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1.
Some Results about Duality and Exact Penalization 总被引:1,自引:0,他引:1
In this paper, we introduce the concept of the valley at 0 augmenting function and apply it to construct a class of valley at 0 augmented Lagrangian functions. We establish the existence of a path of optimal solutions generated by valley at 0 augmented Lagrangian problems and its convergence toward the optimal set of the original problem and obtain the zero duality gap property between the primal problem and the valley at 0 augmented Lagrangian dual problem. Moreover, we establish the exact penalization representation results in the framework of valley at 0 augmented Lagrangian. 相似文献
2.
X.X. HUANG K. L. TEO X. Q. YANG 《数学学报(英文版)》2006,22(5):1283-1296
In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed. Necessary and sufficient conditions for approximate strong duality results are derived. Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given. Under certain conditions, it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program. 相似文献
3.
This paper is devoted to developing augmented Lagrangian duality theory in vector optimization. By using the concepts of the supremum and infimum of a set and conjugate duality of a set-valued map on the basic of weak efficiency, we establish the interchange rules for a set-valued map, and propose an augmented Lagrangian function for a vector optimization problem with set-valued data. Under this augmented Lagrangian, weak and strong duality results are given. Then we derive sufficient conditions for penalty representations of the primal problem. The obtained results extend the corresponding theorems existing in scalar optimization. 相似文献
4.
R. S. Burachik A. N. Iusem J. G. Melo 《Journal of Optimization Theory and Applications》2010,147(1):125-140
We consider a problem of minimizing an extended real-valued function defined in a Hausdorff topological space. We study the
dual problem induced by a general augmented Lagrangian function. Under a simple set of assumptions on this general augmented
Lagrangian function, we obtain strong duality and existence of exact penalty parameter via an abstract convexity approach.
We show that every cluster point of a sub-optimal path related to the dual problem is a primal solution. Our assumptions are
more general than those recently considered in the related literature. 相似文献
5.
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. 相似文献
6.
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.) 相似文献
7.
针对一般的非线性规划问题,利用某些Lagrange型函数给出了一类Lagrangian对偶问题的一般模型,并证明它与原问题之间存在零对偶间隙.针对具体的一类增广La- grangian对偶问题以及几类由非线性卷积函数构成的Lagrangian对偶问题,详细讨论了零对偶间隙的存在性.进一步,讨论了在最优路径存在的前提下,最优路径的收敛性质. 相似文献
8.
We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex
constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use
this separation result to provide necessary and sufficient conditions for establishing strong duality between geometric primal
and dual problems. Using the primal function of a constrained optimization problem, we apply our results both in the analysis
of duality schemes constructed using augmented Lagrangian functions, and in establishing necessary and sufficient conditions
for the convergence of penalty methods. 相似文献
9.
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
10.
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. 相似文献
11.
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
12.
A very powerful approach to duality in mathematical programming is the theory of generalised geometric programming. Here we exploit this theory to develop a duality theory for fractional programs. All previous work on duality for such programs uses Lagrangian ideas. 相似文献
13.
We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained
quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition
for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance
measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed
via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature
for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap. 相似文献
14.
In this paper, we introduce a new notion of augmenting function known as indicator augmenting function to establish a minmax
type duality relation, existence of a path of solution converging to optimal value and a zero duality gap relation for a nonconvex
primal problem and the corresponding Lagrangian dual problem. We also obtain necessary and sufficient conditions for an exact
penalty representation in the framework of indicator augmented Lagrangian. 相似文献
15.
《Journal of Mathematical Analysis and Applications》1987,121(1):39-56
A generally nonconvex optimization problem with equality constraints is studied. The problem is introduced as an “inf sup” of a generalized augmented Lagrangian function. A dual problem is defined as the “sup inf” of the same generalized augmented Lagrangian. Sufficient conditions are derived for constructing the augmented Lagrangian function such that the extremal values of the primal and dual problems are equal. Characterization of a class of augmented Lagrangian functions which satisfy the sufficient conditions for strong duality is presented. Finally, some examples of functions and primal-dual problems in the above-mentioned class are presented. 相似文献
16.
In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic
minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints
are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a
single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint
qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality
holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative
for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide
classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures
regularized Lagrangian duality. 相似文献
17.
本文考虑如下带约束广义变分不等式问题的增广Lagrangian对偶理论:寻找一点x∈Γ使满足,〈F(x),y-x〉 φ(x,y)-φ(x,x)≥0,y∈Γ,其中,Γ={y∈X|Θ(y)∈-C}.对于求解这类一般变分不等式问题的基于增广Lagrangian对偶理论分解算法,本文给出了算法的收敛性分析. 相似文献
18.
In this paper, in order to obtain some existence results about solutions of the augmented Lagrangian problem for a constrained
problem in which the objective function and constraint functions are noncoercive, we construct a new augmented Lagrangian
function by using an auxiliary function. We establish a zero duality gap result and a sufficient condition of an exact penalization
representation for the constrained problem without the coercive or level-bounded assumption on the objective function and
constraint functions. By assuming that the sequence of multipliers is bounded, we obtain the existence of a global minimum
and an asymptotically minimizing sequence for the constrained optimization problem. 相似文献
19.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2004,3(4):345-365
A solution concept of fuzzy optimization problems, which is essentially similar to the notion of Pareto optimal solution (nondominated solution) in multiobjective programming problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. We also introduce a concept of fuzzy scalar (inner) product based on the positive and negative parts of fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function for the fuzzy optimization problem are proposed via the concept of fuzzy scalar product. Under these settings, the weak and strong duality theorems for fuzzy optimization problems can be elicited. We show that there is no duality gap between the primal and dual fuzzy optimization problems under suitable assumptions for fuzzy-valued functions. 相似文献
20.
本文考虑带约束的变分不等式系统.提出一个基于增广Lagrangian对偶的分解算法,本文给出了算法的收敛性分析. 相似文献