Differential evolution approach to reliability-oriented optimal design Sara Casciati Struttura Didattica Speciale di Architettura, University of Catania, Siracusa, Italy article info Article history: Received 31 January 2014 Accepted 5 March 2014 Available online 15 March 2014 Keywords: Reliability-based design optimization Moment based methods Nonlinear performance function Multiple limit states Differential evolution algorithm Heuristic optimization methods abstract The adoption of an evolutionary optimization approach, to tackle Reliability Based Design Optimization (RBDO) problems for which classical optimization is not feasible, can potentially lead to expand the use of RBDO in engineering applications. The herein proposed novel approach consists of coupling a differential evolution strategy with the one-level reliability method based on the Karush–Kuhn–Tucker (KKT) conditions. By selecting a few significant examples from the literature, convergence is found within a number of iterations compatible with the current capabilities of standard computational tools. These examples are chosen because lack of convergence is reported in literature when adopting classical gradient-based methods coupled with the reliability assessment. The performance and the efficiency of the algorithm are discussed, together with possible future improvements. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Traditional design procedures of engineering systems are based on deterministic models and parameters. The variability of the loads, the material properties, the geometry, and the boundary conditions is included by means of modeling idealizations and simplifying assumptions, such as the consideration of mean or extreme values and the introduction of safety factors derived from past practice and experience. This approach is not able to realis- tically capture the influence of the uncertainties in the system parameters on the structural performance, which might result to be unsatisfactory. Furthermore, it cannot provide a quantitative measure of the system safety since it is unable to capture a mathematical relation with the risk assessments based on which decisional procedures are carried out [1]. Despite the development of a broad spectrum of different reliability-based design optimization (RBDO) methods in literature over the last 50 years (see, for example, [2]), their practical application to engineering problems is still rather limited in comparison with the deterministic design procedures. This is mainly due to the numerical challenges and computational cost often encountered when explicitly taking into account the effects of the unavoidable uncertainties in the structural performance. In Valdebenito and Schueller [3], the need to improve the current strategies for solving RBDO problems is stated, with particular emphasis on the aspects of numerical efficiency and robustness. The selection of a particular optimization algorithm can be crucial to solve a specific RBDO problem. Indeed, the numerical efficiency, accuracy and stability of the solving algorithm deter- mine the adequacy of the method to practical engineering appli- cations. When traditional gradient-based algorithms are adopted for the objective function minimization, the efficiency related to the moment based methods comes at the price of a limited field of applications, which does not include problems characterized by a large number of random variables and multiple failure criteria considering nonlinear performance functions. Hence, there is a need to re-address the topic in the case that a different solution strategy, such as the one based on evolutionary algorithms of differential type, is pursued. The herein proposed novel approach consists of coupling a differential evolution (DE) strategy [4] with a one-level reliability method based on the Karush–Kuhn–Tucker (KKT) conditions [5]. The adoption of a one-level approach based on the direct introduction of the KKT conditions in the formulation of the general cost minimization problem was proposed in Kuschel and Rackwitz [6] for a single limit state function. Mathematical proof that the solution point of the FORM optimization problem fulfills the KKT conditions is given. The existence of an optimal design point guarantees that the vectors of the Lagrangian multipliers associated to the KKT conditions also exists. Nevertheless, the dependence of the failure domain on the design parameters prevents one from mathematically proving any general conver- gence property of the method. This observation holds also for the two-steps RBDO approaches [7], where two nested optimization problems are formulated, with the inner loop dedicated to the reliability analysis and the outer loop to the cost optimization. Further adaptations were studied in order to handle problems with multiple independent failure modes and/or time-variant sta- tionary loads [5]. The solution was sought by traditional gradient- based optimizers which required the computation of the gradient of the objective as well as the gradient of the constraints. In particular, taking the gradient of the second KKT optimality condition implicitly assumes the existence of the second order derivatives of the limit state functions. Such a requirement was identified as one of the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/probengmech Probabilistic Engineering Mechanics http://dx.doi.org/10.1016/j.probengmech.2014.03.001 0266-8920/& 2014 Elsevier Ltd. All rights reserved. Probabilistic Engineering Mechanics 36 (2014) 72–80