*(English)*Zbl 1177.90377

The article of B. S. Mordukhovich, N. M. Nam and N. D. Yen is a valuable contribution to the mathematical theories of nonlinear and continuous optimization, which at the same time can become very helpful for the numerical solution of such programming problems and, hence, for the solution of various modern real-world applications. In fact, this paper is located on the important interface where global, parametric, bilevel and nonlinear optimization meet; herewith, it covers a great generality of optimization problems. What is more, it is prepared for a general real Banach space setting such that it can also encompass many problems from calculus of variations, optimal control, etc. One key approach of the authors consists in their reference to, modeling with and rigorous investigation of general marginal functions and value functions. Because of the addressed problem classes with their constraints or, in other words, implicit functions or relations implied by composition, the nonsmooth calculus used has had to be very well prepared.

The article is written with great care. Explanatory and formal parts are given in a balanced way; several main results and numerical examples are presented as well.

Indeed, the authors derive new results for computing and estimating Fréchet and limiting subgradients of marginal functions and they specify these results for parametric optimization with smooth and nonsmooth data. Then they use them to establish new calculus rules of generalized differentiation as well as efficient conditions for Lipschitz stability and optimality in nonlinear and nonsmooth programming and in mathematical programs with equilibrium constraints. They compare their results via their dual-space approach with known estimates and optimality conditions.

The paper has five sections, beginning with the Introduction, Preliminaries carefully provided then before turning to Fréchet subgradients of general marginal functions in Banach spaces, and via Fréchet subgradients of value functions in mathematical programming turning to limiting subgradients of value functions and optimality conditions in mathematical programming. There is no particular Conclusion session, but the various explanations and examples them solved serve for a well-integrated and motivated paper that guides the reader and invite him or her to reflect about future analysis and utilization, related with this very significant topic from mathematical optimization, Operations Research and towards an employing of them in various areas of scientific, economical, financial, technological and social life. Here, the reviewers themselves are thinking, e.g., of an OR and data mining study and use for so-called desirability functions in quality analysis and improvement, and of game theoretical applications.

##### MSC:

90C30 | Nonlinear programming |

49J52 | Nonsmooth analysis (other weak concepts of optimality) |

49J53 | Set-valued and variational analysis |

##### Keywords:

variational analysis and optimization; nonsmooth functions and set-valued mappings; generalized differentiation; marginal and value functions; mathematical programming##### References:

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