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**Continuous selections, collectively fixed points and weak Knaster-Kuratowski-Mazurkiewicz mappings in optimization.**
*(English)*
Zbl 1244.90221

The subject of the article under review is located at the interface where functional and convex analysis, geometry as well as topology meet with the applied areas of optimization theory, dynamical systems, game theory and Operational Research. These latter areas require further theoretical foundations to which this paper contributes in a rich and rigorous manner.

Concerning the generality represented in this article, just for one example, continuous selections are still hardly understood in both analysis and operational use, as they go much beyond, e.g., max- or min-type functions and their piecewise combinations. Moreover, this paper contains a set-valued modelling and investigation that goes beyond the traditional scalar-valued ones and, by this, allows to include various expressions of nonsmooth analysis, optimality criteria, etc.

The authors prove theorems on continuous selections, collectivity fixed points, collectivity coincidence points, weak Knaster-Kuratowski-Mazurkiewicz mappings and they provide their applications in various problems which are related with optimization. Each of their theorems is demonstrated based on the preceding assertions. The results contain and improve a number of those in the recent literature. They are shown to be more advantageous in optimization-related applications, too.

This paper is well written and explained, and well structured in five sections: Section 1 on Introduction, Section 2 on Continuous Selection Theorems, Section 3 on Collectively Fixed Points, Collective Coincidence Points and Applications, with the latter ones being Systems of Variational Relations, a Bilevel Optimization Problem and Nash Equilibria, Section 4 on Weak KKM Theorems and Minimax Inequalities, and Section 5 on Concluding Remarks.

In fact, future scientific work on advances of this work, in theory as well as in methods, and future real-world applications certainly seem to be possible and, actually, very useful for a wide range of sciences and their positive impacts on the world of tomorrow.

Concerning the generality represented in this article, just for one example, continuous selections are still hardly understood in both analysis and operational use, as they go much beyond, e.g., max- or min-type functions and their piecewise combinations. Moreover, this paper contains a set-valued modelling and investigation that goes beyond the traditional scalar-valued ones and, by this, allows to include various expressions of nonsmooth analysis, optimality criteria, etc.

The authors prove theorems on continuous selections, collectivity fixed points, collectivity coincidence points, weak Knaster-Kuratowski-Mazurkiewicz mappings and they provide their applications in various problems which are related with optimization. Each of their theorems is demonstrated based on the preceding assertions. The results contain and improve a number of those in the recent literature. They are shown to be more advantageous in optimization-related applications, too.

This paper is well written and explained, and well structured in five sections: Section 1 on Introduction, Section 2 on Continuous Selection Theorems, Section 3 on Collectively Fixed Points, Collective Coincidence Points and Applications, with the latter ones being Systems of Variational Relations, a Bilevel Optimization Problem and Nash Equilibria, Section 4 on Weak KKM Theorems and Minimax Inequalities, and Section 5 on Concluding Remarks.

In fact, future scientific work on advances of this work, in theory as well as in methods, and future real-world applications certainly seem to be possible and, actually, very useful for a wide range of sciences and their positive impacts on the world of tomorrow.

### MSC:

90C30 | Nonlinear programming |

### Keywords:

continuous selections; collectivity fixed points; collectivity coinicdence points; weak Knaster-Kuratowski-Mazurkiewicz theorems; variational relations; bilevel optimization; Nash equilibria; minimax inequalities
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\textit{P. Q. Khanh} et al., J. Optim. Theory Appl. 151, No. 3, 552--572 (2011; Zbl 1244.90221)

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### References:

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