Adaptive algorithms in the parametrized method of centers (Q917455)

The authors consider the problem min\(\{\) f(x): \(x\in {\mathcal D}\}\), where \(x\in H\), H is a Hilbert space, \({\mathcal D}=\{x\in H:\) g(x)\(\leq 0\}\), f and g are convex and limited, \(\overset \circ {\mathcal D}=\{x\in H:\) \(g(x)<0\}\neq \emptyset\). A method is given for solving the problem through minimizing the function \(F(\alpha,\gamma,x)=\max \{\alpha g(x),f(x)-\gamma \}\) with \(\gamma =f(x_ 0)\), \(x_ 0\in {\mathcal D}\), and a corresponding value of \(\alpha >0.\) A proof for the corresponding theorem on the convergence is given.