Structural reliability under uncertainty in moments: distributionally-robust reliability-based design optimization

This paper considers structural optimization under a reliability constraint, where the input distribution is only partially known. Specifically, when we only know that the expected value vector and the variance-covariance matrix of the input distribution belong to a given convex set, we require that, for any realization of the input distribution, the failure probability of a structure should be no greater than a specified target value. We show that this distributionally-robust reliability constraint can be reduced equivalently to deterministic constraints. By using this reduction, we can treat a reliability-based design optimization problem under the distributionally-robust reliability constraint within the framework of deterministic optimization, specifically, nonlinear semidefinite programming. Two numerical examples are solved to show relation between the optimal value and either the target reliability or the uncertainty magnitude.


Introduction
Reliability-based design optimization (RBDO) is a crucial tool for structural design in the presence of uncertainty [2,36,45,49]. It adopts a probabilistic model of uncertainty, and evaluates the probability that a structural design satisfies (or, equivalently, fails to satisfy) performance requirements. An underlying premise is that complete knowledge on statistical information of uncertain parameters is available. In practice, however, it is often difficult to obtain statistical information with sufficient accuracy. This incents recent intensive study of RBDO with incomplete statistical information [9,10,16,21,22,24,25,35,37,38,46,50,51,53].
Another methodology dealing with uncertainty in structural design is robust design optimization [6,20,31]. Although there exist several different concepts in robust design optimization, in this paper we focus attention on the worst-case optimization methodology, which is called robust optimization in mathematical optimization community [4]. This methodology adopts a possibilistic model of uncertainty, i.e., specifies the set of possible values that the uncertain parameters can take. We call this set an uncertainty set. Then, the objective value in the worst case is optimized, under the condition that the constraints are satisfied in the worst cases. This paper deals with RBDO when the input distribution is only partially known. Specifically, we assume that the true expected value vector and the true variancecovariance matrix are unknown (i.e., the true values of the first two moments of the input distribution are unknown), but they are known to belong to a given closed convex set. For example, suppose that the input distribution is a normal distribution, and we only know that each component of the expected value vector and the variance-covariance matrix belongs to a given closed interval. Then, for each possible realization of pair of the expected value vector and the variance-covariance matrix, there exists a single corresponding normal distribution. The set of all such normal distributions is considered as an uncertainty set of the input distribution. 1 As another example, suppose that distribution type of the input distribution is also unknown. Then the uncertainty set is the set of all probability distributions, the expected value vector and the variance-covariance matrix of which belong to a given set. 2 Among probability distributions belonging to a specified uncertainty set defined as above, the worst-case distribution is the one with which the failure probability takes the maximum value. Our methodology is that we require a structure to satisfy the reliability constraint evaluated with the worst-case distribution. In other words, for any probability distribution belonging to the uncertainty set, the failure probability should be no greater than a specified target value. Thus, the methodology guarantees robustness of the structural reliability against uncertainty in the input distribution. 3 The major contribution of this paper is to show, under some assumptions, that this structural requirement is equivalently converted to a form of constraints that can be treated in conventional deterministic optimization. As a result, a design optimization problem under this structural requirement can be solved with a deterministic nonlinear optimization approach.
Recently, RBDO methods with uncertainty in the input distribution have received considerable attention, because in practice it is often that the number of available samples of random variables is insufficient. For example, Gunawan and Papalambros [16] and Youn and Wang [50] proposed Bayesian approaches to compute the confidence that a structural design satisfies a target reliability constraint, when both a finite number of samples and probability distributions of uncertain parameters are available. Noh et al. [37,38] proposed Bayesian methods to adjust an input distribution model to limited data, with a given confidence level. When intervals of input variables are given as input information, Zaman et al. [52] and Zaman and Mahadevan [51] use a family of Johnson distributions to represent the uncertainty. Cho et al. [9] and Moon et al. [35] assume that the input distribution types and parameters follow probability distributions. The failure probability is therefore a random variable, the confidence level of a reliability constraint, i.e., the probability that the failure probability is no greater than a target value, is specified. To reduce computational cost of this method, Jung et al. [25] proposed a so-called reliability approach, inspired by the performance measure approach [33,34]. Subsequently, to further reduce computational cost, Wang et al. [46] proposed to use the second-order reliability method for computation of the failure probability. Ito et al. [21] assume that each of random variable follows a normal distribution with the mean and the variance modeled as random variables, and show that RBDO with a confidence level can be converted to a conventional form of RBDO by altering the target reliability index value. Zhang et al. [53] proposed to use the distributional probability box (the distributional p-box) [41] for RBDO with limited data of uncertain variables. Kanno [29,30] and Jekel and Haftka [23] proposed RBDO methods using order statistics. These methods, based on the order statistics, do not make any assumption on statistical information of uncertain parameters, and use random samples of uncertain parameters directly to guarantee confidence of the target reliability.
As reviewed above, most of existing studies on RBDO with uncertainty in the input distribution [9,21,25,35,46] consider probabilistic models of input distribution parameters and/or distribution types. Accordingly, a confidence level evaluates how the satisfaction of structural reliability is reliable. In contrast, in this paper we consider a possibilistic model of input distribution parameters. Hence, what this approach guarantees is a level of robustness [3] of the satisfaction of structural reliability. A possibilistic model might be, in general, less information-sensitive, and hence useful when reliable statistical information of input distribution parameters is unavailable.
From another perspective referring to Schöbi and Sudret [41], the uncertainty model treated in this paper can be viewed as follows. Uncertainty in a structural system is often divided into aleatory uncertainty and epistemic uncertainty [39]. Aleatory uncertainty, i.e., natural variability, is reflected by an (uncertain) input distribution. Epistemic uncertainty, i.e., state-of-knowledge uncertainty, is reflected by uncertainty in the input distribution moments. Thus, in our model, aleatory uncertainty is probabilistic, while epistemic uncertainty is possibilistic. In other words, state-of-knowledge uncertainty is represented as an uncertainty set of the input distribution moments.
Throughout the paper, we assume that only design variables possess uncertainty, and that variation of a performance requirement can be approximated as a linear function of uncertain perturbations of the design variables. Also, we do not consider an optimization problem with variation of structural topology. As for an uncertainty model of moments of the input distribution, we consider two concrete convex sets. We show that the robust reliability constraint, i.e., constraint that the structural reliability is no less than a specified value for any possible realizations of input distribution moments, can be reduced to a system of nonlinear matrix inequalities. This reduction essentially follows the idea presented by El Ghaoui et al. [12] for computing the worst-case value-at-risk in financial engineering. 4 We can deal with nonlinear matrix inequality constraints within the framework of nonlinear semidefinite programming (nonlinear SDP) [48]. In this manner, we can convert an RBDO problem under uncertainty in the input distribution moments to a deterministic optimization problem. It is worth noting that there exist several applications of linear and nonlinear SDPs, as well as eigenvalue optimization, to robust design optimization of structures [5, 17-19, 26, 28, 32, 43, 44].
The paper is organized as follows. In section 2, we consider the reliability constraint when the input distribution is precisely known, and show some fundamental properties. Section 3 presents the main result; we consider uncertainty in the expected value vector and the variance-covariance matrix of the input distribution, and examine the constraint that, for all possible realizations of the input distribution, the failure probability is no greater than a specified value. Section 4 discusses some extensions of the obtained result. Section 5 presents the results of numerical experiments. Section 6 presents some conclusions.
In our notation, denotes the transpose of a vector or matrix. All vectors are column vectors. We use I to denote the identity matrix. For two matrices X = (X ij ) ∈ R m×n and Y = (Y ij ) ∈ R m×n , we denote by X • Y the inner product of X and Y defined by For a vector x = (x i ) ∈ R n , the notation x 1 , x 2 , and x ∞ designate its 1 -, 2 -, and ∞ -norms, respectively, i.e., For a matrix X = (X ij ) ∈ R m×n , define matrix norms X 1,1 , X F , and X ∞,∞ by Let S n denote the set of n × n symmetric matrices. We write Z 0 if Z ∈ S n is positive semidefinite. Define S n + by S n + = {Z ∈ S n | Z 0}. For a positive definite matrix Z ∈ S n , the notation Z 1/2 designates its symmetric square root, i.e., Z 1/2 ∈ S n satisfying Z 1/2 Z 1/2 = Z. We use Z −1/2 to denote the inverse matrix of Z 1/2 . We use N(µ, Σ) to denote the multivariate normal distribution with an expected value vector µ and a variance-covariance matrix Σ. For a random variable x ∈ R, its expected value

Reliability constraint with specified moments
In this section, we assume that the expected value vector and the variance-covariance matrix of the probability distribution of the design variable vector are precisely known. We first recall the reliability constraint, and then derive its alternative expression that will be used in section 3 to address uncertainty in the probability distribution.
Let x ∈ R n denote a design variable vector, where n is the number of design variables. Assume that performance requirement in a design optimization problem is written in the form where g : R n → R is differentiable. For simplicity, suppose that the design optimization problem has only one constraint; the case where more than one constraints exist will be discussed in section 4.
Assume that x is decomposed additively as where ζ is a random vector andx is a constant (i.e., non-random) vector. Therefore, in a design optimization problem considered in this paper, the decision variable to be optimized isx. We use µ ∈ R n and Σ ∈ S n to denote the expected value vector and the variance-covariance matrix of ζ, respectively, i.e., It is worth noting that Σ is positive definite. Throughout the paper, we assume that, among parameters in a structural system, only ζ possesses uncertainty. Also, we restrict ourselves to optimization without change of structural topology; i.e., we do not consider topology optimization. 5 For simplicity and clarity of discussion, we assume ζ ∼ N(µ, Σ) in section 2 and section 3. In fact, the results established in these sections can be extended to the case that the type of probability distribution is unknown; we then require that the reliability constraint should be satisfied for any probability distribution with moments belonging to a specified set. We defer this case until section 4.
Since x is a random vector, g(x) is a random variable. Therefore, constraint (1) should be considered in a probabilistic sense, which yields the reliability constraint Here, ∈]0, 1] is the specified upper bound for the failure probability. Let g lin (x) denote the first-order approximation of g(x) centered at x =x, i.e., Throughout the paper, we consider an approximation of constraint (3) i.e., Therefore, the corresponding RBDO problem has the following form: Here, f : R n → R is the objective function, X ⊆ R n is a given closed set, and constraint x ∈ X corresponds to, e.g., the side constraints on the design variables. From the basic property of the normal distribution, we can readily obtain the following reformulation of the reliability constraint.
where Φ is the (cumulative) distribution function of the standard normal distribution N(0, 1). Then,x ∈ X satisfies (5) if and only if it satisfies Proof. Since g lin (x) follows the normal distribution, it is standardized by By using this relation, we can eliminate g lin (x) from (4) (i.e., (5)) as This inequality is equivalently rewritten by using the distribution function Φ as By direct calculations, we see that the expected value of g lin (x) is and the variance is which concludes the proof.
In section 3, we deal with the case in which µ and Σ are known imprecisely. To do this, we reformulate κ Σ 1/2 ∇g(x) 2 in (7) into a form suitable for analysis. The following theorem is obtained in the same manner as El Ghaoui et al. [12,Theorem 1].
Theorem 2.2. For κ > 0, Σ ∈ S n + , and ∇g(x) ∈ R n , we have Proof. We first show that the left side of the equation can be reduced to To see this, we apply the Lagrange multiplier method to the equality constrained maximization problem on the right side of (8). Namely, the Lagrangian L 1 : R n × R → R is defined by where µ ∈ R is the Lagrange multiplier. The stationarity condition of L 1 is By solving this stationarity condition, we can find that are optimal. Hence, the optimal value is which is reduced to the left side of (8).
Next, observe that the right side of (8) is further reduced to Here, the last equality follows from the fact that the positive semidefinite constraint is equivalent to the nonnegative constraint on the Schur complement of Σ in the corresponding matrix, i.e., κ 2 − ζ Σ −1 ζ ≥ 0; see [7, appendix A.5.5]. It is worth noting that the last expression in (9) is an SDP problem.
Finally, we shall show that the right side of the proposition in this theorem corresponds to the dual problem of the SDP problem in (9). Since this dual problem is strictly feasible, the proposition follows from the strong duality of SDP [8, section 11.3]. We can derive the dual problem of (9) as follows. The Lagrangian is defined by where z ∈ R, λ ∈ R n , and Λ ∈ S n are the Lagrange multipliers. Indeed, since the positive semidefinite cone satisfies [27,Fact 1.3.17] inf we can confirm that the SDP problem in (9) is equivalent to The dual problem is defined by Since (10) can be rewritten as Therefore, the dual problem in (12) corresponds to the right side of the proposition of the theorem.

Worst-case reliability under uncertainty in moments
In this section, we consider the case that the moments (in this paper, the expected value vector and the variance-covariance matrix) of the design variable vector are uncertain, or not perfectly known. Specifically, they are only known to be in a given set, called the uncertainty set. We require that a structure satisfies the reliability constraint for any moments in the uncertainty set. In other words, we require that the failure probability in the worst case is not larger than a specified value. We show that this requirement can be converted to a form of conventional constraints in deterministic optimization.

Convex uncertainty model of moments
Let U µ ⊂ R n and U Σ ⊂ S n + denote the uncertainty sets, i.e., the sets of all possible realizations, of µ and Σ, respectively. Namely, we only know that µ and Σ satisfy Assume that U µ and U Σ are compact convex sets. For notational simplicity, we write Recall that we are considering the reliability constraint in (5) with a linearly approximated constraint function. The robust counterpart of (5) against uncertainty in µ and Σ is formulated as i.e., we require that the reliability constraint should be satisfied for any normal distribution corresponding to possible realizations of µ and Σ. This requirement is equivalently rewritten as That is, the reliability constraint should be satisfied in the worst case.
The following theorem presents, with the aid of Theorem 2.1 and Theorem 2.2, an equivalent reformulation of (14). (14) if and only if there exists a pair of z ∈ R and Λ ∈ S n satisfying Proof. It follows from Theorem 2.1 that (14) is equivalent to Furthermore, application of Theorem 2.2 yields In the expression above, we see that U is compact and convex, and the feasible set for the minimization is convex. Also, the objective function is linear in µ and Σ for fixed z and Λ, and is linear in z and Λ for fixed µ and Σ. Therefore, the minimax theorem [8,Theorem 8.8] asserts that (18) is equivalent to This inequality holds if and only if there exists a feasible pair of z ∈ R and Λ ∈ S n satisfying which concludes the proof.
The conclusion of Theorem 3.1 is quite abstract in the sense that concrete forms of U µ and U Σ are not specified. To use this result into design optimization in practice, we have to reduce max{∇g( (15) to tractable forms. This is actually performed in section 3.2 and section 3.3, where we consider two specific models of U µ and U Σ .

Uncertainty model with ∞ -norm
Letμ ∈ R n andΣ ∈ S n denote the best estimates of µ and Σ, respectively, whereΣ is positive definite. In this section, we specialize the results of section 3.1 to the case that the uncertainty sets are given as Here, z 1 ∈ R m and Z 2 ∈ S k are unknown vector and matrix reflecting the uncertainty in µ and Σ, respectively, A ∈ R n×m and B ∈ R n×k are constant matrices, and α and β are nonnegative parameters representing the magnitude of uncertainties.
This means that the expected value vector µ belongs to a hypercube centered at the origin, with edges parallel to the axes and with an edge length of 2α. In other words, each component µ j of µ can take any value in [−α, α]. Similarly, a simple example of the uncertainty set in (20) is the one with B = I and k = n, i.e., This means that, roughly speaking, the variance-covariance matrix Σ has componentwise uncertainty. More precisely, for each i, j = 1, . . . , n we havẽ and besides Σ should be positive semidefinite. It is worth noting that, even ifΣ and β satisfyΣ − β11 0 andΣ + β11 0 (here, 1 denotes an all-ones column vector), (21) does not necessarily imply Σ 0. Indeed, as for an example with n = 2, consider Then we haveΣ − β11 = 1 0 0 1 0,Σ + β11 = 5 4 4 5 0, and, for example, we see that To derive the main result in this section stated in Theorem 3.2, we need the two technical lemmas. Lemma 3.1 explicitly computes the value of max{∇g(x) µ | µ ∈ U µ } in (15). Lemma 3.2 converts max{Σ • Λ | Σ ∈ U Σ } to a tractable form.
Proof. Substitution of (19) into the left side yields It is known that the dual norm of the ∞ -norm is the 1 -norm [7, appendix A.1.6], i.e., max t∈R n {s t | t ∞ ≤ 1} = s 1 .
Therefore, we obtain which concludes the proof. Proof. We shall show that the right side corresponds to the dual problem of the SDP problem on the left side. Therefore, this proposition follows from the strong duality of SDP [8, section 11.3], because the dual problem is strictly feasible.
As preliminaries, for a convex cone defined by K = {(s, S) ∈ R × S k | S 1,1 ≤ s}, observe that its dual cone is given by [7,Example 2.25] By using definition (20) of U Σ , the left side of the proposition of this theorem is reduced to The Lagrangian of this optimization problem is defined by where v ∈ R, V ∈ S k , and Ω ∈ S n are the Lagrange multipliers. Indeed, by using (11) and (22), we can confirm that problem (23) is equivalent to The dual problem is then defined by Since (24) can be rewritten as Therefore, the dual problem in (25) is explicitly written as follows: Minimize v∈V, Λ∈S k , Ω∈S kΣ Constraint B (Λ+Ω)B 1,1 ≤ v becomes active at an optimal solution, which concludes the proof.
We are now in position to state the main result of this section. By using Theorem 3.1, Lemma 3.1, and Lemma 3.2, we obtain the following fact.
Theorem 3.2. Let U µ and U Σ be the sets defined by (19) and (20), respectively. Then, x ∈ X satisfies (14) if and only if there exists a pair of z ∈ R and W ∈ S n satisfying Proof. It follows from Lemma 3.1 and Lemma 3.2 that (15) and (16) in Theorem 3.1 are equivalently rewritten as Put W = Λ + Ω to see that this is reduced to This is straightforwardly equivalent to (26) and (27).
It should be emphasized that Theorem 3.2 converts the set of infinitely many reliability constraints in (13) to two deterministic constraints, i.e., (26) and (27). The latter constraints can be handled within the framework of conventional (deterministic) optimization.

Uncertainty model with 2 -norm
In this section, we consider the uncertainty sets defined by Example 3.2. As a simple example, putμ = 0 and A = I with m = n to obtain This means that the expected value vector µ belongs to a hypersphere centered at the origin with radius α. Similarly, putting B = I and k = n we obtain This means that the variance-covariance matrix Σ satisfies and is symmetric positive semidefinite.
In a manner parallel to the proofs of Lemma 3.1 and Lemma 3.2, we can obtain Here, the facts have been used. Accordingly, analogous to Theorem 3.2, we obtain the following conclusion:x ∈ X satisfies (14) if and only if there exists a pair of z ∈ R and W ∈ S n satisfying

Truss optimization under compliance constraint
In this section, we present how the results established in the preceding sections can be employed for a specific RBDO problem. As a simple example, we consider a reliability constraint on the compliance under a static external load. We assume linear elasticity and small deformation. For ease of comprehension, consider design optimization of a truss. In this context, x j denotes the cross-sectional area of truss member j (j = 1, . . . , n), where n is the number of members. We attempt to minimize the structural volume of the truss, c x, under the compliance constraint, where c j denotes the undeformed member length. Let π(x) denote the compliance corresponding to a static external load. The first-order approximation of the compliance constraint is written as whereπ (> 0) is a specified upper bound for the compliance. Accordingly, the design optimization problem to be solved is formulated as follows: Here, the specified lower bound for the member cross-sectional area, denoted byx j (j = 1, . . . , n), is positive, because in this paper we restrict ourselves to optimization problems without variation of structural topology. As for uncertainty sets of the moments, consider, for example, U µ and U Σ studied in section 3.3. For simplicity put A = B = I so as to obtain From the result in section 3.3, we see that problem (28) is equivalently rewritten as follows: Here,x ∈ R n , z ∈ R, and W ∈ S n are variables to be optimized. It is worth noting that problem (29) is a nonlinear SDP problem. The remainder of this section is devoted to presenting a method for solving problem (29) that will be used for the numerical experiments in section 5.
The method sequentially solves SDP problems that approximate problem (29), in a fashion similar to sequential SDP methods for nonlinear SDP problems [32,48]. Letx k denote the incumbent solution obtained at iteration k − 1. Define h k ∈ R n by h k = ∇π(x k ).
At iteration k, we replace ∇π(x) in (29c) and (29d) with h k . Moreover, to deal with π(x) in (29c), we use the fact that s ∈ R satisfies π(x) ≤ s if and only if is satisfied [27, section 3.1], where K(x) ∈ S d is the stiffness matrix of the truss, p ∈ R d is the external load vector, and d is the number of degrees of freedom of the nodal displacements. It is worth noting that, for trusses, K(x) is linear inx. Therefore, (30) is a linear matrix inequality with respect tox and s, and hence can be handled within the framework of (linear) SDP. By this means, we obtain the following subproblem that is solved at iteration k for updatingx k tox k+1 : Since this is a linear SDP problem, we can solve this problem efficiently with a primaldual interior-point method [1].

Extensions
This section discusses some extensions of the results obtained in section 3.

Robustness against uncertainty in distribution type
An important extension is that the obtained results can be applied to the case that, not only the moments, but also the type of probability distribution are unknown. In this case, we consider any combination of all types of probability distributions and all possible moments (expected value vectors and variance-covariance matrices) in the uncertainty set, and require that the failure probability is no greater than a specified value. This robustness against uncertainty in distribution type is important as the input distribution in practice is not necessarily known to be a normal distribution. Recall that, in section 2 and section 3, we assumed that the design variables, x, follows a normal distribution. Then we consider the robust reliability constraint in (14). For the sake of clarity, we restate this problem setting in a slightly different manner. We have assumed that random vector ζ can possibly follow any normal distribution satisfying µ ∈ U µ and Σ ∈ U Σ . We use P N to denote the set of such normal distributions, i.e., In other words, P N is the set of all possible realizations of the input distribution. We write p ∈ P N if p is one of such realizations. With this new notation, (14) can be rewritten equivalently as For U µ and U Σ defined in section 3.2, Theorem 3.2 shows that (33) is equivalent to (26) and (27).
We are now in position to consider any type of probability distribution. Only what we assume is that the input distribution satisfies µ ∈ U µ and Σ ∈ U Σ , where, for a while, we consider U µ and U Σ defined in section 3.2. We use P to denote the set of such distributions, i.e., Then, instead of (33), we consider the following constraint: That is, we require that the reliability constraint should be satisfied for any input distribution p satisfying p ∈ P. A main assertion of this section is that, by simply setting (35) is equivalent to (26) and (27) in Theorem 3.2.
We can show this fact in the following manner. Let P(µ, Σ) denote the set of probability distributions, the expected value vector and the variance-covariance matrix of which are µ and Σ, respectively. Observe that, with P(µ, Σ), (35) can be rewritten equivalently as With relation to the inner supremum, consider the condition sup p∈P(µ,Σ) El Ghaoui et al. [12,Theorem 1] show that (38) holds if and only if (7) of Theorem 2.1 holds with κ defined by (36). Therefore, all the subsequent results established in section 2 and section 3 hold by simply replacing the value of κ with the one in (36). Thus, the robust reliability constraint with unknown distribution type is also reduced to the form in (26) and (27)

Multiple constraints
In section 2 and section 3, we have restricted ourselves to the case that the design optimization problem has a single performance requirement, (1). In this section, we discuss treatment of multiple constraints.
Suppose that the performance requirement is written as The first-order approximation yields where g lin i (x) = g i (x)+∇g i (x) ζ (i = 1, . . . , m). Suppose that we impose a distributionallyrobust reliability constraint for each i = 1, . . . , m independently, i.e., Here, P is the set of possible realizations of the input distribution (i.e., P here is either P N in (32) or P in (34)). It is worth noting that in (39) the worst case distributions are considered independently for each i = 1, . . . , m. Constraint (39) can be straightforwardly dealt with in the same manner as section 3.
In contrast, suppose that we consider a single (i.e., common) worst-case distribution for all i = 1, . . . , m. Then the distributionally-robust reliability constraint is written as Treatment of this constraint remains to be studied as future work. It is worth noting that constraint (39) is conservative compared with constraint (40).

Numerical examples
In section 3.4 we have seen that an optimization problem of trusses under the compliance constraint is reduced to problem (29). In this section we solve this optimization problem numerically.

Example (I): 2-bar truss
Consider a plane truss depicted in Figure 1. The truss has n = 2 members and d = 2 degrees of freedom of the nodal displacements. The elastic modulus of the members is 20 GPa. A vertical external force of 100 kN is applied at the free node. The upper bound for the compliance isπ = 100 J.
We first consider the uncertainty model with the ∞ -norm, studied in section 3.2. In the uncertainty model in (19) and (20)  The magnitude of uncertainty is α = 0.2 and β = 0.01. The specified upper bound for the failure probability is = 0.01. The optimal solution obtained by the proposed method is listed in the row " ∞ -norm unc." of Table 1, where "obj. val." means the objective value at the obtained solution. For comparison, the optimal solution of the nominal optimization problem (i.e., the conventional structural volume minimization under the compliance constraint without considering uncertainty) is also listed. The optimization result was verified as follows. We randomly generate µ ∈ U µ and Σ ∈ U Σ , and then generate 10 6 samples drawn as ζ ∼ N(µ, Σ). Figure 2a and Figure 2b show the samples of x =x + ζ generated in this manner. Figure 2c shows the values of the linearly approximated constraint function, for these samples. Therefore, the ratio of the number of samples of which these function  values are positive divided by the number of all samples (i.e., 10 6 ) should be no greater than (= 0.01). We computed this ratio for each of 10 4 randomly generated samples of µ ∈ U µ and Σ ∈ U Σ , where the continuous uniform distribution was used to generate samples of the components of µ and Σ. Figure 3a shows the histogram of the values of this ratio computed in this manner, i.e., it shows distribution of the failure probability estimated by double-loop Monte Carlo simulation. It is observed in Figure 3a that, for every one of 10 4 probability distribution samples, the failure probability is no greater than . Thus, it is verified that the obtained solution satisfies the distributionally-robust reliability constraint in (14). Indeed, among these samples of the failure probability, the maximum value is 0.009054 (< ). For reference, Figure 3b shows the histogram of failure probabilities computed for the constraint function values without applying the linear approximation, i.e., g(x) = π(x) −π. It is observed in Figure 3b that, in only rare cases, the failure probability exceeds the target value (= 0.01). Figure 4a shows the variation of the optimal value with respect to the upper bound for the failure probability , where α = 0.2 and β = 0.01 are fixed. As decreases, the optimal value increases.
In contrast, Figure 4b shows the variation of the optimal value with respect to α and β, where = 0.01 is fixed. Although in Figure 4b only values of α are shown, values of β ∈ [0, 0.02] are also varied in a manner proportional to α. As the magnitude of uncertainty increases, the optimal value increases. We next consider the uncertainty model with the 2 -norm, studied in section 3.3. The uncertainty set is defined with A, B,μ,Σ, α, and β used above. The specified upper bound for the failure probability is = 0.01. The obtained optimal solution is listed in the row " 2 -norm unc." of Table 1. It can be observed that the objective value is small compared with the solution with the ∞ -norm uncertainty model. This is natural, because, with the common values of α and β, the uncertainty set with the 2 -norm is included in the uncertainty set with the ∞ -norm. The optimization result is verified in the same manner as above. Namely, Figure 5 shows 10 4 samples of the failure probability, each of which was computed with 10 6 samples of ζ. Among these samples, the maximum failure probability is 0.009850 (< ), which verifies that the obtained solution satisfies distributionally-robust reliability constraint (14). Figure 6a and Figure 6b show the variations of the optimal value with respect to the failure probability, , and the magnitude of uncertainty, α and β, respectively. These variations show trends similar to the ones with the ∞ -norm uncertainty model in Figure 4a and Figure 4b.
Finally, as discussed in section 4.1, we consider, not only the normal distributions, but all the probability distributions with µ and Σ belonging to the uncertainty set. That is, the set of possible realizations of probability distributions is given by (34). Figure 7 collects the variations of the optimal value with respect to the failure probability, , and the magnitude of uncertainty, α and β (in the same manner as above, values of β ∈ [0, 0.02] are varied in a manner proportional to α). Compared with the results for normal distributions in Figure 4 and Figure 6, the optimal value in Figure 7 is large, as expected. Moreover, as decreases, the optimal value in Figure 7a

Example (II): 29-bar truss
Consider a plane truss depicted in Figure 8, where n = 29 and d = 20. The elastic modulus of the members is 20 GPa. Vertical external forces of 100 kN are applied at two nodes as shown in Figure 8. The upper bound for the compliance isπ = 1000 J. The lower bounds for the member cross-sectional areas arex j = 200 mm 2 (j = 1, . . . , n).
As for the uncertainty model, we consider both the model with the 2 -norm, putting  where 1 ∈ R n is an all-ones column vector. The magnitude of uncertainty is α = 0.2 and β = 0.01. The specified upper bound for the failure probability is = 0.01.
The optimization results obtained by the proposed method are listed in Table 2. Figure 9a and Figure 9b show the variations of the optimal value with respect to the failure probability and the magnitude of uncertainty, respectively.
As done in section 4.1, we next require that the reliability constraint should be satisfied for all the probability distributions satisfying µ ∈ U µ and Σ ∈ U Σ , i.e., for any probability distribution belonging to P in (34). For the 2 -norm uncertainty, Figure 10a and Figure 10b report the variations of the optimal value with respect to the failure probability and the magnitude of uncertainty, respectively. Figure 11 collects the optimal solutions of the optimization problem without uncertainty, as well as the distributionally-robust RBDO problems with the two uncertainty models. Here, the width of each member in the figures are proportional to its cross-sectional area.

Conclusions
This paper has dealt with reliability-based design optimization (RBDO) of structures, in which knowledge of the input distribution that the design variables follow is imprecise. Specifically, we only know that the expected value vector and the variance-covariance matrix of the input distribution belong to a specified convex set, and do not know their true values. Then we attempt to optimize a structure, under the constraint that, even for the worst-case input distribution, the failure probability of the structure is no greater than the specified value. This constraint, called the distributionally-robust reliability constraint, is equivalent to infinitely many reliability constraints corresponding to all possible realizations of the input distribution. Provided that change of a constraint function value is well approximated as a linear function of uncertain perturbations of the design variables, this paper has presented a tractable reformulation of the distributionally-robust reliability constraint. This paper has established the concept of distributionally-robust RBDO, and developed fundamental results. Much remains to be studied. For instance, in this paper we have considered uncertainty only in the design variables. Other sources of uncertainty in structural optimization can be explored. Also, as discussed in section 4.2, multiple performance requirement in the form of (40) remains to be studied. Extension to topology optimization is of great interest. Moreover, this paper relies on the assumption that quantity of interest is approximated, with sufficient accuracy, as a linear function of uncertainty perturbations of the design variables. Extension to nonlinear cases can be attempted. Finally, development of a more efficient algorithm for solving the optimization problem presented in this paper can be studied.