Papers by Youssef Elfoutayeni
The linear complementarity problem LCP is to find vector in satisfying , and , or showing... more The linear complementarity problem LCP is to find vector in satisfying , and , or showing that no such vector exists, where as a matrix and as a vector, are given data. This problem has been characterized by several authors who have treated cases associated with a . The aim of this paper is to give a general characterization of this problem. Furthermore, through this paper, we can provide the solution (if it exists) of a linear complementarity problem in a straightforward manner and according to the data. We close our paper with some numerical examples to illustrate our theoretical results.
The importance of prey switching in the dynamics of marine ecosystems has been underlined at all ... more The importance of prey switching in the dynamics of marine ecosystems has been underlined at all levels of the trophic chain: zooplankton species, fish, whales, birds and mammals. Most of the models of foraging behavior emphasize the effects of spatial distribution of populations, much more than the trophic structure of food webs. We propose an economic approach coupling mass balance relationships with the principles of optimal foraging theory and the related ideal free distribution theory. In this model, populations optimize the 'utility' of their diet (the energy gain), being constrained by balance equations between biomasses and trophic flows. This results in a Generalized Nash Equilibrium Problem. More precisely, We are interested in equilibrium of mathematical game given by the situation where all species try to optimize their strategies according to the strategies of all other species.
The Linear Complementarity Problem LCP(M,q) is to find a vector x in IRⁿ satisfying x≥0, Mx+q≥0 a... more The Linear Complementarity Problem LCP(M,q) is to find a vector x in IRⁿ satisfying x≥0, Mx+q≥0 and x^{T}(Mx+q)=0, where M as a matrix and q as a vector, are given data. In this paper we show that the linear complementarity problem is completely equivalent to finding the fixed point of the map x=max(0,(I-M)x-q); to find an approximation solution to the second problem, we propose an algorithm starting from any interval vector X⁽⁰⁾ and generating a sequence of the interval vector (X^{(k)})_{k=1,..} which converges to the exact solution of our linear complementarity problem. We close our paper with some examples which illustrate our theoretical results.
In the past ten years an increasing number of articles, books and conferences have raised the sub... more In the past ten years an increasing number of articles, books and conferences have raised the subject of the bio-economic models of fishery. Most of these models deal with the case of a one fish population. Recently, everyone has been trying to see what happens in the case of two or three competing fish populations. This paper presents a bio-economic model for several fish populations taking into consideration the fact that the prices of fish populations vary according to the quantity harvested. These fish populations compete with each other for space or food. The natural growth of each one is modeled using a logistic law. The objective of this work is multiple, it consists in defining the mathematical model; studying the existence and stability of the equilibrium point; calculating the fishing effort that maximizes the income of the fishing fleet exploiting all fish populations.
Most bio-economic models do not take into account the variational of the price of fish population... more Most bio-economic models do not take into account the variational of the price of fish population. Usually, the existing models consider that the prices of the fish populations are constants. In this work, we will take that the price of fish population depends on quantity harvested; for this we propose to define a bio-economic model that merges a model of competition and a model of prey-predator of three fish populations. More specifically, we assume that on the one hand, the evolution of the first and second fish population is described by a density dependent model taking into account the competition between fish populations which compete with each other for space or food; on the other hand, the evolution of the second and third fish population is described by a Lotka-Volterra model. The objective of this work is to maximize the income of the fishing fleet that exploits the three fish populations, but we have to respect two constraints, the first one is the sustainable management of the resources and the second one is the preservation of the biodiversity. The existence of the steady states and its stability are studied using eigenvalue analysis. The problem of determining the equilibrium point that maximizes the income is then solved by using the linear complementarity problem. Finally, some numerical simulations are discussed.
In this work we define a bio-economic equilibrium model for several fishermen who catch two fish ... more In this work we define a bio-economic equilibrium model for several fishermen who catch two fish species; these species compete with each other for space or food. The natural growth of each species is modeled using a logistic law. The objective of the work is to find the fishing effort that maximizes the profit of each fisherman constrained by the con-servation of the biodiversity. The existence of the steady states and its stability are studied using eigenvalue analysis. The problem of determining the equilibrium point that maximizes the profit of each fisherman is then solved by using the gene-ralized Nash equilibrium problem. Finally, some numerical simulations are given to illustrate the results.
Applied Mathematical Sciences, Jan 1, 2010
For a given n-vector q and a real square matrix M ∈ IR n×n , the linear complementarity problem, ... more For a given n-vector q and a real square matrix M ∈ IR n×n , the linear complementarity problem, denoted LCP (M, q), is that of finding nonnegative vector z ∈ IR n such that z T (Mz + q) = 0 and Mz + q 0. In this paper we suppose that the matrix M must be a symmetric and positive definite and the set S = {z ∈ IR n / z > 0 and Mz + q > 0}; named interior points set of the LCP (M, q) must be nonempty.
Gauss-Seidel-He method for solving a complementarity problems
The complementarity problem CP(f) is to find a vector z in IRⁿ satisfying 0≤z⊥f(z)≥0 where f is a... more The complementarity problem CP(f) is to find a vector z in IRⁿ satisfying 0≤z⊥f(z)≥0 where f is a given function. This problem can be solved by several methods but the most of these methods require a lot of arithmetic operations, and therefore, it is too difficult, time consuming, or expensive to find an approximate solution of the exact solution. In this paper we give a new method for solving this problem which converges very rapidly relative to most of the existing methods and does not require a lot of arithmetic operations to converge. For this we show that solving the CP(f) is equivalent to solving F(x)=0 where F is a function from IRⁿ into itself defined by F(x)=f(|x|-x)-|x|-x. After we build a sequence of smooth functions F^{(k)}∈C^{∞} which is uniformly convergent to the function F and we show that, an approximation of the solution of the CP(f) is obtained by solving F^{(k)}(x)=0 for a parameter k large enough. For solving the system of nonlinear equations F^{(k)}(x)=0 we use the Gauss-Seidel-He algorithm. We close our paper with some numerical examples to demonstrate the efficiency of our method: The numerical results obtained in this paper are very favorable and showed that our method works well for the problems tested.
Arxiv preprint arXiv:1005.1417, Jan 1, 2010
The linear complementarity problem LCP (M, q) is to find a vector z in IR n satisfying
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Papers by Youssef Elfoutayeni