Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs
Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs
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DOI：
10.1007/s10107-009-0323-4
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发表时间：
2010
影响因子：
2.7
通讯作者：
N. Dinh;B. Mordukhovich;T. Nghia
中科院分区：
文献类型：
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作者：
N. Dinh;B. Mordukhovich;T. Nghia
本文研究了一类新的所谓有限规划的参数优化问题，这类规划一般定义在决策变量的无限维空间上，其中包含无限多个不等式约束。在决策变量为有限维空间的情况下，这些问题归结为半无限规划问题。我们主要研究以凸函数之差为目标的DC无限规划问题。基于变分分析和广义微分的先进工具，主要结果建立了DC无限程序中(本质非光滑)值函数的某些次微分的有效上估计。值/边际函数及其次微分估计在参数优化的许多方面都起着至关重要的作用，包括适定性和敏感性。本文利用所得到的次微分估计，建立了值函数局部Lipschitz连续的可验证条件，并在参数DC无限规划及其显着规范中得到了必要条件。最后，我们利用值函数方法和所建立的次微分估计，在递阶优化的下层和上层研究了具有凸数据的双层有限规划和无限规划。所得结果不仅对所考虑的无穷规划类是新的，而且对它们的半无限规划类也是新的。
The paper concerns the study of new classes of parametric optimization problems of the so-calledinfinite programmingthat are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints,infinitely manyinequality constraints. These problems reduce tosemi-infinite programsin the case of finite-dimensional spaces of decision variables. We focus onDCinfinite programs with objectives given as thedifference of convexfunctions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth)value functionsin DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization includingwell-posednessandsensitivity. In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the localLipschitz continuityof the value functions and derivingnecessary optimality conditionsin parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study ofbilevelfinite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.