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In this paper, we study the convergence of a proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. We extent the method proposed by Moudafi [Proximal point algorithm...

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first-order schemes for constrained convex models under progressively relaxed...

The proximal point algorithm (PPA) has been widely used in convex optimization. Many algorithms fall into the framework of PPA. To guarantee the convergence of PPA, however, existing results conven...

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding comm...

In this paper, the local weak Pareto optimality conditions for nonsmooth vector optimization problems are presented on Hadamard manifolds. Some convergence results of the parallel proximal ...

We consider the convergence rate of the proximal point algorithm (PPA) for finding a minimizer of proper lower semicontinuous convex functions. In the Hilbert space setting, Guler showed that the big-O rate of the PPA can be improved to little-o when the sequence generated by the algorithm converges strongly to a minimizer. In this paper, we establish little-o rate of the PPA in Banach spaces without requiring this assumption. Then we apply the result to give new results on the convergence rate for sequences of alternating and averaged projections.

We propose two forward–backward proximal point type algorithms with inertial/memory effects for determining weakly efficient solutions to a vector optimization problem consisting in vector-minimizing with respect to a given closed convex pointed cone the sum of a proper cone-convex vector function with a cone-convex differentiable one, both mapping from a Hilbert space to a Banach one. Inexact versions of the algorithms, more suitable for implementation, are provided as well, while as a byproduct one can also derive a forward–backward method for solving the mentioned problem. Numerical experiments with the proposed methods are carried out in the context of solving a portfolio optimization problem.

In this paper, the problem of solving generalized fractional programs will be addressed. This problem has been extensively studied and several algorithms have been proposed. In this work, we propose an algorithm that combines the proximal point method with a continuous min–max formulation of discrete generalized fractional programs. The proposed method can handle non-differentiable convex problems with possibly unbounded feasible constraints set, and solves at each iteration a convex program with unique dual solution. It generates two sequences that approximate the optimal value of the considered problem from below and from above at each step. For a class of functions, including the linear case, the convergence rate is at least linear.

We prove the solution existence and propose an iterative algorithm for solving equilibrium problems over the common solutions of a monotone equilibrium problem and the fixed points of a strictly pseudocontractive mapping in Hilbert spaces. The algorithm is a combination of the proximal point and Halpern methods. Strong convergence is established.

The Bregman function-based Proximal Point Algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) Proximal Point Algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding – for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned abo...

We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for a minimization problem. We show the behavioural importance of this proximal point model for the formation of habit in Decision and Making Sciences. The convergence of the sequence generated by our algorithm to a critical-limit point is guaranteed under standard conditions of coercivity and by using the Kurdyka–Łojasiewicz inequality. We present a definition of habit that contemplates that kind of convergence.

Based on the classical proximal point algorithm (PPA), some PPA-based numerical algorithms for general variational inequalities (GVIs) have been developed recently. Inspired by these algorithms, in this article we propose some proximal algorithms for solving linearly constrained GVIs (LCGVIs). The resulted subproblems are regularized proximally, and they are allowed to be solved either exactly or approximately.

A general framework to the over-relaxed proximal point algorithm based on H-maximal monotonicity is introduced, and then it is applied to the approximation solvability of a general class of nonlinear inclusion problems based on the generalized resolvent operator technique. This form of the proximal point algorithm seems to be more application-oriented to inclusion problems.

We apply the proximal point method with the ϕ-divergence given by for the minimization of quasiconvex functions subject to non-negativity constraints. We prove, without the assumption of the boundedness level to the objective function, that the sequence generated by our algorithm is well-defined and it converges to a stationary point when the regularization parameter λ k satisfies , for some . If, in addition, , we then obtain the convergence to an optimal solution. We verify the effectiveness of the proximal algorithm via numerical experiments accomplished with randomly generated test problems.

International audience; An approximation method which combines Tikhonov method with the proximal point algorithm, is presented. Conditions which guarantee the convergence to a particular element of the solution set, are provided. A particular attention is given to the convex and convex-concave optimization cases.

A recent theorem Of W.Takahashi will be pointed out as an equivalent formulation of Ekelands variational principle. This gives rise to study functions having sets of weak sharp minimas in a generalized sense. Connections to the proximal point algorithm in the convex case leads to the basic and still open question: How to use Ekelands principle numerically