Editing Multidisciplinary design optimization (section)

=== Gradient-based methods ===
There were two schools of structural optimization practitioners using [[gradient]]-based methods during the 1960s and 1970s: optimality criteria and [[Mathematical optimization|mathematical programming]]. The optimality criteria school derived recursive formulas based on the [[Karush–Kuhn–Tucker conditions|Karush–Kuhn–Tucker (KKT) necessary conditions]] for an optimal design. The KKT conditions were applied to classes of structural problems such as minimum weight design with constraints on stresses, displacements, buckling, or frequencies [Rozvany, Berke, Venkayya, Khot, et al.] to derive resizing expressions particular to each class. The mathematical programming school employed classical gradient-based methods to structural optimization problems. The method of usable feasible directions, Rosen's gradient projection (generalized reduce gradient) method, sequential unconstrained minimization techniques, sequential linear programming and eventually sequential quadratic programming methods were common choices. Schittkowski et al. reviewed the methods current by the early 1990s.

The gradient methods unique to the MDO community derive from the combination of optimality criteria with math programming, first recognized in the seminal work of Fleury and Schmit who constructed a framework of approximation concepts for structural optimization. They recognized that optimality criteria were so successful for stress and displacement constraints, because that approach amounted to solving the dual problem for [[Lagrange multipliers]] using linear [[Taylor series]] approximations in the reciprocal design space. In combination with other techniques to improve efficiency, such as constraint deletion, regionalization, and design variable linking, they succeeded in uniting the work of both schools. This approximation concepts based approach forms the basis of the optimization modules in modern structural design software.

Approximations for structural optimization were initiated by the reciprocal approximation Schmit and Miura for stress and displacement response functions. Other intermediate variables were employed for plates. Combining linear and reciprocal variables, Starnes and Haftka developed a conservative approximation to improve buckling approximations. Fadel chose an appropriate intermediate design variable for each function based on a gradient matching condition for the previous point. Vanderplaats initiated a second generation of high quality approximations when he developed the force approximation as an intermediate response approximation to improve the approximation of stress constraints. Canfield developed a [[Rayleigh quotient]] approximation to improve the accuracy of eigenvalue approximations. Barthelemy and Haftka published a comprehensive review of approximations in 1993.