Coercivity and generalized proximal algorithms: application-traveling around the world

We present an inexact proximal point algorithm using quasi distances to solve a minimization problem in the Euclidean space. This algorithm is motivated by the proximal methods introduced by Attouch et al., section 4, (Math Program Ser A, 137: 91-129, 2013) and Solodov and Svaiter (Set Valued Anal 7:323-345, 1999). In contrast, in this paper we consider quasi distances, arbitrary (non necessary smooth) objective functions, scalar errors in each objective regularized approximation and vectorial errors on the residual of the regularized critical point, that is, we have an error on the optimality condition of the proximal subproblem at the new point. We obtain, under a coercivity assumption of the objective function, that all accumulation points of the sequence generated by the algorithm are critical points (minimizer points in the convex case) of the minimization problem. As an application we consider a human location problem: How to travel around the world and prepare the trip of a lifetime.