An approximate single-loop chaos control method for reliability based design optimization using conjugate gradient search directions

Single-loop methods based on the Karush–Kuhn–Tucker conditions are considered to be efficient reliability based design optimization (RBDO) methods. The probabilistic performance functions are converted to deterministic functions for reducing the computational burden of reliability analysis. However, the most probable target point (MPTP) estimated using steepest descent search directions diverges or oscillates for highly nonlinear performance functions. Therefore, an approximate single-loop chaos control method is proposed to address this challenge by estimating MPTP using conjugate gradient search directions. An oscillation criterion is also proposed to track the oscillation of MPTP in every iteration. When this criterion is satisfied, the chaos control theory is used to update the current MPTP. The proposed method is tested on six mathematical and two engineering RBDO examples from the literature. Monte Carlo simulations are performed on the obtained solutions for estimating their reliability. The results demonstrate that the proposed method generates the best reliable solution and is also computationally efficient on the chosen set of examples over seven RBDO methods from the literature.


Introduction
Uncertainties in design variables are generally unavoidable while solving real-world engineering problems. Thus, deterministic optimization methods cannot be used for solving such problems. Among the various methods for uncertainty quantification and reliability analysis, reliability based design optimization (RBDO) has gained popularity owing to its accuracy and efficiency. The RBDO model can be expressed as the minimization of an objective function that is subjected to probabilistic constraints or performance functions. These RBDO models can be solved by simulation based methods and analytical methods. The simulation based methods are computationally expensive and therefore many surrogate models such as radial basis functions, Kriging based methods, etc. have been developed in last decade. For example, Mingyang Li and Wang (2019) proposed a surrogate based uncertainty quantification by introducing an equivalent reliability index employing Gaussian process regression to construct a Gaussian mixture model for reliability analysis. Mingyang Li and Wang (2020) developed a Bayesian-enhanced meta-model to handle heterogeneous uncertainties. Fan, Wang, and Hao (2019) proposed a Kriging based surrogate model based on finite-element simulation data to design a crane bridge system. Recently, an adaptive surrogate model and importance sampling based modified sequential optimization and reliability assessment (Song et al. 2021) method has been developed for better computational efficiency. On the other hand, analytical methods such as the first-order reliability method (FORM) (Hasofer and Lind 1974) and the second-order reliability method (SORM) (Breitung 1984) intend to locate the most probable point (MPP) in the standard normal space of variables and then estimate the probability of failure by approximating the performance function at the MPP. For these methods, the accuracy of RBDO methods depends on the estimation of the MPP on the failure surface and their computational efficiency is analysed through the number of function evaluations.
RBDO methods can be classified into double-loop, decoupled-loop and single-loop methods. In double-loop methods, the outer loop performs deterministic optimization that repeatedly calls the inner loop for reliability analysis. The details of the MPP from reliability analysis can either be obtained from the reliability index approach (RIA) (Reddy, Grandhi, and Hopkins 1994) or from the performance measure approach (PMA) (Tu, Choi, and Park 1999). It is found from the literature that PMA shows better numerical stability and computational efficiency than RIA (Ramu et al. 2006). Among the various methods, the advanced mean value (AMV) method (Wu, Millwater, and Cruse 1990) is a widely used PMA based reliability analysis method. In order to improve the computational efficiency of double-loop methods, decoupled-loop methods have been proposed. In these methods, optimization and reliability analysis are decoupled from each other and are performed simultaneously (Du and Chen 2004;Yi, Zhu, and Gong 2016). Since reliability analysis is performed as a sub-optimization problem, decoupled-loop methods have been found to be inefficient. Therefore, single-loop methods are proposed to improve efficiency further, in which the Karush-Kuhn-Tucker (KKT) optimality conditions are used by substituting the probabilistic constraints with equivalent deterministic constraints. Using a similar concept, a single-loop approach (SLA) (Liang, Mourelatos, and Tu 2008) is proposed. Further, in the single-loop single vector (SLSV) method (Xiaoguang Chen, Hasselman, and Neill 1997), a quantile approximation of the probabilistic constraints is made to approximate the MPP. A single-loop single vector using a conjugate gradient (SLSV-CG) (Jeong and Park 2017) is also proposed to increase the robustness of the algorithm. In order to maintain both efficiency and accuracy, a single-loop shifting vector with conjugate gradient (SLShV-CG) (Biswas and Sharma 2021) has been developed in which the single-loop method is coupled with the concept of the shifting vector approach of Sequential Optimization and Reliability Assessment (SORA).
Single-loop methods have been found to be computationally efficient for linear and weakly nonlinear convex functions. However, sometimes these methods do not converge for highly nonlinear concave functions. Therefore, some hybrid methods have been developed that utilize the concepts of different RBDO methods. For example, in an adaptive hybrid approach (AHA) (Gang Li, Meng, and Hu 2015), the single-loop method and the double-loop method are selected based on the performance function type. The hybrid mean value (HMV) method (Youn, Choi, and Park 2003) and the enhanced hybrid mean value (EHMV) method (Youn, Choi, and Du 2005) have also been developed based on the nature of the probabilistic constraints (convex or concave). These methods improved the numerical stability of the AMV method. However, many of these methods still fail to achieve convergence for highly nonlinear performance function (Keshtegar, Hao, and Meng 2017). Therefore, to overcome the issue of divergence, oscillation or bifurcation of the solution, the chaos control (CC) method (Yang and Yi 2009), the hybrid chaos control method (HCC) , the adaptive chaos control method (ACC) (Gang Li, Meng, and Hu 2015), the self adaptive modified chaos control method (SMCC) (Keshtegar, Hao, and Meng 2017) and the chaotic conjugate stability transformation method (CCSTM) (Keshtegar 2016) have been proposed. An enhanced chaos control method (ECC)  has been proposed in which the control factor is updated by the Wolfe-Powell criterion. Further, an enhanced step length adjustment iterative algorithm with a second-order reliability method based stepped-up SORA approach (SSORA-SORM) ) was proposed to boost the efficiency of RBDO. An SLA confidence based design optimization (CBDO)  was developed for uncertainty modelling caused by insufficient data by coupling complete PMA, SORM and a single-loop strategy, which increases the efficiency and robustness of CBDO. In another study, an augmented step size adjustment (ASSA) (Hao et al. 2021) algorithm was proposed with a bi-stage RBDO framework with isogeometric analysis.
All these methods enhance the stability of PMA, thereby improving the performance of reliability analysis. In recent years, several single-loop methods based on the concept of chaos control have also been proposed for increasing their efficacy. In the adaptive conjugate single-loop method (AC-SLA) (Meng and Keshtegar 2019), the MPP is approximated by a dynamic conjugate scalar factor, whereas in an adaptive hybrid single-loop method (AH-SLM) (Jiang et al. 2017) the MPP is estimated by an iterative control strategy. In an enhanced single-loop method (ESM) (Keshtegar and Hao 2018), the performance of approximating the MPP is increased by an adaptive step size. In a chaotic single-loop approach (CSLA) (Meng et al. 2018), the robustness is enhanced by automatic selection of a chaos control factor. However, CLSA sometimes has low efficiency in solving highly nonlinear performance functions because it is sensitive to the initial chaos control factor (Meng et al. 2018).
It is found from the literature that single-loop methods using chaos control theory have certain limitations. First, the solution gets stuck in a periodic oscillation owing to the use of steepest descent search directions while estimating MPTPs for highly nonlinear performance functions. Secondly, the efficiency of these methods is sensitive to the initial chaos control factor value. Thirdly, there is no proper oscillation criterion for tracking oscillation during the convergence of MPTPs. Therefore, a new RBDO method using the single-loop method with chaos control theory is proposed to address these issues. The following are the contributions of this article.
• An approximate single-loop chaos control (ASLCC) method is proposed to handle periodic oscillation among MPTPs. In every iteration, MPTPs are estimated using conjugate gradient search directions. When oscillation among these points is observed, the current MPTP is updated using chaos control theory. • An oscillation criterion is also proposed to track the oscillation in the convergence of MPTPs.
The vector difference between the three most recent MPTPs is used to check the criterion in the standard normal space of variables. • The performance assessment of the proposed method is tested on six mathematical and two engineering RBDO examples from the literature. The ASLCC method is compared with the AMV (Tu, Choi, and Park 1999), CGA (Ezzati, Mammadov, and Kulkarni 2015), CC (Yang and Yi 2009), HCC , SORA (Du and Chen 2004), ASORA (Yi, Zhu, and Gong 2016) and SLA (Liang, Mourelatos, and Tu 2008) methods, and the reliability of the solutions obtained is verified using Monte Carlo simulations.
The organization of the article is as follows. A brief description of the RBDO formulation, along with the details of reliability analysis and chaos control theory based methods, are presented in Section 2. The proposed method is presented in Section 3 with the help of algorithmic steps. Eight RBDO examples are solved using the proposed method and the results are compared with other RBDO methods from the literature in Section 4. Conclusions are drawn in Section 5 with a note on future work.

Basic RBDO formulation
The RBDO formulation (Lee, Choi, and Gorsich 2010;Jeong and Park 2017) is presented in Equation (1): where f (μ X ) is the objective function, G i (d, μ X ) is the probabilistic inequality constraint, also known as the performance function, such that G i (d, μ X ) > 0 represents failure, h j (d) is the deterministic inequality constraint, μ X is the mean value of the random variable X, μ (L) X and μ (U) X are the lower and upper limits of μ X , d = [d 1 , d 2 , . . . , d n ] T is the deterministic design vector, and d (L) and d (U) are the lower and upper limits of design vector d, respectively. P is the probability operator and P T f i is the probability of failure of the ith performance function, which is evaluated by the target reliability . P T f i represents the standard normal cumulative distribution function. The failure probability of a performance function is expressed as a multidimensional integral (Tu, Choi, and Park 1999;Youn, Choi, and Park 2003) as given in Equation (2): where f X (X) is the joint probability distribution function of the random variable X and F G i (0, d) is the representation of the cumulative distribution function. The probabilistic constraint of Equation (2) can be evaluated by the RIA or the PMA as discussed in Section 1. From the literature, it is found that PMA is widely used because of its efficiency and accuracy. It is described in the online supplemental data, which can be accessed at https://doi.org/10.1080/0305215X.2021.2007242.

The single-loop approach (SLA)
In the SLA (Liang, Mourelatos, and Tu 2008), the probabilistic constraints are converted into approximate deterministic constraints. This is done by eliminating the reliability analysis loop by using KKT optimality conditions for estimating the approximated MPTP. Both the target reliability index β T i and sensitivity analysis play a vital role in approximating the MPTP. The formulation of the SLA is given in Equation (3): where X (k) i represents the vector of the approximated MPTP of the ith performance function at the kth iteration, which is calculated by the mean value vector μ (k) X of the random variable X (k) i , the target reliability index (β T i ) of the ith performance function, the standard deviation (σ X ) of the random variable X (k) i and the steepest descent direction vector (α (k) i ) of the ith performance function. When G i (d, X) is less than or equal to zero, the constraint function is reliable. This approach shows good efficiency for convex problems, however it fails to converge for highly nonlinear concave functions (Jeong and Park 2017).

The chaos control (CC) method
The chaos control (Yang and Yi 2009) method demonstrates promising numerical stability for periodic oscillation of the solution and is developed from the concept of the stability transformation method (STM) (Pingel, Schmelcher, and Diakonos 2004). The MPTP search of the chaos control method is updated by the following formulation: where C is the n × n involutory matrix. For simplicity, this matrix is considered to be the identity matrix, I. The chaos control factor is chosen as λ ∈ [0, 1]. When λ is equal to one, the formulation of the CC method is similar to that of the AMV method, which can cause the same issues discussed earlier. Therefore, a small value of λ is considered for stable convergence. F is the vector of the response function with respect to the iterative values of u (k) . The nonlinear mapping (F(u (k) AMV ) is used to control the unstable fixed points of the iterative algorithm.

The adaptive chaos control (ACC) method
To improve the efficiency of the CC method, the following formulations were proposed by the modified chaos control  method: whereñ k is the modified search direction based on the chaos control on performance measure function. The modification is done by extending the search to β-circle using Equation (5).
In the ACC method, a function-type criterion is used to determine the difference between convex and concave performance functions. When the function is convex, the AMV method is used. Otherwise, a modified chaos control (MCC)  method is used to control the oscillation of the MPTP for concave functions. Moreover, the control factor λ is determined from the angle updating criterion, which is given by Equation (6): where θ (k) and θ (k−1) are the angles between (U (k+1) , U (k) ) and (U (k) , U (k−1) ), respectively.

The proposed approximate single-loop chaos control (ASLCC) method
It can be observed from the previous sections that chaos control theory is effective in dealing with the MPTP when it starts oscillating in the search space. However, the use of chaos control theory in the douple-loop structure makes the RBDO method inefficient. Moreover, the chaos control factor is a sensitive parameter for convergence. Therefore, in the proposed method, the single-loop method is coupled with chaos control theory for addressing these issues. In the following section, the proposed method is discussed in detail.

The proposed ASLCC method
As discussed in Section 2.2, the SLA employs KKT conditions to approximate the MPTP from the standard normal space to the normal space. This enhances the efficiency of the RBDO method by eliminating the reliability analysis loop. The deterministic optimization that is used in the RBDO formulation is as follows: where f (μ X ) is the objective function, G i is the ith deterministic constraint, d is the deterministic design variable and μ X is the mean value of the random variable vector X. The approximated MPTP is expressed as where U is the vector of random variables in the standard normal space, σ X is the standard deviation of the random variables and X (k) i,MPTP is the approximated MPTP for the ith constraint in the normal space at the kth iteration. Since the accuracy of the reliable optimal solution depends on X (k) i,MPTP , this article thus focuses on evaluating the accurate U (k) i,ASLCC given in Equation (8) using the proposed method. An oscillation criterion is also proposed to track MPTPs. When these MPTPs start oscillating, the current MPTP of Equation (8) is updated using chaos control theory. The oscillation criterion is given in Equation (9): i,ASLCC ) are almost in the same direction, signifying there is convergence as shown in Figure 1(a). In this case, the conjugate gradient search direction is used to estimate the MPTP because it is found to be robust (Jeong and Park 2017). On the other hand, if the dot product is negative, the two vectors (U i,ASLCC ) are moving away from each other. The angle between these vectors thus increases, signifying oscillation among MPTPs as shown in Figure  1(b). In this case, chaos control theory is utilized to estimate the new MPTP, which helps to eliminate oscillation of MPTPs for highly nonlinear performance functions.

Estimation of the MPTP using the conjugate gradient search direction
The formulation to approximate the MPTP is as follows: where α (k) i is the normalized direction at the kth iteration, which is estimated by the conjugate gradient search direction, D (k) i . It is given as Figure 2 shows the iterative updating of the MPTP using conjugate gradient search directions.

Estimation of the MPTP using chaos control theory
After calculating X (k) i,MPTP using Equation (10), it is transformed to the standard normal space to find the corresponding value of U (k) i,MPTP using Equation (13). Thereafter, U i,ASLCC is updated using Equation (14): It can be seen that U (k) i,ASLCC is the same as U (k) i,MPTP when k < 2 or γ i > 0 of Equation (9). Otherwise, U (k) i,ASLCC is evaluated separately using chaos control theory. The chaos control factor λ (k) i is changed in each iteration using the angle condition given in Equation (6). If the angle (θ (k) ) as shown in Figure  1 is increasing from the previous angle (θ (k−1) ), λ (k) i is decreased to 0.2 × λ (k) i , otherwise λ (k) i is kept constant. The converging and diverging scenarios of the MPTP in the standard normal space are shown in Figures 1(a) and 1(b), respectively. It can also be seen in Figure 1(a) that the consecutive MPTPs are in the same direction and therefore achieve stable convergence. On the other hand, if the direction of the vectors changes, it signifies divergence or oscillation as shown in Figure 1(b).
The proposed method is referred to as the approximate single-loop chaos control method (ASLCC) because the chaos control method is used when MPTPs start oscillating. The following are the steps for the ASLCC method.
Step 1. Initialize with mean values of the random variables, μ (0) X , the standard deviation of the random variables, σ X , the target reliability index of the performance function, β T , the chaos control factor, λ (k) i , and the involutory matrix, C, which is considered to be the identity matrix, I.
Step 2. Set k = 0, initial design variables Step 3. Perform deterministic optimization as and generate the mean values of random variables μ (k+1) X .
Step 4. Calculate D (k) i using Equation (16): where α (k) i is estimated with conjugate gradient search direction D Step 5. Update X i,MPTP in the normal space as and calculate U (k+1) i,MPTP = T(X (k+1) i,MPTP ) and proceed Step 6. Check the value of oscillation criterion γ (k) when k > 1. If the value is positive or k ≤ 1, estimate U (k+1) i,ASLCC as U (k+1) i,MPTP and go to Step 8. Otherwise, proceed to Step 7.
Step 9. Evaluate the angle between two vectors as and calculate the iterative chaos control factor for k ≥ 1 as Step 10. Check the iteration counter k. If k > 2, calculate the oscillation criterion using Equation (22). Otherwise, go to Step 11.
Step 11. Update the MPTP, X i,MPTP , using U (k+1) i,ASLCC as given in Equation (23) and update k = k + 1 and go to Step 3: The flowchart of the ASLCC is shown in Figure S1 of the online supplemental data.

Results and discussion of numerical examples
The proposed method is tested on six mathematical and two engineering RBDO examples. Monte Carlo simulation (MCS) with a sample size of 10 6 is performed at the optimal solution for measuring its reliability for each of the performance functions. The computational efficiency is measured through the number of function calls (nfc), of which f FC is the measure of objective function calls and G FC is the measure of performance function calls during optimization. The results of the proposed method are compared with those of the AMV (Tu, Choi, and Park 1999), CGA (Ezzati, Mammadov, and Kulkarni 2015), CC (Yang and Yi 2009), HCC , SORA (Du and Chen 2004), ASORA (Yi, Zhu, and Gong 2016) and SLA (Liang, Mourelatos, and Tu 2008) methods from the literature. In this section, two variants of the ASLCC method are presented. In Equations (16) and (17), conjugate gradient and steepest descent search directions are used to calculate the MPTP, X (k+1) i,MPTP .
Step 4 discussed in Section 3.1.2 suggests that, when k ≥ 2, the search direction (D (k) i ) is calculated using Equation (16), otherwise use Equation (17). This variant is referred to as ASLCC-2. However, when D (k) i is estimated only using Equation (17) in every iteration, this variant is referred to as ASLCC-1. All the methods are initialized with the same design point and are terminated with the same termination conditions. The MATLAB R2016b platform is used for developing the methods and fmincon is used as an optimizer that uses the sequential quadratic programming method. The termination criterion for fmincon is set to the default values, whereas the termination criterion for the developed methods is set to

Example 1
A highly nonlinear RBDO example (Meng and Keshtegar 2019) is shown in Equation (24). The example has two independent random variables with normal distributions and three performance functions (G 1 (X), G 2 (X) and G 3 (X)). The performance function G 2 (X) is concave and other two are convex in nature. The initial design point is considered as μ (0) x = [5.0, 5.0] T , and the target reliability index for each constraint is considered to be β T i = 3.5.
Table 1 presents the solutions obtained from the RBDO methods. The objective value (f * ) corresponds to the obtained optimum solution (μ * x ). Here, 'β T MCS ' represents the target reliability achieved for each optimum solution as listed in the sixth column of the table. The constraint function calls (G FC ) denote the total number of times the constraint function is called by fmincon and for gradient calculation. The term 'f FC ' represents the total number of objective function calls in the entire optimization process.
It can be seen from column six of Table 1 that the ASLCC-2 method generates a reliable solution. However, the SLSVCG, CGA and ASLCC-1 methods could not generate any reliable solution. The rest of the methods fail to converge for the given RBDO example. Figure 3 shows the optimal solution obtained by the ASLCC-2 method. It can be seen that the performance functions G 1 (X) and G 2 (X) are active at the solution and the performance function G 3 (X) is inactive. The same observation can be seen in the last column of Table 1 for G 3 (X), where the value of β becomes infinity.
The computational efficiency of the methods is quantified using the function evaluations presented in the fourth and fifth columns of Table 1. ASLCC-2 is found better than CGA and ALSCC-1. However, it requires more function evaluations than SLSVCG. ALSCC-1 is also found to be more computationally efficient than the CGA method. Table 1. RBDO results for Example 1 with β T = 3.5 (λ = 0.5).   The convergence of the ALSCC-1, ALSCC-2 and CGA methods is shown in Figure 4. A smooth convergence of ASLCC-2 can be seen, whereas ASLCC-1 faces difficulty while converging to the solution. ALSCC-1, thus, requires almost seven time more iterations than ALSCC-2. The convergence of all methods for the performance function (G 2 (X)) with respect to the number of iterations is shown in Figure S2 in the online supplemental data. It can be seen that ALSCC-2 shows smooth convergence while dealing with the concave performance function. Other methods show oscillation because the MPTPs are estimated using the steepest descent search directions.
The chaos control factor (λ) plays a vital role in generating the solution for chaos control based RBDO methods. Therefore, two λ values are considered; one nearer to zero (λ = 0.2) and the other closer to one (λ = 0.8). Table 2 presents the solutions obtained by these methods. Here, the CC and HCC methods achieve convergence for λ = 0.2 but show chaos for λ = 0.8. ASLCC-1 also demonstrates similar chaos for λ = 0.8. This is because, when λ is closer to one, the method estimates the MPTPs using the steepest descent search directions with chaos control theory and, therefore, becomes unable. However, the ASLCC-2 variant shows no sensitivity towards the initial value of λ. This is because U (k+1) i,MPTP is estimated using the conjugate gradient search directions. Furthermore, the adaptive nature of λ, which depends on the angle condition as presented in Equation (21), also helps ASLCC-2 to become insensitive to λ values.

Example 2
This nonlinear RBDO example consists of a highly nonlinear concave performance function with a nonlinear objective function. The RBDO problem (Jeong and Park 2017) formulation is given in the online supplemental data. This example consists of two independent random normal variables with standard deviation 0.6, and the initial design point is considered to be μ (0) x = [5.0, 5.0] T . The RBDO results obtained by different methods are summarized in Table 3. It can be seen from the last column of the table that the ALSCC-2, SLShV-CG, CGA, CC and HCC methods generate reliable solutions. However, SORA is unable to generate any reliable solution. Among them, ASLCC-2 generates the best reliable solution. The solution obtained by the ASLCC-2 method is shown in Figure 5. It can be seen that the performance function is active in the solution. The convergence plot of methods is shown in Figure 6, in which all methods converge smoothly. Table 3 also presents the computational efficiency in the fourth and fifth columns. ALSCC-2 is found to be the most efficient method, followed by the HCC and CGA methods. Table 4 presents results for different values of λ for chaos control based RBDO methods. Among them, ASLCC-2 generates the best reliable solution with fewer function evaluations. Moreover, it is found to be insensitive to different values of λ.

Speed reducer example
A speed reducer (Ju and Lee 2008) is the first engineering RBDO example considered that is used to rotate the propeller of a light plane. The objective is to reduce the weight of the speed reducer, which is   Table 3. RBDO results for Example 2 with β T = 3.0 (λ = 0.5). subjected to 11 performance functions. These functions are related to physical quantities, like bending stress, contact stress, longitudinal displacement, stress of the shaft, transverse displacement and the geometric conditions. The RBDO example consists of seven independent random variables, such as  gear width (x 1 ), teeth module (x 2 ), number of pinion teeth (x 3 ), the distance between two bearings (x 4 , x 5 ) and axis diameter (x 6 , x 7 ). The RBDO problem formulation and a schematic diagram for the speed reducer ( Figure S3) are given in the online supplemental data. Table 5 presents the results obtained by all the methods. It can be seen that all methods generate optimal solutions equivalent to that of ASLCC-2, with similar reliability. Among the Kriging based methods, IBS is the only method that generates a solution satisfying target reliability for all performance functions. The performance functions G 5 (X), G 6 (X), G 8 (X) and G 11 (X) are the only the active constraints in the optimum solution. The progress of solutions is shown Figure 7, which indicates smooth convergence for all the methods.
The computational efficiency is presented in Table 5. It can be seen from the fourth and fifth columns of the table that ASLCC-1 and ASLCC-2 are the most efficient methods followed by SLA and ASORA. The rest of the methods require relatively many function evaluations.
The results of different initial values of λ for chaos control based methods are listed in Table 6. All methods converge to the same solution. The computational efficiency of the ASLCC-1 and ASLCC-2 methods does not depend on the initial chaos control factor for the given example. However, the efficiency of the CC method changes drastically as the value of λ increases. The HCC method seems to be insensitive to the initial value of λ.

Other examples
Owing to page limitations, four mathematical examples and a spring RBDO example are given in the online supplemental data.

Conclusion
An approximate single-loop chaos control method (ASLCC) using conjugate gradient search directions has been developed in this article. The aim was to track oscillation among MPTPs while solving various RBDO examples. In the proposed method, MPTPs were estimated using conjugate gradient search directions in every iteration. When these points started oscillating, chaos control theory was used to update the current MPTP. An oscillation criterion has also been proposed that estimates chaos among three consecutive MPTPs in the standard normal space. Among the chosen set of RBDO methods, ASLCC-2 generated the best reliable solutions for five out of eight examples. For the other three examples, none of the methods was able to generate a reliable solution. The convergence plots suggest a smooth convergence of ASLCC-2 for all examples, especially for Examples 4.1 and 4.2, while oscillation among MPTPs was observed for the other methods. ALSCC-2 was also computationally efficient, as estimated using the number of function evaluations required to generate a reliable solution. It was found to be computationally efficient on five out of eight examples. The proposed method was also compared with other chaos control based methods for different initial values of λ. ASLCC-2 was found to be insensitive for different λ values while solving all examples. In future work, this method can be made more robust by introducing other search directions so that the method can solve three unsolved examples of this article. Moreover, the proposed method can be extended for solving large-scale real-world RBDO problems.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Data availability statement
Data related to the examples are included in this article. There is no other data associated.