半无限规划的增广拉格朗日对偶理论

本文主要研究带有不等式约束的非凸半无限规划的对偶问题.众所周知,运用标准的拉格朗日函数构造对偶问题通常会存在对偶间隙,为了消除对偶间隙,我们构造一个增广拉格朗日函数,然后讨论对偶性.在合理的假设下,原问题与增广拉格朗日对偶问题之间的强对偶性成立.最后,通过一个算例对结果进行了验证.     

一个使用增广拉格朗日函数的改进的SQP算法

通过引进一个辅助参数改变二次子规划的约束形式,使子问题总存在最优解.使用一个可微增广拉格朗日函数作为效益函数来确定步长,避免了Maratos效应.     

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．     

指数型增广拉格朗日函数在广义半无限规划中的应用

本文利用指数型增广拉格朗日函数将一类广义半无限极大极小问题在一定条件下转化为标准的半无限极大极小问题,使它们具有相同的局部与全局最优解.我们给出了两个转化条件:一个是充分与必要条件,另一个是在实际中易于验证的充分条件.通过这种转化,我们给出了广义半无限极大极小问题的一个新的一阶最优性条件.     

Banach空间中向量优化问题的对称对偶与自身对偶

Ｂａｎａｃｈ空间中向量优化问题的对称对偶与自身对偶董加礼，陈东彦，王连成（吉林工业大学应用数学系，长春，１３００２５）１．引言对称对偶性与自身对偶性是６０年代初关于二次规划研究中提出来的，并且很快被推广到一般的线性规划中，尤其对非线性凸规划问题，这两...     

求解带线性约束的凸优化的一类自适应不定线性化增广拉格朗日方法

增广拉格朗日方法是求解带线性约束的凸优化问题的有效算法.线性化增广拉格朗日方法通过线性化增广拉格朗日函数的二次罚项并加上一个临近正则项,使得子问题容易求解,其中正则项系数的恰当选取对算法的收敛性和收敛速度至关重要.较大的系数可保证算法收敛性,但容易导致小步长.较小的系数允许迭代步长增大,但容易导致算法不收敛.本文考虑求解带线性等式或不等式约束的凸优化问题.我们利用自适应技术设计了一类不定线性化增广拉格朗日方法,即利用当前迭代点的信息自适应选取合适的正则项系数,在保证收敛性的前提下尽量使得子问题步长选择范围更大,从而提高算法收敛速度.我们从理论上证明了算法的全局收敛性,并利用数值实验说明了算法的有效性.     

张量鲁棒主成分分析的增广拉格朗日交替方向方法

张量的鲁棒主成分分析是将未知的一个低秩张量与一个稀疏张量从已知的它们的和中分离出来.因为在计算机视觉与模式识别中有着广阔的应用前景,该问题在近期成为学者们的研究热点.本文提出了一种针对张量鲁棒主成分分析的新的模型,并给出交替方向极小化的求解算法,在求解过程中给出了两种秩的调整策略.针对低秩分量本文对其全部各阶展开矩阵进行低秩矩阵分解,针对稀疏分量采用软阈值收缩的策略.无论目标低秩张量为精确低秩或近似低秩,本文所提方法均可适用.本文对算法给出了一定程度上的收敛性分析,即算法迭代过程中产生的任意收敛点均满足KKT条件.如果目标低秩张量为精确低秩,当迭代终止时可对输出结果进行基于高阶奇异值分解的修正.针对人工数据和真实视频数据的数值实验表明,与同类型算法相比,本文所提方法可以得到更好的结果.     

二次半定规划的增广拉格朗日算法

基于变换X=VV~T,本文将半定规划问题转换为非线性规划问题,提出了解决此问题的增广拉格朗日算法,并证明了算法的线性收敛性.在此算法中,每一次迭代计算的子问题利用最速下降搜索方向和满足wolf条件的线性搜索法求最优解.数值实验表明,此算法是行之有效的,且优于内点算法.     

对偶算法的进一步研究

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.     

求解稀疏逻辑回归问题的嵌套BB算法的分裂增广拉格朗日算法

逻辑回归是经典的分类方法,广泛应用于数据挖掘、机器学习和计算机视觉.现研究带有程。模约束的逻辑回归问题.这类问题广泛用于分类问题中的特征提取,且一般是NP-难的.为了求解这类问题,提出了嵌套BB(Barzilai and Borwein)算法的分裂增广拉格朗日算法(SALM-BB).该算法在迭代中交替地求解一个无约束凸优化问题和一个带程。模约束的二次优化问题.然后借助BB算法求解无约束凸优化问题.通过简单的等价变形直接得到带程。模约束二次优化问题的精确解,并且给出了算法的收敛性定理.最后通过数值实验来测试SALM-BB算法对稀疏逻辑回归问题的计算精确性.数据来源包括真实的UCI数据和模拟数据.数值实验表明,相对于一阶算法SLEP,SALM-BB能够得到更低的平均逻辑损失和错分率.     

Duality and Exact Penalization for Vector Optimization via Augmented Lagrangian

In this paper, we introduce an augmented Lagrangian function for a multiobjective optimization problem with an extended vector-valued function. On the basis of this augmented Lagrangian, set-valued dual maps and dual optimization problems are constructed. Weak and strong duality results are obtained. Necessary and sufficient conditions for uniformly exact penalization and exact penalization are established. Finally, comparisons of saddle-point properties are made between a class of augmented Lagrangian functions and nonlinear Lagrangian functions for a constrained multiobjective optimization problem.     

锥约束非光滑多目标优化问题的对偶及最优性条件

研究了一类涉广义不变凸锥约束非光滑多目标优化问题(记为(MOP)),结合Craven与Yang广义选择定理,建立了该优化问题的Kuhn-Tucker型最优性充分必要条件以及其鞍点与弱有效解之间的关系,给出了(MOP)的Wolfe型与Mond-Weir型弱、强以及逆对偶理论.     

Conjugate Duality in Constrained Set-Valued Vector Optimization Problems

In this article, under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are proposed for a constrained set-valued vector optimization problem. Weak duality, strong duality, and stability criteria are investigated. The inclusion relations between the image sets of the dual problems are also discussed.     

Augmented Lagrangian Theory,Duality and Decomposition Methods for Variational Inequality Problems

In this paper, we develop the augmented Lagrangian theory and duality theory for variational inequality problems. We propose also decomposition methods based on the augmented Lagrangian for solving complex variational inequality problems with coupling constraints.     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

On Duality in Nonconvex Vector Optimization in Banach Spaces Using Augmented Lagrangians

This paper shows how the use of penalty functions in terms of projections on the constraint cones, which are orthogonal in the sense of Birkhoff, permits to establish augmented Lagrangians and to define a dual problem of a given nonconvex vector optimization problem. Then the weak duality always holds. Using the quadratic growth condition together with the inf-stability or a kind of Rockafellar's stability called stability of degree two, we derive strong duality results between the properly efficient solutions of the two problems. A strict converse duality result is proved under an additional convexity assumption, which is shown to be essential.     

Duality in Vector Optimization in Banach Spaces with Generalized Convexity

We consider a vector optimization problem with functions defined on Banach spaces. A few sufficient optimality conditions are given and some results on duality are proved.     

Generalized Symmetric Vector Quasiequilibrium Problems

We establish an existence result for scalar quasiequilibrium problems without any continuity requirement on noncompact subsets of locally convex topological vector spaces. As a consequence, we obtain a solution of symmetric scalar quasiequilibrium problem. Moreover, using a so-called nonlinear scalarization function, existence theorems for vector quasiequilibrium problems and general symmetric vector quasiequilibrium problems are obtained. The authors express their sincere gratitude to the referee for valuable comments and suggestions. This research was in part supported by a grant from IPM (No. 86470016) and the Center of Excellence for Mathematics, University of Isfahan.     

Generalized Invexity and Duality in Multiobjective Programming Problems

In this paper, we are concerned with the multiobjective programming problem with inequality constraints. We introduce new classes of generalized type I vector-valued functions. Duality theorems are proved for Mond–Weir and general Mond–Weir type duality under the above generalized type I assumptions.     

Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential

In this paper, we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential. We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency, which used by the concepts of supremum and infimum of a set. We then establish the weak and strong duality results under this...