Keywords: topology optimization, domain decomposition methods, bilevel optimization problems, Karush-Kuhn-Tucker conditions, necessary optimality conditions.

Consider a typical topology optimization problem, the minimum compliance problem of a linear isotropic elastic continuum structure, in which the constraints are the partial differential equations of linear elasticity. We subdivide the partial differential equations into two subproblems posed on non-overlapping sub-domains. We have obtained an optimization problem [1]. Is such a problem a bilevel or bicriteria optimization problem?

In bicriteria optimization, two functions have to be minimized simultaneously. However, bilevel programming problems (BLPP) are hierarchical optimization problems where an objective function is to be minimized over the graph of the solution set of a second parametric optimization problem.

In fact, on the one hand, the compliance function and the energy functionnal cannot be minimized simultaneously, so, bicriteria optimization is no more suitable, and the hierarchical nature of the two levels is a natural justification for this choice. On the other hand, structural optimization problems have an inherent bilevel form. The upper level objective function measures some performance of the structure, such as the stiffness in our example. The lower level problem describes the behavior of the structure given the choices of the design variables and the external forces acting on it.

The formulation of optimality conditions for bilevel programming problems usually starts with a suitable reformulation of the problem as a one-level one by replacing the lower level optimization problem with its Karush-Kuhn-Tucker (KKT) optimality conditions, which are necessary and sufficient for defining the optimum of the inner level problem only under the convexity condition and a first order constraint qualification. When the inner problem constraints are non convex, the KKT conditions are only necessary. The BLPP can also be reformulated as a single level optimization problem by replacing the lower level problem by its optimal value function, unfortunately, in this case, the value function may not be generally differentiable. The approach used to derive optimality conditions is based on the directional derivatives of the non-differentiable optimal value function [2,3].

As mentioned above, one tool often used to reformulate the bilevel programming problem as an one level problem are the Karush-Kuhn-Tucker conditions. If a regularity condition is satisfied for the lower level problem, then the KKT conditions are necessary optimality conditions. They are also sufficient for the case when the lower level problem is convex, this is the approach used in this paper. So the mathematical analysis of the bilevel formulation is given, and a new single level optimization problem is deduced, from which an optimality system is derived by applying a Lagrange multiplier rule, and an algorithm is proposed for numerical solution.

1

A. Makrizi, B. Radi, A. El Hami, "Solution of the Topology Optimization Problem Based Subdomains Method", Applied Mathematical Sciences, 41(2), 2029-2045, 2008.

2

K. Shimizu, Y. Ishizuka, J.F. Bard, "Nondifferentiable and Two-Level Mathematical Programming", Kluwer Academic Publishers, Boston, 1997.

3

J.J. Ye, "Nondifferentiable multiplier rules for optimization and bilevel optimization problems", SIAM J. Optim, 15(1), 252-274, 2004. doi:10.1137/S1052623403424193

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