A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

Souza, S. S.; Oliveira, Paulo Roberto; Neto, João Xavier da Cruz; Soubeyran, Antoine

doi:10.1016/j.ejor.2009.03.045

Cited by 14 publications

(16 citation statements)

References 20 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Brito

Neto

Lopes

et al. 2012

J Optim Theory Appl

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In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Brito

Neto

Lopes

et al. 2012

J Optim Theory Appl

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“…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].…”

Section: Introductionmentioning

confidence: 48%

“…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.…”

Section: Remark 41mentioning

confidence: 87%

“…On the other hand, several applications in diverse Science and Engineering areas are sufficient motivation to work with nonconvex objective functions and proximal point methods, see for example [3,4,32]. In particular, the class of quasiconvex minimization problems has been receiving special attention from many researchers due to the broad range of applications, for example, in location theory [17], control theory [7] and specially in economic theory [33].…”

Section: Introductionmentioning

confidence: 99%

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

Quiroz

Oliveira

2011

ESAIM: COCV

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Abstract. In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on Hadamard manifolds. To reach this goal, we initially extend the concepts of regular and generalized subgradient from Euclidean spaces to Hadamard manifolds and prove that, in the convex case, these concepts coincide with the classical one. For the minimization problem, assuming that the function is bounded from below, in the quasiconvex and lower semicontinuous case, we prove the convergence of the iterations given by the method. Furthermore, under the assumptions that the sequence of proximal parameters is bounded and the function is continuous, we obtain the convergence to a generalized critical point. In particular, our work extends the applications of the proximal point methods for solving constrained minimization problems with nonconvex objective functions in Euclidean spaces when the objective function is convex or quasiconvex on the manifold.Mathematics Subject Classification. 90C26.

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A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem

et al. 2018

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The purpose of this paper is twofold. First, we examine convergence properties of an inexact proximal point method with a quasi distance as a regularization term in order to find a critical point (in the sense of Toland) of a DC function (difference of two convex functions). Global convergence of the sequence and some convergence rates are obtained with additional assumptions. Second, as an application and its inspiration, we study in a dynamic setting, the very important and difficult problem of the limit of the firm and the time it takes to reach it (maturation time), when increasing returns matter in the short run. Both the formalization of the critical size of the firm in term of a recent variational rationality approach of human dynamics and the speed of convergence results are new in Behavioral Sciences.

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A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

Cited by 14 publications

References 20 publications

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

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

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

A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem

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