A novel generalization of the natural residual function and a neural network approach for the NCP

Method of Alternating Projection for the Absolute Value Equation

“…There are several examples of C-functions [2,17], as well as methods to construct these functions [3]. Popular choices include the natural residual (NR) function and the Fischer-Burmeister (FB) function given respectively by φ NR (s, t) = min(s, t) and φ FB (s, t) = s 2 + t 2 − (s + t).…”

“…, then for sufficiently large k, the sequence lie in IR 2 ++ . Hence, the only subsequential limits of {∇ψ(s k , t k )} ∞ k=1 are the limits of {(s k , 0)} ∞ k=1 and {(0, t k )} ∞ k=1 , which are (s, 0) and (0, t), respectively.…”

“…Let h(w) :=1 2 w − P C 1 (w)2 . Then h is a Lipschitz continuous function with Lipschitz constant 1 and ∇h(w) = w − P C 1 (w).…”

“…2 provided that A and B are not both zero. Intuitively, one can see that if the line C 1 intersects C 2 but does not pass through the origin, then P C 1 (w) / .…”