Normalized domination selection criteria for differential evolution algorithms in constrained optimization for seismic engineering

Optimization is a central aspect of structural engineering, but its practical application hasn't been supported by mathematical and numerical tools because of inner strong non-linear aspects involved. Moreover during last few decades Evolutionary Algorithms (EAs) gives new interest and horizons in this specific topic, thanks to their strong capacity in treatment of these problems more efficiently than standard methods. But a common criticism to EAs is lack of efficiency and robustness in handling constraints, mainly because they were originally developed for unconstraint problems only. For this reason during past decade hy-brid algorithms combining evolutionary computation and constraint-handling techniques have shown to be effective in this specific area. Moreover still now this is a crucial point for practical applications in structural optimization. In this paper a Normalized Domination Se-lection-based (NDS) rule is proposed to solve constrained-handling optimization problems using a modified version of proposed Differential Evolution algorithm (NDS - DEa). The strategy developed doesn't requires any additional parameter, increasing the appeal for a simple implementation in many real problems by structural designer without a specific know-ledge in the field. Mainly it is based on a domination criteria in selection phase. Actually a common way for constrained handling is introducing a specific role for selection step, so that all other phase of EA aren't modified; in this way DE flow chart scheme doesn't present any modification from a standard unconstrained one. Anyway the specific constrained selection scheme plays an important role in solution search efficiency, certainly more than in unconstrained cases. Unconstrained selection is based only on comparing individuals OF values, but in constrained one it seemed somewhat different and complicated. The more simple, common and intuitive way for approaching this phase is the penalty function, where OF values are reduced for those individuals don't satisfying constraints disqualifies (unfeasible in-dividuals). It is immediate (and well known in literature) that depending on penalty low adopted, a more drastic or permissive surviving of unfeasible solutions happened. But this is a central point in this problems, because of in many cases indeed real optimal solutions lies just on one constraints, so that its correct evaluation needs of specific research around the boundary, not only in the feasible space. to develop this strategy the domination concept is related to the specific selection that has to be implemented. If in a unconstrained contest it means simply that the domination coincide with the OF ranking, in the constrained contest the question has to be properly treated. In fact there are three possible scenarios: ->both two individuals are feasible -> selection based on rank -> both two individuals are unfeasible -> selection of the feasible one -> one is feasible and the other is unfeasible -> selection of less unfeasible Moreover the last case presents same some ambiguity because in general there are many con-straints with different scales, so that it is impossible to rank correctly to different unfeasible individuals. For this reason a normalized criteria is here proposed and analyzed with different cross over methodology. A comparative analysis using different test cases is performed.