition number of the corresponding Hessian of the smooth modified Lagrange function with changing parameter in the dual algorithm is proportional to the reciprocal of the parameter,which is very important for the efficiency of the dual algorithm.At last,the numerical experiments are reported to validate the analysis results.

The Second-Order Differential Equation System with the Feedback Controls for Solving Convex Programming

In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.

A NEW TRUST-REGION ALGORITHM FOR NONLINEAR CONSTRAINED OPTIMIZATION

We propose a new trust region algorithm for nonlinear constrained optimization problems. In each iteration of our algorithm, the trial step is computed by minimizing a quadratic approximation to the augmented Lagrange function in the trust region. The augmented Lagrange function is also used as a merit function to decide whether the trial step should be accepted. Our method extends the traditional trust region approach by combining a filter technique into the rules for accepting trial steps so that a trial step could still be accepted even when it is rejected by the traditional rule based on merit function reduction. An estimate of the Lagrange multiplier is updated at each iteration, and the penalty parameter is updated to force sufficient reduction in the norm of the constraint violations. Active set technique is used to handle the inequality constraints. Numerical results for a set of constrained problems from the CUTEr collection are also reported.

Analysis on a Superlinearly Convergent Augmented Lagrangian Method

The augmented Lagrangian method is a classical method for solving constrained optimization.Recently,the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low rank matrix optimization problems.However,most Lagrangian methods use first order information to update the Lagrange multipliers,which lead to only linear convergence.In this paper,we study an update technique based on second order information and prove that superlinear convergence can be obtained.Theoretical properties of the update formula are given and some implementation issues regarding the new update are also discussed.

Optimal cash holding decision with uncertain control

The decision-making of cash holdings is very important for the daily operation of enterprises.This present paper tries to establish uncertain optimal cash holding models with the constraint of safe cash holding area,and discusses the solutions of the models by establishing Lagrange function under KKT condition.On the one hand,this paper enriches the existing cash holding models,on the other hand,it is also a comprehensive discussion on the application of uncertainty theory in cash holding management.

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.