Solving highly nonlinear convex separable programs using successive approximation

Abstract Owing to its rapid convergence, ease of computer implementation[1], and applicability to a wide class of practical problems[2, 3], separable programming is well established among the more useful non-linear programming techniques[2, 4]. Yet at the same time, its impracticality for highly nonlinear problems, pointed out repeatedly[1, 5, 6], constitutes a severe limitation of this important approach. This emerges even more strongly when one observes the essential failure of the method for some of the very small (2 × 2) problems included in this report. In this context of high nonlinearity, we examine the performance of a convergent (to within a given e> 0 of the optimal) alternative procedure based on Refs.[7, 8] which obviates the major difficulties effectively by solving a series of non-heuristic, rigorously determined small separable programs as opposed to a single large one in the standard separable programming technique given, e.g., in Refs.[1, 2, 5]. Specifically, this paper, first, in absence of any such study in the literature, demonstrates the extreme degree of vulnerability of standard separable programming to high nonlinearity, then states the algorithm and some of its important characteristics, and shows its effectiveness for computational examples. Problems requiring up to about 10,000 nonzero elements in their specifications and about 45,000 nonzero elements in the intermediate separable programs, resulting from up to 70 original nonlinear variables and 70 nonlinear constraints are included in these examples.