On the problem of optimal compatibility (Q2755712)

The authors suggest a method of finding global solutions in the class \(P\) of nonconvex optimization problems in a Hilbert space: minimizing \(p\) such that \( p \geq p_0 \) and the system of linear equations \( G(p)x = b(p)\) is compatible in a convex set \(X(p)\). The iterative algorithm for approaching the solution set of the minimization problem is based on the definition of the sequence \( (p_k, x_k), k \geq 0\) where \( x_{k+1} = x_k + \tau_{k+1} (u_{k+1} - x_k)\) and \( \tau_{k+1}, u_{k+1}\) are defined from two other separate optimal problems for the pair \((p_{k+1}, u_{k+1})\) and \(\tau_{k+1}\). The accuracy estimates for approximate solutions are presented together with a constructive regularization algorithm for finding an approximate solution of an arbitrary problem from \(P\) under a perturbed information on the associated functions \(G(.)\) and \(b(.)\).