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A regularized extra-gradient method for solving the equilibrium programming problem with an inexactly specified set. (Russian, English) Zbl 1082.90139
Zh. Vychisl. Mat. Mat. Fiz. 45, No. 4, 650-660 (2005); translation in Comput. Math. Math. Phys. 45, No. 4, 626-636 (2005).
The following equilibrium programming problem is considered: to find the point $$v_{\star}$$ from the conditions $v_{\star} \in W, \Phi (v_{\star}, v_{\star}) \leq \Phi (v_{\star}, w)\quad \forall w \in W, \quad \tag{1}$ where $W= \{ w \in W_0 \mid g_i (w) \leq 0, i=1, 2, \dots,m, \,g_i(w)=0,\;i=m+1,\dots, s\},\tag{2}$ the function $$W_0$$ is a given convex closed set of the Euclidean space $$\mathbb E^n$$, the functions $$\Phi(v,w)$$, $$g_i(w)$$, $$i=1,\dots,s$$ are determined on the set $$W_0$$. The point $$v_{\star}$$ with properties (2) will be called the equilibrium point of the function $$\Phi(v,w)$$ on the set $$W$$. It is known that the problem (1), (2) is unstable in relation to perturbations of the initial data $$\Phi(v,w), g_i(w), i=1,2,\dots,s$$ and for its solution it is necessary to use regularizing methods. Here, a regularized extra-gradient method of solution for the problem (1), (2) under conditions when the set $$W$$ is also know approximately is proposed. The convergence of the method is analyzed and the regularizing operator is constructed.
##### MSC:
 90C52 Methods of reduced gradient type 90C25 Convex programming 91B50 General equilibrium theory
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