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A minimax inequality with applications to existence of equilibrium point and fixed point theorems

Xie Ding, Kok-Keong Tan (1992)

Colloquium Mathematicae

Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an...

A modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration

Zainab Hassan Ahmed, Mohamed Hbaib, Khalil K. Abbo (2024)

Applications of Mathematics

The Fletcher-Reeves (FR) method is widely recognized for its drawbacks, such as generating unfavorable directions and taking small steps, which can lead to subsequent poor directions and steps. To address this issue, we propose a modification to the FR method, and then we develop it into the three-term conjugate gradient method in this paper. The suggested methods, named ``HZF'' and ``THZF'', preserve the descent property of the FR method while mitigating the drawbacks. The algorithms incorporate...

A Recession Notion for a Class of Monotone Bivariate Functions

Moudafi, A. (2000)

Serdica Mathematical Journal

Using monotone bifunctions, we introduce a recession concept for general equilibrium problems relying on a variational convergence notion. The interesting purpose is to extend some results of P. L. Lions on variational problems. In the process we generalize some results by H. Brezis and H. Attouch relative to the convergence of the resolvents associated with maximal monotone operators.

A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets

Gilbert Crombez (2006)

Czechoslovak Mathematical Journal

The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this...

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