Optimization of Fitzpatrick Functions and a Numerical Minimization Algorithm

We consider optimization problems for a class of convex functions on H × H introduced by Simon Fitzpatrick, where H is a real Hilbert space. We show that the minimization problem of Fitzpatrick functions can be transformed from solving of the correspondent differential inclusions (d.i) on H × H, to solving simplified d.i. on H. By using the idea of optimization of Fitzpatrick functions we introduce a numerical algorithm for solving convex smooth optimization problems by reducing the number of the independent variables. We present a comparative study with numerical examples. Finally, we show that Fitzpatrick functions are closely related to Lyapunov functions.