非线性优化修正Frisch函数方法的乘子映射

对于同时含有等式与不等式约束的非线性优化问题的修正Frisch函数方法,给出其乘子映射和解映射的导数的估计.将得到的估计用于建立修正Frisch函数方法的线性收敛速率.在线性无关的约束规范,严格互补条件和二阶充分性条件成立的前提下,证得该收敛率与1/c成正比.本文的收敛性分析依赖于矩阵的奇异值分解,其方法可以用来分析其他的修正Lagrange方法.     

Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail.     

Generalized multivalued nonlinear quasivariational inclusions

In this paper, we introduce and study a few classes of generalized multivalued nonlinear quasivariational inclusions and generalized nonlinear quasivariational inequalities, which include many classes of variational inequalities, quasivariational inequalities and variational inclusions as special cases. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for finding the approximate solutions of these classes of quasivariational inclusions and quasivariational inequalities. We establish the existence of solutions for this generalized nonlinear quasivariational inclusions involving both relaxed Lipschitz and strongly monotone and generalized pseudocontractive mappings and obtain the convergence of iterative sequences generated by the algorithms. Under certain conditions, we derive the existence of a unique solution for the generalized nonlinear quasivariational inequalities and obtain the convergence and stability results of the Noor type perturbed iterative algorithm. The results proved in this paper represent significant refinements and improvements of the previously known results in this area.     

A nonlinear Lagrangian for constrained optimization problems

A novel nonlinear Lagrangian is presented for constrained optimization problems with both inequality and equality constraints, which is nonlinear with respect to both functions in problem and Lagrange multipliers. The nonlinear Lagrangian inherits the smoothness of the objective and constraint functions and has positive properties. The algorithm on the nonlinear Lagrangian is demonstrated to possess local and linear convergence when the penalty parameter is less than a threshold (the penalty parameter in the penalty method has to approximate zero) under a set of suitable conditions, and be super-linearly convergent when the penalty parameter is decreased following Lagrange multiplier update. Furthermore, the dual problem based on the nonlinear Lagrangian is discussed and some important properties are proposed, which fail to hold for the dual problem based on the classical Lagrangian. At last, the preliminary and comparing numerical results for several typical test problems by using the new nonlinear Lagrangian algorithm and the other two related nonlinear Lagrangian algorithms, are reported, which show that the given nonlinear Lagrangian is promising.     

Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.     

An accelerated multiplier method for nonlinear programming

This paper describes an accelerated multiplier method for solving the general nonlinear programming problem. The algorithm poses a sequence of unconstrained optimization problems. The unconstrained problems are solved using a rank-one recursive algorithm described in an earlier paper. Multiplier estimates are obtained by minimizing the error in the Kuhn-Tucker conditions using a quadratic programming algorithm. The convergence of the sequence of unconstrained problems is accelerated by using a Newton-Raphson extrapolation process. The numerical effectiveness of the algorithm is demonstrated on a relatively large set of test problems.This work was supported by the US Air Force under Contract No. F04701-74-C-0075.     

A gradient projection-multiplier method for nonlinear programming

This paper describes a gradient projection-multiplier method for solving the general nonlinear programming problem. The algorithm poses a sequence of unconstrained optimization problems which are solved using a new projection-like formula to define the search directions. The unconstrained minimization of the augmented objective function determines points where the gradient of the Lagrangian function is zero. Points satisfying the constraints are located by applying an unconstrained algorithm to a penalty function. New estimates of the Lagrange multipliers and basis constraints are made at points satisfying either a Lagrangian condition or a constraint satisfaction condition. The penalty weight is increased only to prevent cycling. The numerical effectiveness of the algorithm is demonstrated on a set of test problems.The author gratefully acknowledges the helpful suggestions of W. H. Ailor, J. L. Searcy, and D. A. Schermerhorn during the preparation of this paper. The author would also like to thank D. M. Himmelblau for supplying a number of interesting test problems.     

Picard’s iterative method for nonlinear advection-reaction-diffusion equations

Picard’s iterative method for the solution of nonlinear advection-reaction-diffusion equations is formulated and its convergence proved. The method is based on the introduction of a complete metric space and makes uses of a contractive mapping and Banach’s fixed-point theory. From Picard’s iterative method, the variational iteration method is derived without making any use at all of Lagrange multipliers and constrained variations. Some examples that illustrate the advantages and shortcomings of the iterative procedure presented here are shown.     

A class of nonlinear Lagrangians for nonconvex second order cone programming

This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

求解非凸半定规划的一个非线性Lagrange算法及其收敛性分析

本文提出了一个求解非凸半定规划的非线性Lagrange算法,当二阶充分条件以及严格互补条件成立时,证明了这一算法的收敛性定理.收敛结果表明,当惩罚参数小于某个阀值时,算法是局部收敛的;此外,还给出了解的一个依赖于惩罚参数的误差界.     

A new subspace correction method for nonlinear unconstrained convex optimization problems

This paper gives a new subspace correction algorithm for nonlinear unconstrained convex optimization problems based on the multigrid approach proposed by S. Nash in 2000 and the subspace correction algorithm proposed by X. Tai and J. Xu in 2001. Under some reasonable assumptions, we obtain the convergence as well as a convergence rate estimate for the algorithm. Numerical results show that the algorithm is effective.     

一类带反凸约束的非线性比式和问题的全局优化算法

本文针对一类带有反凸约束的非线性比式和分式规划问题,提出一种求其全局最优解的单纯形分支和对偶定界算法.该算法利用Lagrange对偶理论将其中关键的定界问题转化为一系列易于求解的线性规划问题.收敛性分析和数值算例均表明提出的算法是可行的.     

一类广义非线性隐拟变分包含

引入和研究了一类新的涉及集值极大单调映象的广义非线性隐拟变分包含.对此类变分包含在没有紧性假设下证明了解的存在定理.为寻求此类变分包含的近似解,建议和分析了一个新的迭代算法.给出了由新算法产生的迭代序列的收敛性.作为特殊情形,也讨论了在此领域内的某些已知结果.     

A Nonlinear Lagrange Algorithm for Minimax Problems with General Constraints

This article presents a novel nonlinear Lagrange algorithm for solving minimax optimization problems with both inequality and equality constraints, which eliminates the nonsmoothness of the considered problems and the numerical difficulty of the penalty method. The convergence of the proposed algorithm is analyzed under some mild assumptions, in which the sequence of the generated solutions converges locally to a Karush-Kuhn-Tucker solution at a linear rate when the penalty parameter is less than a threshold and the error bound of the solutions is also obtained. Finally, the detailed numerical results for several typical testproblems are given in order to show the performance of the proposed algorithm.     

Uniform exponential stabilization of nonlinear systems in Banach spaces

This paper presents necessary and sufficient conditions for uniform exponential stabilization of a class of nonlinear systems in Banach state spaces. The stabilization assumptions are formulated in terms of integral estimates involving the control operator and the state of the uncontrolled version of the system at hand. An explicit estimate of the convergence speed is given. Applications to feedback stabilization of affine control systems are given. Illustrative examples are further provided.     

Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity

Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.     

Finite convergence of algorithms for nonlinear programs and variational inequalities

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.The research of the first author was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173.The authors are indebted to Professors Olvi Mangasarian, Garth McCormick, Jong-Shi Pang, Hanif Sherali, and Hoang Tuy for helpful comments and suggestions and to two anonymous referees for constructive remarks and for bringing to their attention the results in Refs. 13 and 14.     

用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

Steepest descent preconditioning for nonlinear GMRES optimization

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.