Primal and dual convergence of a proximal point exponential penalty method for linear programming

Dual Convergence for Penalty Algorithms in Convex Programming

“…Finally, in Sect. 4 we apply our general approach to obtain the first result of this type in a nonlinear framework for the so-called penalty-proximal algorithms, which extends significantly previous work on this subject where primal convergence has been already established; see [3,[8][9][10][11]. Indeed, for purely primal penalty-proximal algorithms, to the best of our knowledge, this is the first dual convergence result beyond the very restrictive case of Linear Programming.…”

“…The following convergence result generalizes that of [3,9], which was devoted to the very restrictive case of Linear Programming. Here we only require that the functions f i be convex and (locally) differentiable around the limit point x ∞ .…”

“…More precisely, in the setting of Linear Programming, a similar dual convergence result is proved in [9] for the particular case where θ ≡ exp(·) is the exponential penalty, and replacing (A 3 ) by the more general h k = +∞, together with the following additional hypotheses: (r k ) k is nonincreasing, ( r k−1 −r k r k−1 h k ) is bounded, ε k h k → 0, and e k ≡ 0 in (A 2 ). In [3], and still in the setting of Linear Programming, two dual convergence results are proved for a wide class of penalty barrier functions.…”

“…In view of already known primal convergence results (see, for instance, [3,[8][9][10][11]), natural but not sufficient assumptions for (A 1 ) to hold are r k 0 and ε k 0 as well as both residues η k and e k tend to 0 in R d as k → ∞, (A 2 ) where we recall that r k 0 means (r k ) k goes to 0 with positive values.…”

“…First, in the specific case where we impose η k = e k = 0, (PPPA) is exactly the method whose convergence is studied in [10] (see also [18][19][20]). More generally, for η k > 0 and e k = 0, (PPPA) was considered in [3,9]. In order to allow e k = 0, some variants of the proximal point algorithm have been proposed in the literature.…”