We present a generalized numerical embedding algorithm for solvillg nonsmooth equations based on the results in [1]. Convergence of the algorithm is proved care- fully and implementation is discussed. Application of the algorithm to the com- plementarity problem, variational inequalities and nonlinear optimization problem is discussed.
In 1994, O’leary and Yeremin extended the quasi-Newton method for minimizing a collection of functions with a common Hessian matrix to the block version,and discussed some algebraic properties of this block quasi-Newton method. In thispaper, we derive compact representations of the block BFGS’s updating matrices.These representations allow us to efficiently implement limited memory methods,e.g., the limited memory BFGS method, for minimizing a collection of functionswith a common Hessian matrix. The method relieves the requirement for the storage counts and has the savings in the operation counts, in particular, for large scaleproblems. The numerical experiments for the multiple unconstrained optimizationproblems show that the method works efficiently. Compared with O’Leary’s multiple version of BFGS method, our multiple version of the limited memory BFGSmethod is more efficient in the total operation counts and the storage counts.
