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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.
90C52 Methods of reduced gradient type
90C25 Convex programming
91B50 General equilibrium theory
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