Abstract
Studying Bregman distance iterative methods for solving optimization problems has become an important and very interesting topic because of the numerous applications of the Bregman distance techniques. These applications are based on the type of convex functions associated with the Bregman distance. In this paper, two different extragraident methods were proposed for studying pseudomonotone variational inequality problems using Bregman distance in real Hilbert spaces. The first algorithm uses a fixed stepsize which depends on a prior estimate of the Lipschitz constant of the cost operator. The second algorithm uses a self-adaptive stepsize which does not require prior estimate of the Lipschitz constant of the cost operator. Some convergence results were proved for approximating the solutions of pseudomonotone variational inequality problem under standard assumptions. Moreso, some numerical experiments were also given to illustrate the performance of the proposed algorithms using different convex functions such as the Shannon entropy and the Burg entropy. In addition, an application of the result to a signal processing problem is also presented.
Highlights
Let H be a real Hilbert space with norm k · k and inner product h·, ·i
We assume that the following assumptions hold
We introduce two Bregman subgradient extragradient method for solving variational inequalities with pseudomonotone and Lipschitz continuous operator in a real Hilbert space
Summary
Let H be a real Hilbert space with norm k · k and inner product h·, ·i. We study the Variational. An effort in solving this problem for the case of monotone VIP was recently introduced by Hieu and Cholamjiak [33] They introduced an extragradient method with Bregman distance which does not require a prior estimate of the Lipschitz constant. When the cost operator is not monotone, the result of [33] can not be applied to such VIP, for instance, see Example 3.12 and 4.2 of [19] Motivated by these results, in the present paper, we first introduce two Bregman extragradient methods of Popov’s type for solving pseudomonotone variational inequalities in real Hilbert spaces. The stepsize requires a prior estimate of the Lipschitz constant This extends the results of [30,32] from monotone VIP to pseudomonotone VIP in real Hilbert spaces.
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