A Switching Criterion in Hybrid Quasi-Newton BFGS - Steepest Descent Direction

Two modified methods for unconstrained optimization are presented. The methods employ a hybrid descent direction strategy which uses a linear convex combination of quasi-Newton BFGS and steepest descent as search direction. A switching criterion is derived based on the First and Second order Kuhn-Tucker condition. The switching criterion can be viewed as a way to change between quasi-Newton and steepest descent step by matching the Kuhn-Tucker condition. This is to ensure that no potential feasible moves away from the current descent step to the other one that reduced the value of the objective function. Numerical results are also presented, which suggest that an improvement has been achieved compared with the BFGS algorithm