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11.
In this paper, by virtue of two intermediate derivative-like multifunctions, which depend on an element in the intermediate
space, some exact calculus rules are obtained for calculating the derivatives of the composition of two set-valued maps. Similar
rules are displayed for sums. Moreover, by using these calculus rules, the solution map of a parametrized variational inequality
and the variations of the feasible set of a parametrized mathematical programming problem are studied.
This research was partially supported by the National Natural Science Foundation of China (Grant numbers: 10871216 and 60574073). 相似文献
12.
Jean-Paul Penot 《Journal de Mathématiques Pures et Appliquées》2008,89(6):523-537
We introduce new methods for defining generalized sums of monotone operators and generalized compositions of monotone operators with linear maps. Under asymptotic conditions we show these operations coincide with the usual ones. When the monotone operators are subdifferentials of convex functions, a similar conclusion holds. We compare these generalized operations with previous constructions by Attouch–Baillon–Théra, Revalski–Théra and Pennanen–Revalski–Théra. The constructions we present are motivated by fuzzy calculus rules in nonsmooth analysis. We also introduce a convergence and a closure operation for operators which may be of independent interest. 相似文献
13.
For a mathematical programming problem, we consider a Lagrangian approach inspired by quasiconvex duality, but as close as possible to the usual convex Lagrangian. We focus our attention on the set of multipliers and we look for their interpretation as generalized derivatives of the performance function associated with a simple perturbation of the given problem. We do not use quasiconvex dualities, but simple direct arguments. 相似文献
14.
Jean-Paul Penot 《Optimization》2019,68(7):1411-1427
AbstractWe survey the role of generalized dualities when dealing with generalized monotone operators, observing that for many conjugacies the coupling function is neither bilinear nor finitely valued. We also make a comparison with the use of bifunctions considered in a similar perspective. We introduce a class of operators close to the class of accretive operators and we raise some open questions. 相似文献
15.
Jean-Paul Penot 《Optimization Letters》2014,8(3):1191-1200
Second-order derivatives of Chaney’s type are introduced for an arbitrary subdifferential. Their uses for necessary optimality conditions and sufficient optimality conditions are put in light when the subdifferential satisfies a weak sum rule. 相似文献
16.
Given convergent sequences of functions (f
n
) and (g
n
), we look for conditions ensuring that the sequences (f
n
+g
n
), (max(f
n
,g
n
)) and (f
n
g
n
) converge, being the infimal convolution. The convergences we use are variational convergences. This study is motivated by applications to Hamilton–Jacobi equations. 相似文献
17.
Jean-Paul Penot 《Set-Valued Analysis》1994,2(1-2):259-274
We examine some connections between convergence theories and optimization. In particular we study the Lipschitzian character of the infimal value function with respect to variations of the objective function. We also study the approximate solution multifunction for convex and nonconvex objective functions.Dedicated to the many real and putative authors of La Marseillaise, on its bicentennial. 相似文献
18.
Jean-Paul Penot 《Mathematical Programming》1994,67(1-3):225-245
New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat's rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki (1988) and Cominetti (1989); but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. We point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. Our conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral we recover the classical conditions of Ben-Tal—Zowe (1982) and Cominetti (1990). 相似文献
19.
We present characterizations of some generalized convexity properties of functions with the help of a general subdifferential. We stress the case of lower semicontinuous functions. We also study the important case of marginal functions and we provide representation results. 相似文献
20.
Jean-Paul Penot 《Set-Valued Analysis》2008,16(4):429-442
We introduce a notion of continuity for multimaps (or set-valued maps) which is mild. It encompasses both lower semicontinuity
and upper semicontinuity. We give characterizations and we consider some permanence properties. This notion can be used for
various purposes. In particular, it is used for continuity properties of subdifferentials and of value functions in parametrized
optimization problems. We also prove an approximate selection theorem.
相似文献