## Abstract

We consider a class of “generalized equations,” involving point-to-set mappings, which formulate the problems of linear and nonlinear programming and of complementarity, among others. Solution sets of such generalized equations are shown to be stable under certain hypotheses; in particular a general form of the implicit function theorem is proved for such problems. An application to linear generalized equations is given at the end of the paper; this covers linear and convex quadratic programming and the positive semidefinite linear complementarity problem. The general nonlinear programming problem is treated in Part II of the paper, using the methods developed here.

Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS74-20584 A02.

## Preview

Unable to display preview. Download preview PDF.

## References

H. Brézis,

*Opérateurs maximaux monotones*(North-Holland, Amsterdam, 1973).R.W. Cottle, “Nonlinear programs with positively bounded Jacobians”,

*SIAM Journal on Applied Mathematics*14 (1966) 147–158.J.W. Daniel, “Stability of the solution of definite quadratic programs”,

*Mathematical Programming*5 (1973) 41–53.G.B. Dantzig and R.W. Cottle, “Positive (semi-) definite programming”, in: J. Abadie, ed.,

*Nonlinear programming*(North-Holland, Amsterdam, 1968) 55–73.S. Kakutani, “A generalization of Brouwer’s fixed point theorem”,

*Duke Mathematical Journal*8 (1941) 457–459.O.L. Mangasarian,

*Nonlinear programming*(McGraw-Hill, New York, 1969).J.J. Moré, “Coercivity conditions in nonlinear complementarity problems”,

*SIAM Review*16 (1974) 1–16.J.J. Moré, “Classes of functions and feasibility conditions in nonlinear complementarity problems”,

*Mathematical Programming*6 (1974) 327–338.H. Nikaido,

*Convex structures and economic theory*(Academic Press, New York and London, 1968).S.M. Robinson, “An implicit-function theorem for generalized variational inequalities”, Technical Summary Report No. 1672, Mathematics Research Center, University of Wisconsin-Madison, 1976; available from National Technical Information Service under Accession No. AD A031952.

S.M. Robinson, “A characterization of stability in linear programming”,

*Operations Research*25 (1977) 435–447.R.T. Rockafellar, “Local boundedness of nonlinear, monotone operators”,

*Michigan Mathematical Journal*16 (1969) 397–407.R.T. Rockafellar,

*Convex analysis*(Princeton University Press, Princeton, NJ, 1970).R.T. Rockafellar, “Monotone operators and the proximal point algorithm”,

*SIAM Journal of Control and Optimization*14 (1976), 877–898.H. Scarf,

*The computation of economic equilibria*(Yale University Press, New Haven and London, 1973).

## Author information

### Authors and Affiliations

## Editor information

## Rights and permissions

## Copyright information

© 1979 The Mathematical Programming Society

## About this chapter

### Cite this chapter

Robinson, S.M. (1979). Generalized equations and their solutions, Part I: Basic theory. In: Huard, P. (eds) Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120850

### Download citation

DOI: https://doi.org/10.1007/BFb0120850

Received:

Published:

Publisher Name: Springer, Berlin, Heidelberg

Print ISBN: 978-3-642-00797-2

Online ISBN: 978-3-642-00798-9

eBook Packages: Springer Book Archive