# The pseudomonotone polar for multivalued operators

@article{Bueno2016ThePP, title={The pseudomonotone polar for multivalued operators}, author={Orestes Bueno and John Edwin Cotrina}, journal={Optimization}, year={2016}, volume={66}, pages={691 - 703} }

In this work, we study the pseudomonotonicity of multivalued operators from the point of view of polarity, in an analogous way as the well-known monotone polar due to Martínez-Legaz and Svaiter, and the quasimonotone polar recently introduced by Bueno and Cotrina. We show that this new polar, adapted for pseudomonotonicity, possesses analogous properties to the monotone and quasimonotone polar, among which are a characterization of pseudomonotonicity, maximality and pre-maximality. Furthermore…

## 4 Citations

### On Maximality of Quasimonotone Operators

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We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martínez-Legaz and Svaiter. We first recover several properties…

### On Maximality of Quasimonotone Operators

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We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martínez-Legaz and Svaiter. We first recover several properties…

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ABSTRACT In this paper, we deal with three aspects of p-cyclically monotone operators. First, we introduce a notion of monotone polar adapted for p-cyclically monotone operators and study these kinds…

### Maximality and first-order criteria of r-monotone operators

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ABSTRACT In this article, the new notions of maximal r-monotone operator and r-polar are introduced and studied. Then, first-order characterizations for smooth and locally Lipschitz r-monotone maps…

## References

SHOWING 1-10 OF 15 REFERENCES

### On Maximality of Quasimonotone Operators

- Mathematics
- 2016

We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martínez-Legaz and Svaiter. We first recover several properties…

### Continuity and Maximality Properties of Pseudomonotone Operators

- Mathematics
- 2003

£is called pseudomonotone (in Karamardian’s sense) if for all (x;x £ ) and (y;y £ ) in its graph, hx £ ;y † xi µ 0 implies hy £ ;y † xi µ 0. We deflne an equivalence relation on the set of…

### Pseudomonotone Operators: A Survey of the Theory and Its Applications

- MathematicsJ. Optim. Theory Appl.
- 2012

The purpose of this survey paper is to present the most fundamental results in this field, starting from the earliest developments and reaching the latest results and some open questions.

### On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality

- Mathematics
- 2006

Paramonotone maps are monotone maps that satisfy a mild additional condition. They were introduced in order to ensure convergence of certain algorithms for solving the Stampacchia variational…

### Stability of Quasimonotone Variational Inequality Under Sign-Continuity

- MathematicsJ. Optim. Theory Appl.
- 2013

It is proved that regularity results for the solution maps can be obtained under some very weak regularity hypothesis on the set-valued operator, namely the lower or upper sign-continuity.

### Adjusted Sublevel Sets, Normal Operator, and Quasi-convex Programming

- MathematicsSIAM J. Optim.
- 2005

A new notion of "adjusted sublevel set" of a function is introduced and studied, which is both quasi-monotone and, in the case of quasi-convex functions, cone upper-semicontinuous.

### Complementarity problems over cones with monotone and pseudomonotone maps

- Mathematics
- 1976

The notion of a monotone map is generalized to that of a pseudomonotone map. It is shown that a differentiable, pseudoconvex function is characterized by the pseudomonotonicity of its gradient.…

### A Note on Minty Variational Inequalities and Generalized Monotonicity

- Mathematics
- 2001

Some notions of generalized monotonicity for multi-valued mappings are characterized in terms of properties of the associated Minty variational inequalities. In particular, it is shown that the Minty…

### A geometrical insight on pseudoconvexity and pseudomonotonicity

- MathematicsMath. Program.
- 2010

A representation of generalised monotone maps allows us to obtain a symmetry between maps and their inverses and maximality of generalisation monot one maps is analysed.

### Monotone Operators Representable by l.s.c. Convex Functions

- Mathematics
- 2005

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone…