Error Bounds in Constrained Best Approximation

1965 ◽  
Vol 2 (3) ◽  
pp. 473-479 ◽  
Author(s):  
M. A. Hanson
Keyword(s):  
Error Bounds ◽  
Best Approximation ◽  
Constrained Best Approximation

Constrained best approximation in Hilbert space

1990 ◽  
Vol 6 (1) ◽  
pp. 35-64 ◽  
Author(s):  
Charles K. Chui ◽  
Frank Deutsch ◽  
Joseph D. Ward
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Constrained Best Approximation

scholarly journals Constrained best approximation in Hilbert space, II

1992 ◽  
Vol 71 (2) ◽  
pp. 213-238 ◽  
Author(s):  
Charles K Chui ◽  
Frank Deutsch ◽  
Joseph D Ward
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Constrained Best Approximation

scholarly journals Lagrange multiplier characterizations of constrained best approximation with nonsmooth nonconvex constraints

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4669-4684
Author(s):  
H. Mohebi
Keyword(s):  
Lagrange Multiplier ◽  
Best Approximation ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Dual Cone ◽  
The Best Approximation ◽  
Constraint Set ◽  
Nonconvex Constraint ◽  
Constrained Best Approximation

In this paper, we consider the constraint set K := {x ? Rn : gj(x)? 0,? j = 1,2,...,m} of inequalities with nonsmooth nonconvex constraint functions gj : Rn ? R (j = 1,2,...,m).We show that under Abadie?s constraint qualification the ?perturbation property? of the best approximation to any x in Rn from a convex set ?K := C ? K is characterized by the strong conical hull intersection property (strong CHIP) of C and K, where C is an arbitrary non-empty closed convex subset of Rn: By using the idea of tangential subdifferential and a non-smooth version of Abadie?s constraint qualification, we do this by first proving a dual cone characterization of the constraint set K. Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set ?K is closed and convex, we show that the Lagrange multiplier characterizations of constrained best approximation holds under a non-smooth version of Abadie?s constraint qualification. The obtained results extend many corresponding results in the context of constrained best approximation. Several examples are provided to clarify the results.

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