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SUMMARYA new and powerful mathematical programming method is described, which is capable of solving a broad class of structural optimization problems. The method employs mixed directlreciprocal design variables in order to get conservative, first-order approximations to the objective function and to the constraints. By this approach the primary optimization problem is replaced with a sequence of explicit subproblems. Each subproblem being convex and separable, it can be efficiently solved by using a dual formulation. An attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs. Examples of application to real-life aerospace structures are offered to demonstrate the power and generality of the approach presented. BACKGROUNDIt is now widely recognized that many optimal sizing problems can be accurately approximated by a mathematical programming problem having a simple algebraic structure: linear objective function and separable constraints.'T2 This explicit subproblem is generated by linearizing the behaviour constraints with respect to the reciprocals of the design variables. On the other hand, it is often useful, for fabricational reasons, to link the design variables through linear inequality constraints. Therefore at each stage of the iterative optimization process, the approximate subproblem to be dealt with exhibits the following explicit form:In these expressions the x:s denote the design variables, which correspond to the transverse sizes of the structural members (bar cross-sectional areas, membrane thicknesses). The structural weight (I) is a linear objective function, because the weight coefficients wi are prescribed parameters related to the material mass density and to geometrical quantities (bar lengths, membrane areas).

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The development of methods to take into account uncertainties in structural analysis and in design optimization is attracting both the scienti c and the industrial communities. In this domain possibilistic methods in which uncertainties are de ned by fuzzy numbers appear as an alternative to the classical probabilistic methods such as the Monte Carlo simulations or the stochastic nite element method. The principal dif culty of possibilistic methods is that they lead to the resolution of systems of equations in which the coef cients are de ned by intervals. Several approaches for the direct solution of such interval linear equations systems are presented and compared. The vertex method is taken as reference. It is shown that the problems that are solved are mathematically different according to the method used. For static linear analysis a cost-effective iterative solution of the vertex is proposed. It is based on Neumann series expansions and solves an optimization subproblem to compute extrema of the structural responses. The effectiveness of the method is illustrated by the solution of truss problems, classical in the optimization literature. Extension of the method to inverse problems of design is also considered.