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讨论非线性半定规划的四个专题,包括半正定矩阵锥的变分分析、非凸半定规划问题的最优性条件、非凸半定规划问题的扰动分析和非凸半定规划问题的增广Lagrange方法. 相似文献
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讨论非线性半定规划的四个专题, 包括半正定矩阵锥的变分分析、非凸半定规划问题的最优性条件、非凸半定规划问题的扰动分析和非凸半定规划问题的增广Lagrange方法. 相似文献
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非可微二层凸规划的最优性条件 总被引:3,自引:0,他引:3
本文考虑的是构成函数为非可微凸函数的二层规划问题(NDBP),得到了下层极值函数和上层复合目标函数的方向导数和次微分的估计式,给出非可微二层凸规划(NDBP)最优解的几种最优性条件。 相似文献
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金照林 《数学的实践与认识》2023,(4):43-51
提出使用凸松弛的方法求解二层规划问题,通过对一般带有二次约束的二次规划问题的半定规划松弛的探讨,研究了使用半定规划(SDP)松弛结合传统的分枝定界法求解带有凸二次下层问题的二层二次规划问题,相比常用的线性松弛方法,半定规划松弛方法可快速缩小分枝节点的上下界间隙,从而比以往的分枝定界法能够更快地获得问题的全局最优解. 相似文献
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增广Lagrange方法是求解非线性规划的一种有效方法.从一新的角度证明不等式约束非线性非光滑凸优化问题的增广Lagrange方法的收敛性.用常步长梯度法的收敛性定理证明基于增广Lagrange函数的对偶问题的常步长梯度方法的收敛性,由此得到增广Lagrange方法乘子迭代的全局收敛性. 相似文献
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在经营管理、工程设计、科学研究、军事指挥等方面普遍存在着最优化问题,而实际问题中出现的绝大多数问题都被归纳为非线性规划问题之中。作为带等式、不等式约束的复杂事例,最优化问题的求解向来较为繁琐、困难。适当条件下,非线性互补函数(NCP)可以与约束优化问题相结合,其中NCP函数的无约束极小解对应原约束问题的解及其乘子。本文提出了一类新的NCP函数用于解决等式和不等式约束非线性规划问题,结合新的NCP函数构造了增广Lagrangian函数。在适当假设条件下,证明了增广Lagrangian函数与原问题的解之间的一一对应关系。同时构造了相应算法,并证明了该算法的收敛性和有效性。 相似文献
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用次微分及法锥表达的对偶问题 总被引:3,自引:0,他引:3
考虑下述非可微凸规划问题: (P)min f(x), 约束条件:g(x)=(g_1(x),…,g_m(x))≤0,x∈C, 其中f,g_i,i=1,…,m为有限值的定义在IR~n上的凸函数,C为IR~n中的凸集,y~t为向量y(视为列向量)的转置. 如果f,g,…,g_m是可微的,Wolfe建立了一个对偶问题: 相似文献
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In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program. 相似文献
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In this paper, we consider the convergence of the generalized alternating direction method of multipliers(GADMM) for solving linearly constrained nonconvex minimization model whose objective contains coupled functions. Under the assumption that the augmented Lagrangian function satisfies the Kurdyka-Lojasiewicz inequality, we prove that the sequence generated by the GADMM converges to a critical point of the augmented Lagrangian function when the penalty parameter in the augmented Lagrangian function is sufficiently large. Moreover, we also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm. 相似文献
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This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed. 相似文献
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A new algorithm for solving nonconvex, equality-constrained optimization problems with separable structures is proposed in the present paper. A new augmented Lagrangian function is derived, and an iterative method is presented. The new proposed Lagrangian function preserves separability when the original problem is separable, and the property of linear convergence of the new algorithm is also presented. Unlike earlier algorithms for nonconvex decomposition, the convergence ratio for this method can be made arbitrarily small. Furthermore, it is feasible to extend this method to algorithms suited for inequality-constrained optimization problems. An example is included to illustrate the method.This research was supported in part by the National Science Foundation under NSF Grant No. ECS-85-06249. 相似文献
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The problem of finding the best rank-one approximation to higher-order tensors has extensive engineering and statistical applications. It is well-known that this problem is equivalent to a homogeneous polynomial optimization problem. In this paper, we study theoretical results and numerical methods of this problem, particularly focusing on the 4-th order symmetric tensor case. First, we reformulate the polynomial optimization problem to a matrix programming, and show the equivalence between these two problems. Then, we prove that there is no duality gap between the reformulation and its Lagrangian dual problem. Concerning the approaches to deal with the problem, we propose two relaxed models. The first one is a convex quadratic matrix optimization problem regularized by the nuclear norm, while the second one is a quadratic matrix programming regularized by a truncated nuclear norm, which is a D.C. function and therefore is nonconvex. To overcome the difficulty of solving this nonconvex problem, we approximate the nonconvex penalty by a convex term. We propose to use the proximal augmented Lagrangian method to solve these two relaxed models. In order to obtain a global solution, we propose an alternating least eigenvalue method after solving the relaxed models and prove its convergence. Numerical results presented in the last demonstrate, especially for nonpositive tensors, the effectiveness and efficiency of our proposed methods. 相似文献
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N.Q. Huy 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(1):163-176
In this paper the pseudo-Lipschitz property of the constraint set mapping and the Lipschitz property of the optimal value function of parametric nonconvex semi-infinite optimization problems are obtained under suitable conditions on the limiting subdifferential and the limiting normal cone. Then we derive sufficient conditions for the strong duality of nonconvex semi-infinite optimality problems and a criterion for exact penalty representations via an augmented Lagrangian approach. Examples are given to illustrate the obtained results. 相似文献
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Weak stability and strong duality of a class of nonconvex infinite programs via augmented Lagrangian
In this paper we deal with weak stability and duality of a class of nonconvex infinite programs via augmented Lagrangian. Firstly, we study a concept of weak-subdifferential of an extended real valued function on a topological linear space. Augmented Lagrangian functions and a concept of weak-stability are constructed. Next, relations between weak-stability and strong duality of problems via augmented Lagrangians are investigated. Applications for convex infinite programs are discussed. Saddle point theorems are established. An illustrative example is given. 相似文献
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In this paper, we present new convergence results of augmented Lagrangian methods for mathematical programs with complementarity
constraints (MPCC). Modified augmented Lagrangian methods based on four different algorithmic strategies are considered for
the constrained nonconvex optimization reformulation of MPCC. We show that the convergence to a global optimal solution of
the problem can be ensured without requiring the boundedness condition of the multipliers. 相似文献
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This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical since it is readily testable given only a gradient (or subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for proximal point algorithms: in particular, it uses a single relative tolerance parameter, rather than a summable parameter sequence. Our analysis first describes an abstract version of the criterion within Rockafellar’s general parametric convex duality framework, and proves a global convergence result for the resulting algorithm. Specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented Lagrangian method with a novel inexact solution condition for the subproblems. Finally, we present computational results drawn from the CUTE test set—including many nonconvex problems—indicating that the approach works well in practice. 相似文献
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Augmented Lagrangian Theory,Duality and Decomposition Methods for Variational Inequality Problems 总被引:2,自引:0,他引:2
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. 相似文献