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1.
Tim Hoheisel 《Journal of Mathematical Analysis and Applications》2008,337(1):292-310
We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush-Kuhn-Tucker conditions do not provide necessary optimality criteria. In order to obtain reasonable first order conditions under very weak assumptions, we introduce several MPVC-tailored constraint qualifications, discuss their relation, and prove an optimality condition which may be viewed as the counterpart of what is called M-stationarity in the MPEC-field. 相似文献
2.
We consider a class of optimization problems with switch-off/switch-on constraints, which is a relatively new problem model.
The specificity of this model is that it contains constraints that are being imposed (switched on) at some points of the feasible
region, while being disregarded (switched off) at other points. This seems to be a potentially useful modeling paradigm, that
has been shown to be helpful, for example, in optimal topology design. The fact that some constraints “vanish” from the problem
at certain points, gave rise to the name of mathematical programs with vanishing constraints (MPVC). It turns out that such
problems are usually degenerate at a solution, but are structurally different from the related class of mathematical programs
with complementarity constraints (MPCC). In this paper, we first discuss some known first- and second-order necessary optimality
conditions for MPVC, giving new very short and direct justifications. We then derive some new special second-order sufficient
optimality conditions for these problems and show that, quite remarkably, these conditions are actually equivalent to the
classical/standard second-order sufficient conditions in optimization. We also provide a sensitivity analysis for MPVC. Finally,
a relaxation method is proposed. For this method, we analyze constraints regularity and boundedness of the Lagrange multipliers
in the relaxed subproblems, derive a sufficient condition for local uniqueness of solutions of subproblems, and give convergence
estimates.
Research of the first author was supported by the Russian Foundation for Basic Research Grants 07-01-00270, 07-01-00416 and
07-01-90102-Mong, and by RF President’s Grant NS-9344.2006.1 for the support of leading scientific schools. The second author
was supported in part by CNPq Grants 301508/2005-4, 490200/2005-2 and 550317/2005-8, by PRONEX-Optimization, and by FAPERJ. 相似文献
3.
Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.Communicated by Z. Q. Luo 相似文献
4.
Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem. 相似文献
5.
Tim Hoheisel Ji?í V. Outrata 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(5):2514-2526
A mathematical program with vanishing constraints (MPVC) is a constrained optimization problem arising in certain engineering applications. The feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated. This, in turn, implies that standard penalty functions are typically non-exact for MPVCs. We therefore develop a new MPVC-tailored penalty function which is shown to be exact under reasonable assumptions. This new penalty function can then be used to derive (or recover) suitable optimality conditions for MPVCs. 相似文献
6.
7.
Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers. 相似文献
8.
9.
In this paper, we develop the sufficient conditions for the existence of local and global saddle points of two classes of
augmented Lagrangian functions for nonconvex optimization problem with both equality and inequality constraints, which improve
the corresponding results in available papers. The main feature of our sufficient condition for the existence of global saddle
points is that we do not need the uniqueness of the optimal solution. Furthermore, we show that the existence of global saddle
points is a necessary and sufficient condition for the exact penalty representation in the framework of augmented Lagrangians.
Based on these, we convert a class of generalized semi-infinite programming problems into standard semi-infinite programming
problems via augmented Lagrangians. Some new first-order optimality conditions are also discussed.
This research was supported by the National Natural Science Foundation of P.R. China (Grant No. 10571106 and No. 10701047). 相似文献
10.
在弧连通锥-凸假设下讨论Hausdorff局部凸空间中的一类数学规划的最优性条件问题.首先,利用择一定理得到了锥约束标量优化问题的一个必要最优性条件.其次,利用凸集分离定理证明了无约束向量优化问题关于弱极小元的标量化定理和一个一致的充分必要条件.所得结果深化和丰富了最优化理论及其应用的内容. 相似文献
11.
集合函数多目标规划的一阶最优性条件 总被引:4,自引:0,他引:4
在文(1)-(4)的基础上,本文通过引入集团函数的伪凸,严格伪凸,拟凸,严格拟凸等新概念,给出了集合函数多目标规划问题有效解的一阶充分条件,弱有效解的阶必要条件,弱有交解的一阶必要条件以及强有效解的一阶充分条件。 相似文献
12.
In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. 相似文献
13.
A. F. Izmailov A. L. Pogosyan 《Computational Mathematics and Mathematical Physics》2009,49(7):1128-1140
A new class of optimization problems is discussed in which some constraints must hold in certain regions of the corresponding space rather than everywhere. In particular, the optimal design of topologies for mechanical structures can be reduced to problems of this kind. Problems in this class are difficult to analyze and solve numerically because their constraints are usually irregular. Some known first- and second-order necessary conditions for local optimality are refined for problems with vanishing constraints, and special Newton-type methods are developed for solving such problems. 相似文献
14.
In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized
equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related
problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming.
We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type
and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational
analysis and generalized differentiation.
Dedicated to Jiří V. Outrata on the occasion of his 60th birthday.
This research was partly supported by the National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the
Australian Research Council under grant DP-0451168. 相似文献
15.
J. Doležal 《Journal of Optimization Theory and Applications》1981,35(1):143-148
It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions. 相似文献
16.
17.
First and second order analysis of nonlinear semidefinite programs 总被引:14,自引:0,他引:14
Alexander Shapiro 《Mathematical Programming》1997,77(1):301-320
In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions
for such problems are presented. A second-order analysis is also given. Second-order necessary and sufficient optimality conditions
are derived. Finally, sensitivity analysis of such programs is discussed. 相似文献
18.
T. Q. Bao P. Gupta B. S. Mordukhovich 《Journal of Optimization Theory and Applications》2007,135(2):179-203
We study multiobjective optimization problems with equilibrium constraints (MOPECs) described by parametric generalized equations
in the form
where both mappings G and Q are set-valued. Such models arise particularly from certain optimization-related problems governed by variational inequalities
and first-order optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the
general problems under consideration and for their important specifications by using modern tools of variational analysis
and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical
example from bilevel programming with convex while nondifferentiable data. 相似文献
19.
Second-Order Optimality Conditions in Set Optimization 总被引:3,自引:0,他引:3
J. Jahn A. A. Khan P. Zeilinger 《Journal of Optimization Theory and Applications》2005,125(2):331-347
In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks. 相似文献