Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators

Kaplan, A.; Tichatschke, R.

doi:10.1007/s11228-004-4379-2

Cited by 8 publications

(8 citation statements)

References 21 publications

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“…This condition is rather restrictive, in particular, a maximal monotone operator associated with a Lagrangian of a smooth convex programming problem is paramonotone only if all constraints are not active ( [15,28]). …”

Section: Bregman-function-based Methodsmentioning

confidence: 99%

“…Proceeding from the general framework (2) and the convergence results in [22,25,26,27,28], we revise here some ideas originally developed for the improvement of certain proximal-like methods and applications to some classes of problems. On this way, these ideas can be extended to a wide class of proximal methods as well as to the APP.…”

Section: Remarkmentioning

confidence: 99%

“…For the GPMs, considered in [24,25,28], the assumption on strong convexity of T] can be replaced by the weaker assumption B8 from Section 7.1 below (function D in B8 has to be taken with rj in place of h).…”

Section: Uc = \{Lo-mentioning

confidence: 99%

“…In [28] we study the convergence of G P M for two classes of variational inequalities. The first one arises from problem m i n { / ( x ) : Pi(x) < 0, z = l , .…”

Section: \-^ Ci{z)\\v -Z\\ -C{z)mentioning

confidence: 99%

“…The convergence analysis uses essentially the following modification of Proposition 4 in lusem/Kallio [16] (which was proved for a VI with singlevalued, monotone and continuous operator): The second class of Vis considered in [28] is the complementarity problem hnd u* eW^, qeQ{u*):…”

Section: \-^ Ci{z)\\v -Z\\ -C{z)mentioning

confidence: 99%

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Some Results About Proximal-Like Methods

Kaplan

Tichatschke

Lecture Notes in Economics and Mathematical Systems

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Summary. We discuss some ideas for improvement, extension and appUcation of proximal point methods and the auxihary problem principle to variational inequalities in Hilbert spaces. These methods are closely related and will be joined in a general framework, which admits a consecutive approximation of the problem data including applications of finite element techniques and the ^-enlargement of monotone operators. With the use of a "reserve of monotonicity" of the operator in the variational inequality, the concepts of weak-and elliptic proximal regularization are developed. Considering Bregman-function-based proximal methods, we analyze their convergence under a relaxed error tolerance criterion in the subproblems. Moreover, the case of variational inequalities with non-paramonotone operators is investigated, and an extension of the auxiliary problem principle with the use of Bregman functions is studied. To emphasize the basic ideas, we renounce all the proofs and sometimes precise descriptions of the convergence results and approximation techniques. Those can be found in the referred papers.

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Section: Bregman-function-based Methodsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Uc = \{Lo-mentioning

confidence: 99%

“…In [28] we study the convergence of G P M for two classes of variational inequalities. The first one arises from problem m i n { / ( x ) : Pi(x) < 0, z = l , .…”

Section: \-^ Ci{z)\\v -Z\\ -C{z)mentioning

confidence: 99%

Section: \-^ Ci{z)\\v -Z\\ -C{z)mentioning

confidence: 99%

See 3 more Smart Citations