初始点任意的一个非线性优化的广义梯度投影法

广义投影算法的优点是避免转轴运算。它成功地给出了线性约束问题、初始点任意的只带非线性不等式约束问题，以及利用辅助规划来处理带等式与不等式约束问题的算法．后者完满地解决了投影算法对于非线性等式约束问题的处理，但要求满足不等式约束的初始点．本文据此利用广义投影与罚函数技巧给出了一个初始点任意的等式与不等式约束问题的算法，省去了求初始解的计算，并保持了上述方法的优点，证明了算法的全局收敛性     

解带线性或非线性约束最优化问题的三项记忆梯度Rosen投影算法

利用Rosen投影矩阵，建立求解带线性或非线性不等式约束优化问题的三项记忆梯度Rosen投影下降算法，并证明了算法的收敛性．同时给出了结合FR，PR，HS共轭梯度参数的三项记忆梯度Rosen投影算法，从而将经典的共轭梯度法推广用于求解约束规划问题．数值例子表明算法是有效的。     

带非线性源项的变分不等式的区域分解法及其收敛速度分析

本文考虑一类带非线性源项的变化不等式。针对其有限元离散问题,我们构造了乘性与加性Ｓｃｈｗａｒｚ算法,其产生的上解序列或下解序列不仅单调收敛于有限元解,而且具有限元网格ｈ无关的收敛率．     

关于非线性不等式组Levenberg-Marquardt算法的收敛性(英文)

本文研究了一类非线性不等式组的求解问题.利用一列目标函数两次可微的参数优化问题来逼近非线性不等式组的解,光滑Levenberg-Marquardt方法来求解参数优化问题,在一些较弱的条件下证明了文中算法的全局收敛性,数值实例显示文中算法效果较好.     

凸可行问题的块迭代次梯度投影算法

本文,针对由非线性不等式系统构成的凸可行问题,提出了序列块迭代次梯度投影算法和平行块迭代次梯度投影算法.将非线性不等式系统分成若干个子系统,然后将当前迭代点在子系统各个子集上的次梯度投影的凸组合作为当前迭代点在这个子系统上的近似投影.在较弱条件下证明了两种算法的收敛性.     

广义集值强非线性混合似变分不等式的辅助原理和三步迭代算法

使用辅助原理技巧研究了一类广义集值强非线性混合变分不等式．证明了此类集值强非线性混合变分不等式辅助问题解的存在性和唯一性;构建了一个新的三步迭代算法,通过辅助原理技巧,构建并计算此类非线性混合变分不等式的近似解,进一步证明非线性混合变分不等式解的存在性以及由算法产生的三个序列的收敛性．所得结论推广了近年来许多混合变分不等式和准变分不等式以及他们的有关结果．     

广义集值强非线性混合似变分不等式解的迭代逼近

延拓辅助原理的技巧研究一类取非紧值的集值映象的广义强非线性混合似变分不等式．证明了这类广义强非线性混合似变分不等式的辅助问题解的存在性．利用该存在性结果,给出了解这类广义强非线性混合似变分不等式的迭代算法,最终证明了这类广义强非线性混合似变分不等式解的存在性及由算法生成的迭代序列的收敛性．     

一类优化问题的非单调信赖域算法

本文提出了一类带不等式约束和简单边界的非线性优化问题的非单调信赖域算法,在一定的条件下,证明了算法的全局收敛性,并通过数值实验验证了算法的合理性。     

带非线性不等式约束优化问题的信赖域算法

借助于KKT条件和NCP函数,提出了求解带非线性不等式约束优化问题的信赖域算法.该算法在每一步迭代时,不必求解带信赖域界的二次规划子问题,仅需求一线性方程组系统.在适当的假设条件下,它还是整体收敛的和局部超线性收敛的.数值实验结果表明该方法是有效的.     

解含非线线源项的变分不等式的加性广义Schwartz算法

１引言近年来,变分不等式区域分解算法取得了许多成果,就线性算子变分不等式而言,读者可参见［１］、［２］、［３］等参考文献中关于重叠型的Ｓｃｈｗａｒｔｚ算法的分析以及文献［４］中关于非重叠型的Ｓｃｈｗａｒｔｚ算法的讨论,就非线性算子变分不等式而言,读者可参见文献〔５〕、［６］、［７］中的有关结果．最近,在文献［８］中针对线性变分不等式问题,提出了一种加性广义Ｓｃｈｗａｒｔｚ算法,其数值算例表明,这种算法可通过调节参数从而使算法的收敛速度大大加快,较经典的加性和乘性Ｓｃｈｗａｒｔｚ算法,具有明显的数…     

The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

In this paper, we consider the least l ₂-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l ₂-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.     

非线性不等式组的光滑近似方法及其收敛性

将非线性不等式组的求解转化成非线性最小二乘问题,利用引入的光滑辅助函数,构造新的极小化问题来逐次逼近最小二乘问题.在一定的条件下,文中所提出的光滑高斯-牛顿算法的全局收敛性得到保证.适当条件下,算法的局部二阶收敛性得到了证明.文后的数值试验表明本文算法有效.     

On proto-differentiability of generalized perturbation maps

This paper is devoted to the sensitivity analysis in optimization problems and variational inequalities. The concept of proto-differentiability of set-valued maps (see [R.T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 449-482]) plays the key role in our investigation. It is proved that, under some suitable qualification conditions, the generalized perturbation maps (that is, the solution set map to a parameterized constraint system, to a parameterized variational inequality, or to a parameterized optimization problem) are proto-differentiable.     

Comments on: Farkas’ Lemma: three decades of generalizations for mathematical optimization

In these comments on the excellent survey by Dinh and Jeyakumar, we briefly discuss some recently developed topics and results on applications of extended Farkas’ lemma(s) and related qualification conditions to problems of variational analysis and optimization, which are not fully reflected in the survey. They mainly concern: Lipschitzian stability of feasible solution maps for parameterized semi-infinite and infinite programs with linear and convex inequality constraints indexed by arbitrary sets; optimality conditions for nonsmooth problems involving such constraints; evaluating various subdifferentials of optimal value functions in DC and bilevel infinite programs with applications to Lipschitz continuity of value functions and optimality conditions; calculating and estimating normal cones to feasible solution sets for nonlinear smooth as well as nonsmooth semi-infinite, infinite, and conic programs with deriving necessary optimality conditions for them; calculating coderivatives of normal cone mappings for convex polyhedra in finite and infinite dimensions with applications to robust stability of parameterized variational inequalities. We also give some historical comments on the original Farkas’ papers.     

基于动力系统的线性不等式组的解法

本文提出了一种新的求解线性不等式组可行解的方法-基于动力系统的方法．假设线性不等式组的可行域为非空,在可行域的相对内域上建立一个非线性关系表达式,进而得到一个结构简单的动力系统模型．同时,定义了穿越方向。文章最后的数值实验结果表明此算法是有效的．     

求解线性不等式组的方法

本提出了一个新的求解线性不等式组可行解的方法－－无约束极值方法。通过在线性不等式组的非空可行域的相对内域上建立一个非线性极值问题，根据对偶关系，得到了一个对偶空间的无约束极值及原始，对偶变量之间的简单线性映射关系，这样将原来线性不等式组问题的求解转化为一个无约束极值问题。中主要讨论了求解无约束极值问题的共轭梯度算法。同时，在寻找不等式组可行解的过程中，定义了穿越方向，这样大大减少计算量。中最后数值实验结果表明此算法是有效的。     

Convergence of multisplitting method for a symmetric positive definite matrix

The auxiliary principle is used to suggest and analyze some iterative methods for solving solving hemivariational inequalities under mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique. Since hemivariational inequalities include variational inequalities and nonlinear optimization problems as special cases, our results continue to hold-for these problems.     

基于参数化水平集法的材料非线性子结构拓扑优化

针对利用传统水平集法进行非线性结构拓扑优化计算过程复杂及计算效率低等问题,将参数化水平集方法引入材料非线性结构拓扑优化中。通过全局径向基函数插值初始水平集函数,建立了以插值系数为设计变量、结构的应变能最小为目标函数、材料用量为约束条件的材料非线性结构拓扑优化模型,利用有限元分析对材料非线性结构建立平衡方程,并用迭代法求解。同时,采用子结构法划分设计区域为若干个子区域,将全自由度平衡方程的求解分解为缩减的平衡方程和多个子结构内部位移的求解,减小了计算成本。算例表明,这种处理非线性关系的方法可以在保证数值稳定的同时提高计算效率,得到边界清晰、结构合理的拓扑优化构形。     

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.