Smooth representation of a parametric

polyhedral convex set with application

to sensitivity in optimization

Author:
Dinh The Luc

Journal:
Proc. Amer. Math. Soc. **125** (1997), 555-567

MSC (1991):
Primary 52A20, 90C31

DOI:
https://doi.org/10.1090/S0002-9939-97-03507-7

MathSciNet review:
1343711

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show in this paper that if a polyhedral convex set is defined by a parametric linear system with smooth entries, then it possesses local smooth representation almost everywhere. This result is then applied to study the differentiability of the solutions and the marginal functions of several classes of parametric optimization problems.

**1.**J.-P. Aubin and H. Frankowska,*Set-valued Analysis*, Birkhäuser, Boston, MA, 1990. MR**91d:49001****2.**B. Bank, J. Guddat, D. Klatte, B. Kummer, K. Tammer,*Nonlinear Parametric Optimization*, Akademie-Verlag, Berlin, 1982, and Birkhäuser, Boston, MA, 1983. MR**84i:90147****3.**C. Berge,*Topological Spaces*, Macmillan, New York, 1963. MR**21:4401**(French original).**4.**G. B. Dantzig, J. Folkman and N. Shapiro,*On the continuity of the minimum sets of a continuous function*, J. Math. Anal. Appl.**17**(1967), 519-548. MR**34:7241****5.**A. V. Fiacco,*Introduction to Sensitivity and Stability Analysis in Nonlinear Programming*, Academic Press, New York, 1983. MR**85b:90072****6.**R. Horst and H. Tuy,*Global Optimization*, Springer-Verlag, Berlin, 1990. MR**92d:90002****7.**A. B. Levy and R. T. Rockafellar,*Sensitivity analysis of solutions to generalized equations*, Trans. Amer. Math. Soc.**345**(1994), 661-671. MR**95a:90164****8.**D. T. Luc,*Random version of the theorems of the alternative*, Math. Nach.**129**(1986), 149-155. MR**88d:46087****9.**D. T. Luc and P. H. Dien, Differentiable selection of optimal solutions in parametric linear programming, Proc. Amer. Math. Soc. (in press).**10.**J.-P. Penot,*Preservation of persistence and stability under intersections and operations*, Parts I, II, J. Optim. Theory Appl.**79**(1993), 525-550, 551-561.**11.**J.-P. Penot,*Compact nets, filters and relations*, J. Math. Anal. Appl.**93**(1983), 400-417. MR**84h:49032****12.**S. M. Robinson,*Stability theory for systems of inequalities*. Part I: Linear systems, SIAM J. Numer. Anal.**12**(1975), 754-769. MR**53:14270****13.**S. M. Robinson,*An implicit-function theorem for a class of nonsmooth functions*, Math. Oper. Res.**16**(1991), 292-304. MR**92g:58013****14.**R. T. Rockafellar, Convex Analysis, Princeton Univ Press, Princeton, NJ, 1970. MR**43:445**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
52A20,
90C31

Retrieve articles in all journals with MSC (1991): 52A20, 90C31

Additional Information

**Dinh The Luc**

Affiliation:
Université d’Avignon, 33 rue Louis Pasteur, Avignon, France

Address at time of publication:
Institute of Mathematics, P. O. Box 631, Hanoi, Vietnam

DOI:
https://doi.org/10.1090/S0002-9939-97-03507-7

Keywords:
Polyhedral convex set,
representing point and direction,
sensitivity analysis,
parametric linear problem,
parametric concave problem,
parametric polyhedral problem

Received by editor(s):
January 25, 1995

Received by editor(s) in revised form:
May 17, 1995

Additional Notes:
The author is on leave from the Institute of Mathematics, Hanoi, Vietnam

Communicated by:
Joseph S. B. Mitchell

Article copyright:
© Copyright 1997
American Mathematical Society