CONLIN: An efficient dual optimizer based on convex approximation concepts

Abstract

The Convex Linearization method (CONLIN) exhibits many interesting features and it is applicable to a broad class of structural optimization problems. The method employs mixed design variables (either direct or reciprocal) in order to get first order, conservative approximations to the objective function and to the constraints. The primary optimization problem is therefore replaced with a sequence of explicit approximate problems having a simple algebraic structure. The explicit subproblems are convex and separable, and they can be solved efficiently by using a dual method approach.

In this paper, a special purpose dual optimizer is proposed to solve the explicit subproblem generated by the CONLIN strategy. The maximum of the dual function is sought in a sequence of dual subspaces of variable dimensionality. The primary dual problem is itself replaced with a sequence of approximate quadratic subproblems with non-negativity constraints on the dual variables. Because each quadratic subproblem is restricted to the current subspace of non zero dual variables, its dimensionality is usually reasonably small. Clearly, the Hessian matrix does not need to be inverted (it can in fact be singular), and no line search process is necessary.

An important advantage of the proposed maximization method lies in the fact that most of the computational effort in the iterative process is performed with reduced sets of primal variables and dual variables. Furthermore, an appropriate active set strategy has been devised, that yields a highly reliable dual optimizer.