# Brief note on equality constraints in pure dual SAO settings

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## Abstract

We reflect on using equality constraints in a pure dual setting, to augment the predominant use of inequality constraints only in pure dual statements in structural optimization. Notwithstanding the fact that equalities have previously not been used in pure dual statements, their inclusion is sound from a theoretical point of view. This was already shown by Falk as early as 1965, but not really exploited in the structural optimization community. Hence, we herein mainly elaborate on some possible interesting numerical results. To overcome the difficulties associated with dual variables going to infinity if a subproblem is infeasible, we rely on the so-called bounded dual approach to achieve convergence. While the theme of this brief note is a pure dual method which can accommodate equality constraints, we depart with a sequential approximate optimization (SAO) algorithm, which provides for both equality and inequality constraints. The SAO algorithm in essence solves a sequence of approximate quadratic programs (QPs), but the approximate Hessian is diagonal. This means that the separable QP subproblems can then easily be solved using a pure dual method, which is attractive if the number of design variables *n* ≫ *m*, with *m* the total number of constraints. When *m* > *n*, the subproblems can efficiently be solved using a sequential quadratic programming (SQP)-like approach with an approximate diagonal Hessian matrix, but the resulting iteration paths are identical to that of the dual formulation; only the cost of solving the subproblems themselves is affected.

## Keywords

Duality Equality constraints Simultaneous analysis and design (SAND) Sequential approximate optimization (SAO)## Notes

### Acknowledgment

We would like to thank the anonymous reviewers of a first draft of our note for their helpful and insightful comments.

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