A convergent tangential approximation procedure for a decomposable nonlinear system

Abstract

A procedure is developed to solve a concave decomposable system. The general structure is treated in the same way as the classical Dantzig-Wolfe decomposition and Benders' partitioning procedures. Nonlinearities are permitted provided the objective function and coupling constraints are separable by subsystem. The decomposition is achieved by introducing a vector parameter and a new set of constraints.

Finally convergence theory that may be applied to investigate the convergence of cutting plane algorithms that do not require exact cuts is presented. This theory is of interest when the parameters of the cuts can not be calculated with a finite number of arithmetic operations and function evaluations. The theory indicates when one may truncate the calculations of the parameters after only a finite number of arithmetic operations and function evaluations without affecting convergence. The theory is applied to one aspect of the Tangential Approximation Procedure to indicate its possibilities.

A procedure is also developed under which inactive contraints can be dropped from cutting plane algorithms applied to problems with concave objective functions and constraints. This extends the work of Topkis, and Eaves and Zangwill who required a strict quasiconcavity assumption on the objective function.

Using Topkis' notion of a limiting cutting plane function, it is shown that any accumulation point of the sequence generated by the procedure is an optimal solution to the original problem.

The Tangential Approximation Procedure presented extends Geoffrion's study by improving the approximation to the objective function as well as to the implicitly defined feasible region of the parameter in the coordinating master program at each stage.