曲率障碍下四阶变分不等式的交替方向乘子法北大核心CSCD

对于重调和算子和曲率障碍表示的变分不等式,提出了自适应交替方向乘子数值解法(SADMM).对问题引入一个辅助变量表示曲率函数的增广Lagrange函数,导出一个约束极小值问题,并且该问题等价于一个鞍点问题.然后采用交替方向乘子法(ADMM)求解这个鞍点问题.通过采用平衡原理和迭代函数,得到了自动调整罚参数的自适应法则,从而提高了计算效率.证明了该方法的收敛性,并给出了利用迭代函数近似罚参数的具体方法.最后,用数值计算结果验证了该方法的有效性.     

向量优化中广义增广拉格朗日对偶理论及应用

作者介绍了一种基于向量值延拓函数的广义增广拉格朗日函数,建立了基于广义增广拉格朗日函数的集值广义增广拉格朗日对偶映射和相应的对偶问题,得到了相应的强对偶和弱对偶结果,将所获结果应用到约束向量优化问题.该文的结果推广了一些已有的结论.     

具有线性化技术的三块非凸不可分优化问题BregmanADMM收敛性分析

交替方向乘子法是求解两块可分离凸优化问题的有效方法,但是对于三块不可分的非凸优化问题的交替方向乘子法的收敛性可能无法保证.该文主要研究的是用线性化广义Bregman交替方向乘子法(L-G-BADMM)求解目标函数是三块不可分的非凸极小化问题的收敛性分析.在适当假设条件下,对算法中子问题进行求解并构建满足Kurdyka-Lojasiewicz性质的效益函数,经过理论证明可以得到该算法的收敛性.     

两大部类扩大再生产中的广义拉格朗日乘子

在一个带有非负和不等式约束的优化问题有最优解的情形下,存在着广义拉格朗日乘子即资源的影子价格．本文探索给出马克思两大部类扩大再生产中的影子价格,为经典的马克思扩大再生产理论增添新的重要内容．首先使用“价值系数法”替代单纯形法,简便地求得了扩大再生产优化问题的最优解．然后运用库恩一塔克条件,确立了关于最优解与广义拉格朗日乘子的互补松弛条件的三个不等式组．进而利用这些不等式组和已知的最优解,简便地解出广义拉格朗日乘子,即两大部类扩大再生产中的影子价格．最后引用和借鉴《资本论》中的两个举例,对所获得的影子价格和目标函数最优值做了计算验证．     

集值映射向量优化问题的e-真有效解

引入了集值映射向量优化问题的αe-弱有效解、e-真有效解、e-真鞍点概念,在近似广义C-次似凸条件下,建立了e-真有效解的标量化定理、Lagrang乘子定理和e-真鞍点定理,并讨论了集值映射向量优化问题的αe-弱有效解的标量化定理和Laugrange乘子定理,推广了已有结果。     

求解一般约束优化问题的一个全局收敛的混合不精确SQP算法

对于一般约束优化问题,本文通过一种特殊的耦合策略,把一个局邵超线性收敛的不精确SQP算法与广义梯度投影法相结合,从而给出了一个混合算法.该算法无需计算拉格朗日函数的海色矩阵,并且在适当的假设下,算法具有全局和局部超线性收敛性.     

Toeplitz矩阵填充的 $\ell$-步修正增广拉格朗日乘子算法[英文]

基于 Toeplitz矩阵填充(TMC)的修正增广拉格朗日乘子(MALM)算法, 本文给出此算法的一种加速策略, 提出Toeplitz矩阵填充的 $\ell$-步修正增广拉格朗日乘子算法. 该方法通过削减原 MALM算法中每一步迭代的频繁数据传输, 提高算法的运行效率. 同时也证明了新算法的收敛性. 最后以数值实验表明 $\ell$-步修正增广拉格朗日乘子算法比原 MALM算法更有效.     

Toeplitz矩阵填充的?-步修正增广拉格朗日乘子算法（英文）

基于Toeplitz矩阵填充(TMC)的修正增广拉格朗日乘子(MALM)算法,本文给出此算法的一种加速策略,提出Toeplitz矩阵填充的?-步修正增广拉格朗日乘子算法.该方法通过削减原MALM算法中每一步迭代的频繁数据传输,提高算法的运行效率.同时也证明了新算法的收敛性.最后以数值实验表明?-步修正增广拉格朗日乘子算法比原MALM算法更有效.     

一种广义共轭下的拉格朗日函数及鞍点定理

本文根据文［４］所提出的广义共轭概念，作者给出了一个新的拉格朗日函数，得到了拉格朗日函数的一些结论；导出了拉格朗日函数的鞍点最优性条件；最后就一簇特殊的Ｈ给出了拉格朗日函数的特殊形式以及约束规划问题的最优性条件，而这正是一般规划论书中所讲相关内容的推广．     

对等式约束非线性规划问题的Hestenes-Powell增广拉格朗日函数的进一步研究

本文对用无约束极小化方法求解等式约束非线性规划问题的Hestenes-Powell 增广拉格朗日函数作了进一步研究．在适当的条件下,我们建立了Hestenes-Powell增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系,并且也给出了Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系．因此,从理论的观点来看,原约束问题的解和对应的拉格朗日乘子值不仅可以用众所周知的乘子法求得,而且可以通过对Hestenes-Powell 增广拉格朗日函数在原问题变量和乘子变量的积空间上执行一个单一的无约束极小化来获得．     

Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.     

Unified theory of augmented Lagrangian methods for constrained global optimization

We classify in this paper different augmented Lagrangian functions into three unified classes. Based on two unified formulations, we construct, respectively, two convergent augmented Lagrangian methods that do not require the global solvability of the Lagrangian relaxation and whose global convergence properties do not require the boundedness of the multiplier sequence and any constraint qualification. In particular, when the sequence of iteration points does not converge, we give a sufficient and necessary condition for the convergence of the objective value of the iteration points. We further derive two multiplier algorithms which require the same convergence condition and possess the same properties as the proposed convergent augmented Lagrangian methods. The existence of a global saddle point is crucial to guarantee the success of a dual search. We generalize in the second half of this paper the existence theorems for a global saddle point in the literature under the framework of the unified classes of augmented Lagrangian functions.     

Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems

Using an augmented Lagrangian approach, we study the existence of augmented Lagrange multipliers of a semi-infinite programming problem and discuss their characterizations in terms of saddle points. In the case of a sharp Lagrangian, we obtain a first-order necessary condition for the existence of an augmented Lagrange multiplier for the semi-infinite programming problem and some first-order sufficient conditions by assuming inf-compactness of the data functions and the extended Mangasarian–Fromovitz constraint qualification. Using a valley at 0 augmenting function and assuming suitable second-order sufficient conditions, we obtain the existence of an augmented Lagrange multiplier for the semi-infinite programming problem.     

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.     

Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.     

An Exact Augmented Lagrangian Function for Nonlinear Programming with Two-Sided Constraints

This paper is aimed toward the definition of a new exact augmented Lagrangian function for two-sided inequality constrained problems. The distinguishing feature of this augmented Lagrangian function is that it employs only one multiplier for each two-sided constraint. We prove that stationary points, local minimizers and global minimizers of the exact augmented Lagrangian function correspond exactly to KKT pairs, local solutions and global solutions of the constrained problem.     

Existence of augmented Lagrange multipliers for cone constrained optimization problems

In this paper, by using an augmented Lagrangian approach, we obtain several sufficient conditions for the existence of augmented Lagrange multipliers of a cone constrained optimization problem in Banach spaces, where the corresponding augmenting function is assumed to have a valley at zero. Furthermore, we deal with the relationship of saddle points, augmented Lagrange multipliers, and zero duality gap property between the cone constrained optimization problem and its augmented Lagrangian dual problem.     

Subgradient Methods for Saddle-Point Problems

We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.     

Convergence Rate of the Augmented Lagrangian SQP Method

In this paper, the augmented Lagrangian SQP method is considered for the numerical solution of optimization problems with equality constraints. The problem is formulated in a Hilbert space setting. Since the augmented Lagrangian SQP method is a type of Newton method for the nonlinear system of necessary optimality conditions, it is conceivable that q-quadratic convergence can be shown to hold locally in the pair (x, ). Our interest lies in the convergence of the variable x alone. We improve convergence estimates for the Newton multiplier update which does not satisfy the same convergence properties in x as for example the least-square multiplier update. We discuss these updates in the context of parameter identification problems. Furthermore, we extend the convergence results to inexact augmented Lagrangian methods. Numerical results for a control problem are also presented.     

A Modified Barrier-Augmented Lagrangian Method for Constrained Minimization

We present and analyze an interior-exterior augmented Lagrangian method for solving constrained optimization problems with both inequality and equality constraints. This method, the modified barrier—augmented Lagrangian (MBAL) method, is a combination of the modified barrier and the augmented Lagrangian methods. It is based on the MBAL function, which treats inequality constraints with a modified barrier term and equalities with an augmented Lagrangian term. The MBAL method alternatively minimizes the MBAL function in the primal space and updates the Lagrange multipliers. For a large enough fixed barrier-penalty parameter the MBAL method is shown to converge Q-linearly under the standard second-order optimality conditions. Q-superlinear convergence can be achieved by increasing the barrier-penalty parameter after each Lagrange multiplier update. We consider a dual problem that is based on the MBAL function. We prove a basic duality theorem for it and show that it has several important properties that fail to hold for the dual based on the classical Lagrangian.