Convergence of a proximal algorithm for solving the dual of a generalized fractional program

Abstract

We propose to use the proximal point algorithm to regularize a "dual" problem of generalized fractional programs (GFP). The proposed technique leads to a new dual algorithm that generates a sequence which converges from below to the minimal value of the considered problem. At each step, the proposed algorithm solves approximately an auxiliary problem with a unique dual solution whose every cluster point gives a solution to the dual problem. In the exact minimization case, the sequence of dual solutions converges to an optimal dual solution. For a class of functions, including the linear case, the convergence of the dual values is at least linear.

Augmented Lagrangian dual for nonconvex minimax fractional programs and proximal bundle algorithms for its resolution

“…The rate of convergence of Algorithm 1 is linear under the following weak sharp minima assumption introduced in [39,1,23,22,3]. To announce this assumption, let S * denote the optimal solutions set of (D).…”

“…In the literature, there have been two types of algorithms for solving a GFP, primal Dinkelbach-type algorithms [10,17,18,19,21,40,39,51,28,13]; and dual algorithms [2,3,7,8,9,11,12,14,15,16,20,22,23,31]. See also [45,46,50,49,47,48] for more references on fractional programming.…”

Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs

“…This idea is encouraged by the recent works [3,4,16,15,2,8] dealing with duality to solve generalized fractional programming problems, and motivated by the following well known result.…”

“…For surmounting these numerical difficulties, several primal and dual algorithms were proposed (for primal algorithms, see e.g. [20,39,43,1,7], and [16,15,2] for dual algorithms) by using the proximal point algorithm (see e.g. [33,30,37,21]).…”

“…Our objective in this paper is to analyze a dual vision of the bundle method given in [43] for solving GFP, by considering the dual proximal point algorithm proposed in [16,15] which is, in turn, based on the continuous reformulation, also known to as partial dual counterpart, of GFP introduced in [3].…”