Finite termination of a dual Newton method for convex best C 1 interpolation and smoothing

Regularity and well-posedness of a dual program for convex best C 1-spline interpolation

“…On the other hand, the result [16, Lemma 3.2] states that F is regular atp if |I(p)| = 0, no need of an extra strongly nondegenerate index to ensure the regularity. That is, in the case |I(p)| = 0, Proposition 3.4 is stronger (not weaker) than [16,Lemma 3.2]. This motivate us to investigate if Proposition 3.4 can be strengthened when |I(p)| = 0.…”

“…We also note that violation of condition (15) may lead to singular Jacobian of F atp. The following example, taken from [16], illustrates such a possibility. Proposition 3.4 gives a sufficient condition for the regularity of F at a point where degeneracy is in presence.…”

“…The inequality uses the relations (16)(17)(18). Hence, u j V j (u j ) T ≥ 0 and equality holds if and only if u = 0 for all = i j −1 , .…”

“…By well-posedness, we mean that (7) has a unique solution towards which every minimizing sequence converges. In [16], we established the following result on regularity. Let…”

“…Despite this, numerical experiments [1,18,19,21] show that the Newton method terminates in a small number of steps (averaging 3-5 steps) if a sufficiently good starting point is used. In [16], by making use of the piecewise linearity property of F , we explained that the lack of twice differentiability of L does not present any difficulty in analyzing the convergence of Newton's method. We achieved this by recasting the classical Newton method as a generalized Newton method, which generates the next iterate p + from the current iterate p as follows…”