We present a framework of algorithms for the solution of 
continuous optimization and variational inequality problems.  In the general 
algorithm, a search direction finding auxiliary problem is obtained by
replacing the original cost function with an approximating monotone cost 
function.  Alternately, a line search is made in the search
direction in order to obtain a decrease in a merit function.
The proposed framework encompasses algorithm classes presented earlier by 
Cohen, Dafermos, Migdalas, and Tseng, and includes numerous
descent and successive approximation type methods, such as
Newton methods, Jacobi and Gauss-Seidel type decomposition methods for 
problems defined over Cartesian product sets, and proximal point methods, 
among others. 
The auxiliary problem of the general algorithm also induces equivalent 
optimization reformulations and descent methods for asymmetric
variational inequalities. 

We study the convergence properties of the general
algorithm when applied to unconstrained optimization, nondifferentiable
optimization, constrained differentiable optimization, and variational 
inequalities; the emphasis of the convergence analyses is placed on basic
convergence results, convergence using different line search strategies and
truncated subproblem solutions, and convergence rate results.
This analysis offers a unification of known results; moreover, it provides 
strengthenings of convergence results for many existing algorithms, and 
indicates possible improvements of their realizations.