Convergence of convex-concave saddle functions: applications to convex programming and mechanics

Azé, D.; Attouch, Hédy; Wets, Roger J.-B.

doi:10.1016/s0294-1449(16)30335-3

Cited by 23 publications

(3 citation statements)

References 24 publications

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“…One could also use Theorem 1.4 in studying epi-hypo-convergence of saddlefunctions as defined by Attouch and Wets [7]; see also Azé, Attouch and Wets [10]. However, this is more complicated than the study of epi-convergence and it is not clear whether Theorem 4.2 has a direct generalization to the saddle-function case.…”

Section: Connections With Epi-convergencementioning

confidence: 99%

Graphical convergence of sums of monotone mappings

Pennanen¹,

Rockafellar

Théra

2002

Proc. Amer. Math. Soc.

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Abstract. This paper gives sufficient conditions for graphical convergence of sums of maximal monotone mappings. The main result concerns finitedimensional spaces and it generalizes known convergence results for sums. The proof is based on a duality argument and a new boundedness result for sequences of monotone mappings which is of interest on its own. An application to the epi-convergence theory of convex functions is given. Counterexamples are used to show that the results cannot be directly extended to infinite dimensions.

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Section: Connections With Epi-convergencementioning

confidence: 99%

Graphical convergence of sums of monotone mappings

Pennanen¹,

Rockafellar

Théra

2002

Proc. Amer. Math. Soc.

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“…This class is also called finite-valued bifunctions on rectangles and designated by fv-biv (X' Y). It is important because typical bifunctions in many mathematical models of practical problems are Lagrange functions 14,15 in constrained optimization, Hamilton functions in variational calculus and optimal control, and payoff functions in zero-sum games, all belong to this class. Epi/hypo-convergence of finite-valued bifunctions on rectangles was studied 16,17 .…”

Section: Introductionmentioning

confidence: 99%

Characterizations of variational convergence of bifunctions defined on products of two sets

Huynh¹

2020

Sci. Tech. Dev. J.- Eng. Tech.

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We present definitions of types of variational convergence of finite-valued bifunctions defined on rectangular domains and establish characterizations of these convergences. In the introduction, we present the origins of the research on variational convergence and then we lead to the specific problem of this paper. The content of the paper consists of 3 parts: variational convergance of fucntion; variational convergance of bifunction; and characterizations of variational convergence of bifunction, this part is the main results of this paper. In section 2, we presented the definition of epi convergence and presented a basic property problem that will be used to extend and develop the next two sections. In section 3, we start to present a new definition, the definition of convergence epi / hypo, minsup and maxinf. To clearly understand of these new definitions we have provided comments (remarks) and some examples which reader can check these definitions. The above contents serve the main result of this paper will apply in part 4. Now, we will explain more detail for this part as follows. Firstly, variational convergence of bifunctions is characterized by the epi- and hypo-convergence of related unifunctions, which are slices sup- and inf-projections. The second characterization expresses the equivalence of variational convergence of bifunctions and the same convergence of the so-called proper bifunctions defined on the whole product spaces. In the third one, the geometric reformulation, we establish explicitly the interval of all the limits by computing formulae of the left- and right-end limit bifunctions, and this is necessary and sufficient conditions of the sequence bifunctions to attain epi / hypo, minsup and maxinf convergence.

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“…For instance, since each triple π = (a, b, c) could be identified with a couple (C, c) ∈ 2 R n+1 × R n , where C is a certain closed set (e.g., either the compact set {(a t , b t ) , t ∈ T } or the closure of the characteristic cone of π defined in Section 2), it is possible to consider Π equipped with the Hausdorff topology, the bounded Hausdorff topology ( [3], [2], [31]), or any other topology on the space of closed sets ( [4]). We prefer to use the topology of the uniform convergence in the parameter space Π first, because this topology makes sense in practice (so that it has been extensively analyzed) and second, because the representation of π in 2 R n+1 × R n affects the dual problem, i.e., this approach is only suitable for the stability analysis of the primal problem (in particular, the stability of the primal feasible set has been analyzed in [27] taking as C the intersection of the closure of the characteristic cone of π with the closed unit ball, obtaining results which are not valid for the topology of the uniform convergence).…”

mentioning

confidence: 99%

Primal-dual stability in continuous linear optimization

Goberna

Todorov²

2007

Math. Program.

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Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the nonhomogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.

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Convergence of convex-concave saddle functions: applications to convex programming and mechanics

Cited by 23 publications

References 24 publications

Graphical convergence of sums of monotone mappings

Graphical convergence of sums of monotone mappings

Characterizations of variational convergence of bifunctions defined on products of two sets

Primal-dual stability in continuous linear optimization

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