Dual Methods for Large-scale, Partially-separable Nonlinear Optimization

Abstract

In nonlinear optimization, the dual problem is generally not easier to solve than the primal problem. Convex separable optimization problems, frequently arising in electrical and mechanical engineering, constitute a notable exception to the rule. The dual problem is to optimize the dual objective function l over a non-negative orthant, and the evaluation of l reduces to the execution of independent linear searches only. To generalize the idea, we consider partially-separable problems with objective and constraint functions such that the Hessians are block-diagonal matrices with at most 2*2 blocks. The evaluation of the dual objective function is accordingly reduced to a number of independent planar searches. Obviously, 3*3 blocks would lead to spatial searches, etc. Finally, we consider the potential of the dual approach for execution on parallel computers.