Residual and quasisolution methods with an extended set for searching for Nash equilibrium points in a bilinear two-person game with nonzero sum (Q1878774)

The paper considers the following two-person game. The cost of the first player is \[ \Phi_1(x_1,x_2)=\langle x_1,C_1x_2 +c_1\rangle +(1/2) \langle B_1x_1,x_1\rangle, \] and the set of his strategies is \[ x_1\in X_1=\{x_1\in \mathbb R^{n_1}\,:\, x_1\geq 0,\, A_1x_1\leq b_1\}; \] the cost of the second player is \[ \Phi_2(x_1,x_2)=\langle C_2x_1 +c_2,x_2\rangle +(1/2) \langle B_2x_2,x_2\rangle, \] and the set of his strategies is \[ x_2\in X_2=\{x_2\in \mathbb R^{n_2}\,:\, x_2\geq 0,\, A_2x_2\leq b_2\}. \] The first player tries to minimize his cost, and the second player also tries to minimize his cost. The problem is to find Nash equilibrium points in the game. The authors suggest two methods mentioned in the title for finding these points and prove their convergence.