Regularized continuous extragradient method for a multicriterial equilibrium programming problem (Q2429700)

Consider the following problem: find a point \[ u_*= (w_*, \lambda^*, p^*)\in \Omega= W_0\times E^{m_2}_+\times E^{m_2}_+ \] satisfying the conditions \[ \begin{aligned} w_* &\in \text{Argmin}\{\langle\lambda^*, f(w)\rangle_{E^{m_2}}\mid g(w)\leq T_2 p^*,\,w\in W_0\},\\ \lambda^* &\in \text{Argmin}\{-\langle\lambda, f(w_*)- (1/2) T_1\lambda\rangle\mid\lambda\geq 0\},\\ p^* &\in \text{Argmin}\{-\langle p,g(w)_*)- (1/2) T_2p\rangle\mid p\geq 0\},\end{aligned} \] where \(W_0\) is a given closed convex set in the Euclidean space \(E^{m_1}\), \(f(w)= (f^1(w), f^2(w),\dots,\) \(f^{m_2}(w))^T\) and \(g(w)= (g^1(w), g^2(w),\dots, g^{m_3}(w))^T\) are continuously differentiable functions on \(W_0\) (\(T\) stands for transposition), \(\lambda= (\lambda^1,\lambda^2,\dots, \lambda^{m_2})^T\in E^{m_2}\) are variables \(p (p^1,p^2,\dots p^{m_2})^T\in E^{m_3}\) are Lagrange multipliers, \(T_1\) and \(T_2\) are square matrices of size \(m_2\) and \(m_3\), respectively, \[ \langle a,b\rangle_{E^m}= \sum^m_{i=1} a^i b^i \] is the inner product in \(E^m\), \(|a|= (\sum^m_{i=1} (a^i)^2)^{1/2}\) is the norm in \(E^m\), \(E^m_+= \{a= (a^1,\dots,a^m)^T: a^1\geq 0,\dots, a^m\geq 0\}\), and \(\text{Argmin}\{h(z)\mid t z\in Z\}\) is the set of points of minimum of a function \(h(z)\) on a set \(Z\). This problem includes as special cases the mathematical programming problem, the equilibrium (Pareto) multicriterial programming problem, and the minimization problem with an equilibrium choice of the feasible set. To solve the above problem, the authors suggest a regularized version of the continuous extragradient method, analyze its convergence, and construct a regularizing operator.