Nonlinear Least Squares for Inverse Problems by Guy Chavent

Author:Guy Chavent
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

(4.2)

and we recall the definition of stationary points:

Definition 4.0.9

A point is a stationary point of problem (4.1) if it satisfies the first-order necessary optimality condition:

(4.3)

A point is a parasitic stationary point if satisfies (4.3) but is not a solution of (4.1).

A local minimum of J on C is a stationary point.

Our objective is to find additional conditions on C, φ, and z which ensure that (4.1) is both wellposed and optimizable: A wellposed problem in the sense of Hadamard has a unique solution, which depends continuously on the right-hand side of the equation. For the optimization problem (4.1) of interest, uniqueness of the solution can hold only if the data z has a unique projection on D = φ(C), which cannot be true for anyz when D is not convex! So the existence, uniqueness, and stability properties will be required only to hold for data z that stay in some neighborhood of D = φ(C) in F.

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