The estimation of failure probabilities as a false optimization problem Jorge E. Hurtado ⇑ , Juliana Ramírez Universidad Nacional de Colombia - Sede Manizales, Apartado 127, Manizales, Colombia article info Article history: Received 28 March 2011 Received in revised form 23 August 2011 Accepted 19 July 2013 Available online 30 August 2013 Keywords: Structural reliability Monte Carlo simulation False optimization Particle Swarm Optimization Reliability plot abstract This paper introduces a new regard and a powerful method for estimating small failure probabilities. It consists in considering the reliability problem as a false constrained optimization of a function. The opti- mization is called false because the minimum of the function is known beforehand. However, the process of computing such a minimum yields the samples located in the failure domain as a by-product, thus allowing the computation of the failure probability in a very simple manner. An algorithm based on an ad-hoc modification of the well-known Particle Swarm Optimization technique is proposed. It is charac- terized by the fact that it may deliver the same value of the failure probability as simple Monte Carlo simulation. In addition, the algorithm yields a visualization of all the computed samples in bidimensional plot, from which the critical realizations of the random variables can be drawn. These are the samples that mark the boundary between the safety and failure domains and therefore constitute a highly valu- able information for design and diagnosis. The excellent accuracy and low computational cost of the pro- posed approach are illustrated with several examples. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction As is well known, the basic problem of structural reliability can be defined as the estimation of the probability mass of a failure do- main F defined by a limit state function GðxÞ of a set of input ran- dom variables x. The complement to the failure domain is called the safe domain S. In transforming the variables x to the standard Normal space u, the problem reads [1] P f ¼ Z F f u ðuÞdu ð1Þ where f u ðuÞ is the joint probability density function of the basic variables in the n-dimensional space u. In performing this transfor- mation the limit state function GðxÞ is transformed into a function gðuÞ and the failure domain becomes F¼fu : gðuÞ 0g. The methods proposed to solve this problem can be grouped into two categories, namely those based on first- and second order approximations of gðuÞ (FORM and SORM), on the one hand, and simulation methods, on the other [2]. The former are based on the Taylor expansion of the limit state function about the point on the surface gðuÞ¼ 0 that is the closest to the origin. This, so- called design point, is assumed to be the critical vector of random variables, as it is characterized by the minimum Euclidean distance to the origin of coordinates. However, it has been observed that as the number of dimensions of the reliability problem increase, the design point gradually looses such a critical meaning and remains only a geometrical indicator that points to the failure zone in an abstract manner [3,4]. Among the simulation methods one can count the Importance Sampling [5, e.g.], Directional Simulation [1, e.g.], Line Sampling [6,7], Subset Simulation [8], Radial-based Importance Sampling [9,10] and others. A common characteristic of these simulation techniques is their purpose of reducing the computational cost of the Simple Monte Carlo Simulation (SMCS), which consists in gener- ating a large population from density f u ðuÞ, calculating the value of gðuÞ for each sample and, finally, estimating the failure probability as the fraction of cases for which gðuÞ is less than or equal to zero. In this paper a new interpretation of the simulation approach is proposed. It is called False Optimization because its central idea is to conceive the simulation as the search of the minimum of a func- tion whose value is known in advance. The worth of such a compu- tation is the collection of samples detected in the failure domain, which are henceforth used for the estimation of the failure prob- ability. In addition, the proposed approach yields another highly useful information, namely a visualization in two dimensions of the samples actually computed. From this plot a set of critical rea- lizations in the safe domain that mark its boundary with the failure domain can be selected. These critical realizations serve, therefore, as meaningful substitutes of the design point, whose shortcomings in this regard have already been mentioned and are discussed and illustrated in the sequel. The paper is divided as follows. First, the central idea of False Optimization (denoted as FO hereinafter) is exposed in detail. Sec- 0167-4730/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.strusafe.2013.07.010 ⇑ Corresponding author. Tel.: +57 68879300. E-mail address: jehurtadog@unal.edu.co (J.E. Hurtado). Structural Safety 45 (2013) 1–9 Contents lists available at ScienceDirect Structural Safety journal homepage: www.elsevier.com/locate/strusafe