zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Subgradients of marginal functions in parametric mathematical programming. (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:
90C30Nonlinear programming
49J52Nonsmooth analysis (other weak concepts of optimality)
49J53Set-valued and variational analysis
References:
[1]Aubin J.-P. (1984). Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9: 87–111 · Zbl 0539.90085 · doi:10.1287/moor.9.1.87
[2]Auslender A. (1979). Differential stability in nonconvex and nondifferentiable programming. Math. Progr. Study 10: 29–41
[3]Auslender A., Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York
[4]Bonnans J.F., Shapiro A. (2000). Perturbation Analysis of Optimization Problems. Springer, New York
[5]Borwein J.M., Zhu Q.J. (2005). Techniques of Variational Analysis. Springer, New York
[6]Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York
[7]Dien P.H., Yen N.D. (1991). On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints. Appl. Math. Optim. 24: 35–54 · Zbl 0742.90086 · doi:10.1007/BF01447734
[8]Gauvin J., Dubeau F. (1982). Differential properties of the marginal function in mathematical programming. Math. Progr. Study 19: 101–119
[9]Gauvin J., Dubeau F. (1984). Some examples and counterexamples for the stability analysis of nonlinear programming problems. Math. Progr. Study 21: 69–78
[10]Gollan B. (1984). On the marginal function in nonlinear programming. Math. Oper. Res. 9: 208–221 · Zbl 0553.90092 · doi:10.1287/moor.9.2.208
[11]Ha T.X.D. (2005). Lagrange multipliers for set-valued problems associated with coderivatives. J. Math. Anal. Appl. 311: 647–663 · Zbl 1134.90490 · doi:10.1016/j.jmaa.2005.03.011
[12]Ioffe A.D., Tihomirov V.M. (1979). Theory of extremal problems. North-Holland Publishing Co., Amsterdam-New York
[13]Lucet, Y., Ye, J.J.: Sensitivity analysis of the value function for optimization problems with variational inequality constraints. SIAM J. Control Optim. 40, 699–723 (2001); Erratum. SIAM J. Control Optim. 41, 1315–1319 (2002)
[14]Luo Z.Q., Pang J.-S., Ralph D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge
[15]Maurer H., Zowe J. (1979). First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Prog. 16: 98–110 · Zbl 0398.90109 · doi:10.1007/BF01582096
[16]Minchenko L.I. (2003). Multivalued analysis and differential properties of multivalued mappings and marginal functions. Optimization and related topics. J. Math. Sci. 116: 3266 · Zbl 1056.49021 · doi:10.1023/A:1023669004408
[17]Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, pp. 32–46, SIAM Publications (1992)
[18]Mordukhovich B.S. (2006). Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin
[19]Mordukhovich B.S. (2006). Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin
[20]Mordukhovich B.S., Nam N.M. (2005). Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30: 800–816 · Zbl 05279643 · doi:10.1287/moor.1050.0147
[21]Mordukhovich B.S., Nam N.M., Yen N.D. (2007). Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55: 685–708 · Zbl 1121.49017 · doi:10.1080/02331930600816395
[22]Mordukhovich B.S., Shao Y. (1996). Nonsmooth analysis in Asplund spaces. Trans. Am. Math. Soc. 348: 1230–1280 · Zbl 0881.49009 · doi:10.1090/S0002-9947-96-01543-7
[23]Outrata J.V., Koĉvara M., Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht
[24]Phelps R.R. (1993). Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, Berlin
[25]Robinson S.M. (1979). Generalized equations and their solutions, I: Basic theory. Math. Progr. Study 10: 128–141
[26]Rockafellar R.T. (1982). Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Progr. Study 17: 28–66
[27]Rockafellar R.T. (1985). Extensions of subgradient calculus with applications to optimization. Nonlinear Anal. 9: 665–698 · Zbl 0593.49013 · doi:10.1016/0362-546X(85)90012-4
[28]Rockafellar R.T., Wets R.J.-B. (1998). Variational Analysis. Springer, Berlin
[29]Thibault L. (1991). On subdifferentials of optimal value functions. SIAM J. Control Optim. 29: 1019–1036 · Zbl 0734.90093 · doi:10.1137/0329056
[30]Ye J.J. (2001). Multiplier rules under mixed assumptions of differentiability and Lipschitz continuity. SIAM J. Control Optim. 39: 1441–1460 · Zbl 0994.90138 · doi:10.1137/S0363012999358476