Saddle problem and optimization problem as an integrated system (Q735640)

Considered is a system of optimization problems with one of them being a saddle problem induced by convex programming and the other, an optimization problem on a convex set: \[ p^*,w^*\in \text{ArgSdl}\{f(w)\mid g(w)\leq y^*,\;w\in W_{0}\}, \tag{1} \] \[ y^*\in \text{ArgMax}\{\langle p^*,y\rangle \mid y\in Y\}, \tag{2} \] where \(f(w)\) is a scalar function, \(g(w)\) is a vector function, \(p\) belongs to the positive orthant \(\mathbb R_+^m\), \(W\subset\mathbb R^n\) and \(Y\subset\mathbb R_+^m\) are convex closed sets. It is assumed that \(f(w)\) and all the components of the function \(g(w)\) are convex. The author examines the properties if this system, the sphere of its application and methods for its solution. The convergence of the proposed methods is established.