Optimization Letters

, Volume 13, Issue 2, pp 399–417 | Cite as

Characterizing nonconvex constrained best approximation using Robinson’s constraint qualification

  • Hossein MohebiEmail author
  • Morteza Sheikhsamani
Original Paper


Extending a powerful fundamental result of constrained best approximation, we show under a suitable condition that the “perturbation property” of the best approximation \(x_0\) to any \(x \in {{\mathbb {R}}}^n\) from a convex set \({\tilde{K}}:=C \cap K\) is characterized by the strong conical hull intersection property (CHIP) of C and K at \(x_0\). The set \(C \subseteq {{\mathbb {R}}}^n\) is closed and convex and the set K has the representation that \(K:=\{x\in {{\mathbb {R}}}^n : -g(x) \in S \}\), where the function \(g: {{\mathbb {R}}}^n \longrightarrow {{\mathbb {R}}}^m\) is continuously Fréchet differentiable that is not necessarily convex. We prove this by first establishing a dual cone characterization of the constraint set K. Our results show that the convex geometry of \({\tilde{K}}\) is critical for the characterization rather than the representation of K by convex inequalities, which is commonly assumed for the problems of best approximation from a convex set. In the special case where the set K is convex, we show that the Lagrange multiplier characterization of best approximation holds under the Robinson’s constraint qualification. The lack of representation of K by convex inequalities is supplemented by the Robinson’s constraint qualification, but the characterization, even in this special case, allows applications to problems with \(g:=(g_1, g_2, \ldots , g_m)\), where \(g_1, g_2, \ldots , g_m\) are quasi-convex functions, as it guarantees the convexity of K.


Lagrange multiplier Constrained best approximation Strong conical hull intersection property Robinson’s constraint qualification Perturbation property Constraint set 



The authors are very grateful to the anonymous referee and the associate editor for their useful suggestions regarding an earlier version of this paper. The comments of the referee and the associate editor were very useful and they helped us to improve the paper significantly. Also, many thanks to the associate editor for remembrance the paper [15]. This research has partially supported by Mahani Mathematical Research Center.


  1. 1.
    Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. 86, 135–160 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Deutsch, F.: The role of conical hull intersection propertyin convex optimization and approximation. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory IX. Vanderbilt University Press, Nashville (1998)Google Scholar
  5. 5.
    Deutsch, F.: Best Approximation in Inner Product Spaces. Springer, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Deutsch, F., Li, W., Swetits, J.: Fenchel duality and the strong conical hull intersection property. J. Optim. Theory Appl. 102, 681–695 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deutsch, F., Li, W., Ward, J.D.: A dual approach to constrained interpolation from a convex subset of Hilbert space. J. Approx. Theory 90, 385–444 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deutsch, F., Li, W., Ward, J.D.: Best approximation from the intersection of a closed convex set and a polyhedron in Hilbert space, weak Slater conditions, and the strong conical hull intersection property. SIAM J. Optim. 10, 252–268 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jeyakumar, V., Mohebi, H.: A global approach to nonlinearly constrained best approximation. Numer. Funct. Anal. Optim. 26(2), 205–227 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, C., Jin, X.: Nonlinearly constrained best approximation in Hilbert spaces: the strong CHIP, and the basic constraint qualification. SIAM J. Optim. 13(1), 228–239 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, C., Ng, K.F.: On best approximation by nonconvex sets and perturbation of nonconvex inequality systems in Hilbert spaces. SIAM J. Optim. 13, 726–744 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, C., Ng, K.F.: Constraint qualification, the strong CHIP and best approximation with convex constraints in Banach spaces. SIAM J. Optim. 14, 584–607 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
  14. 14.
    Mordukhovich, B.S., Nghia, T.T.A.: Nonsmooth cone-constrainted optimization with application to semi-infinite programming. Math. Oper. Res. 39(2), 301–324 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mordukhovich, B.S., Phan, H.M.: Tangential extremal principles for finite and infinite systems of sets II: applications to semi-infinite and multiobjective optimization. Math. Program. Ser. B 136, 31–63 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, New York (1970)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Mahani Mathematical Research CenterShahid Bahonar University of KermanKermanIran
  2. 2.Department of MathematicsKerman Graduate University of Advanced TechnologyKermanIran

Personalised recommendations