Regularity and Stability for Convex Multivalued Functions

Stephen M. Robinson
Stephen M. Robinson
Mathematics Research Center, University of Wisconsin-Madison, Madison, Wisconsin 53706
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Mathematics Research Center, University of Wisconsin-Madison, Madison, Wisconsin 53706

Published Online:1 May 1976https://doi.org/10.1287/moor.1.2.130

Abstract

Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.

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- Convergence of Finite Element Approximations to State Constrained Convex Parabolic Boundary Control Problems
  SIAM Journal on Control and Optimization, Vol. 27, No. 4
- On Some Mathematical Tools for Local Controllability
  IFAC Proceedings Volumes, Vol. 22, No. 4
- Remarks on multifunctions with convex graph
  Archiv der Mathematik, Vol. 52, No. 5
- Further properties of the regularity of inclusion systems
  Nonlinear Analysis: Theory, Methods & Applications, Vol. 13, No. 11
- Metric regularity, openness and Lipschitzian behavior of multifunctions
  Nonlinear Analysis: Theory, Methods & Applications, Vol. 13, No. 6
- Stability of solutions for a class of nonlinear cone constrained optimization problems, part 1: Basic theory
  Numerical Functional Analysis and Optimization, Vol. 10, No. 11-12
- Stability of solutions for a class of nonlinear cone constrained optimization problems, part 2: Application to parameter estimation
  Numerical Functional Analysis and Optimization, Vol. 10, No. 11-12
- Convexity of the optimal multifunctions and its consequences in vector optimization
  Optimization, Vol. 20, No. 6
- Convergence of SQP-Like Methods for Constrained Optimization
  SIAM Journal on Control and Optimization, Vol. 27, No. 1
- Stable approximations of set-valued maps
  Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Vol. 5, No. 6
- Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps
  Journal of Mathematical Analysis and Applications, Vol. 134, No. 2
- An inversion theorem for set-valued maps
  17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 37, No. 3
- On constraint controllability of linear systems in Banach spaces
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- Differentiability of Set‐Valued Maps in Banach Spaces
  11 November 2006 | Mathematische Nachrichten, Vol. 139, No. 1
- Regularity, calmness and support principle
  Optimization, Vol. 19, No. 1
- Necessary Optimality Conditions for the Control of Semilinear Hyperbolic Boundary Value Problems
  SIAM Journal on Control and Optimization, Vol. 25, No. 5
- Solvability results for sublinear functions and operators
  Zeitschrift für Operations Research, Vol. 31, No. 3
- On midpoint convex set-valued functions
  Aequationes Mathematicae, Vol. 33, No. 1
- Automatic continuity and openness of convex relations
  Proceedings of the American Mathematical Society, Vol. 99, No. 1
- Extensions of Radstrom's lemma with application to stability theory of mathematical programming
  Journal of Mathematical Analysis and Applications, Vol. 117, No. 2
- A relaxed projection method for variational inequalities
  Mathematical Programming, Vol. 35, No. 1
- Stability and regular points of inequality systems
  Journal of Optimization Theory and Applications, Vol. 48, No. 1
- Norm duality for convex processes and applications
  Journal of Optimization Theory and Applications, Vol. 48, No. 1
- Nonsmooth analysis of vector-valued mappings with contributions to nondifferentiable programming
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- Openness, Closedness and Duality in Banach Spaces with Applications to Continuous Linear Programming
- Nonsmooth Analysis and Optimization for a Class of Nonconvex Mappings
- A convex sets in general position
  Linear Algebra and its Applications, Vol. 64
- Conjugate convex operators
  Journal of Mathematical Analysis and Applications, Vol. 102, No. 2
- Local convex analysis
  Journal of Soviet Mathematics, Vol. 26, No. 4
- Generalized linear complementarity problems treated without fixed-point theory
  Journal of Optimization Theory and Applications, Vol. 43, No. 3
- A Ritz Method for the Numerical Solution of a Class of State Constrained Control Approximation Problems
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- A note on the existence of Lagrange multipliers
  Applied Mathematics & Optimization, Vol. 10, No. 1
- Openness of convex measurable correspondences
  Mathematical Notes of the Academy of Sciences of the USSR, Vol. 33, No. 1
- References
- Tangent cones and quasi-interiorly tangent cones to multifunctions
  Transactions of the American Mathematical Society, Vol. 277, No. 2
- The lagrange multiplier set and the generalized gradient set of the marginal function of a differentiable program in a Banach space
  Journal of Optimization Theory and Applications, Vol. 38, No. 3
- On generalized differentials and subdifferentials of Lipschitz vector-valued functions
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  11 April 2018 | The American Mathematical Monthly, Vol. 89, No. 7
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  Journal of Optimization Theory and Applications, Vol. 37, No. 2
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  25 February 2009
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  Journal of Mathematical Analysis and Applications, Vol. 80, No. 1
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  25 February 2009
- Nonsmooth analysis: differential calculus of nondifferentiable mappings
  Transactions of the American Mathematical Society, Vol. 266, No. 1
- Subdifferentials of compactly lipschitzian vector-valued functions
  Annali di Matematica Pura ed Applicata, Vol. 125, No. 1
- Further properties of Lagrange multipliers in nonsmooth optimization
  Applied Mathematics & Optimization, Vol. 6, No. 1
- Differential stability in infinite-dimensional nonlinear programming
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  15 May 2007 | Numerical Functional Analysis and Optimization, Vol. 2, No. 6
- On intersections of multifunctions
  27 June 2007 | Mathematische Operationsforschung und Statistik. Series Optimization, Vol. 11, No. 1
- Sufficient Conditions for Kuhn–Tucker Vectors in Convex Programming
  SIAM Journal on Control and Optimization, Vol. 17, No. 6
- Dualität und Stabilität
- References
- Nonconvex minimization problems
  Bulletin of the American Mathematical Society, Vol. 1, No. 3
- A remark on a regularity assumption for the mathematical programming problem in Banach spaces
  Journal of Optimization Theory and Applications, Vol. 25, No. 3
- Differentiable Perturbations of Infinite Optimization Problems
- A MODIFIED NEWTON ALGORITHM FOR FUNCTIONS OVER CONVEX SETS
- Symmetric duality for structured convex programs
  Transactions of the American Mathematical Society, Vol. 245, No. 0
- Multivalued convexity and optimization: A unified approach to inequality and equality constraints
  Mathematical Programming, Vol. 13, No. 1

cover image Mathematics of Operations Research

Volume 1, Issue 2

May 1976

Pages 97-196

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Published Online:May 01, 1976

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Stephen M. Robinson, (1976) Regularity and Stability for Convex Multivalued Functions. Mathematics of Operations Research 1(2):130-143.

https://doi.org/10.1287/moor.1.2.130

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Regularity and Stability for Convex Multivalued Functions

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Volume 1, Issue 2

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