A proximal point algorithm with a ϕ-divergence for quasiconvex programming

Abstract

We use the proximal point method with the ϕ-divergence given by ϕ(t) = t−log t− 1 for the minimization of quasiconvex functions subject to nonnegativity constraints. We establish that the sequence generated by our algorithm is well-defined in the sense that it exists and it is not cyclical. Without any assumption of boundedness level to the objective function, we obtain that the sequence converges to a stationary point. We also prove that when the regularization parameters go to zero, the sequence converges to an optimal solution.

Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints

“…However, only few can be found for the quasiconvex case, i.e., when the objective function in the minimization problem is quasiconvex. We describe next the recent studies [10][11][12][13] concerning the quasiconvex case. In [10,11,13], the proximal method for minimizing smooth quasiconvex functions was studied, where [13] works with a class of separated Bregman distances and [10,11] with a particular φ-divergence distance.…”

“…We describe next the recent studies [10][11][12][13] concerning the quasiconvex case. In [10,11,13], the proximal method for minimizing smooth quasiconvex functions was studied, where [13] works with a class of separated Bregman distances and [10,11] with a particular φ-divergence distance. In [12], the proximal method with Bregman distances was studied and convergence was proved when f is a convex or quasiconvex function on noncompact Hadamard manifolds.…”

Full convergence of the proximal point method for quasiconvex functions on Hadamard manifolds

“…So we believe that our algorithm is more practical than previous works in proximal methods with quasiconvex functions, see Cunha et al [12], Chen and Pan [11], Souza et al [32] and Papa Quiroz and Oliveira [26]. Proof.…”

“…Attouch and Teboulle [5], with a regularized Lotka-Volterra dynamical system, have proved the convergence of the continuous method to a point which belongs to a certain set which contains the set of optimal points; see also Alvarez et al [2], that treats a general class of dynamical systems that includes the one of Attouch and Teboulle [5], and includes also the case of quasiconvex objective functions in connection with continuous in time models of generalized proximal point algorithms. Cunha et al [12] and Chen and Pan [11], with a particular φ-divergence distance, have proved the full convergence of the proximal method to the KKT-point of the problem when parameter λ k is bounded and convergence to an optimal solution when λ k → 0. Pan and Chen [23], with the second-order homogeneous distance, and Souza et al [32] with a class of separated Bregman distances, have proved the same convergence result of [11,12].…”

“…Cunha et al [12] and Chen and Pan [11], with a particular φ-divergence distance, have proved the full convergence of the proximal method to the KKT-point of the problem when parameter λ k is bounded and convergence to an optimal solution when λ k → 0. Pan and Chen [23], with the second-order homogeneous distance, and Souza et al [32] with a class of separated Bregman distances, have proved the same convergence result of [11,12].…”