A performance measure approach for risk optimization

Abstract

In recent years, several approaches have been proposed for solving reliability-based design optimization (RBDO), where the probability of failure is a design constraint. The same cannot be said about risk optimization (RO), where probabilities of failure are part of the objective function. In this work, we propose a performance measure approach (PMA) for RO problems. We first demonstrate that RO problems can be solved as a sequence of RBDO sub-problems. The main idea is to take target reliability indexes (i.e., probabilities of failure) as design variables. This allows the use of existing RBDO algorithms to solve RO problems. The idea also extends the literature concerning RBDO to the context of RO. Here, we solve the resulting RBDO sub-problems using the PMA. Sensitivity expressions required by the proposed approach are also presented. The proposed approach is compared to an algorithm that employs the first-order reliability method (FORM) for evaluation of the probabilities of failure. The numerical examples show that the proposed approach is efficient and more stable than direct employment of FORM. This behavior has also been observed in the context of RBDO, and was the main reason for the development of PMA. Consequently, the proposed approach can be seen as an extension of PMA approaches to RO, which result in more stable optimization algorithms.

Appendices

Appendix A: Sensitivity of the RBDO problem

The proposed approach requires the sensitivity of the optimal cost according to target reliability indexes of RBDO problems, .i.e., ∇_βc. This information is required to update the target reliability indexes. For this purpose, we first evaluate the sensitivity considering the RBDO problem from (6)–(7), with constraints imposed directly on target reliability indexes β. In Appendix B, the expressions are extended to the case where the RBDO problem is solved with the PMA from (12)–(13).

The procedure employed to obtain the sensitivity of the optimization problem is similar to that described by Luenberger and Ye (2008). Take the optimization problem: Find $\mathbf {d} \in \mathbb {R}^{n}$ that,

$$ \min c(\mathbf{d},\mathbf{b}) $$

(60)

subject to

$$ \mathbf{g}(\mathbf{d}) \leq \mathbf{b}. $$

(61)

In the RBDO problem from (6)–(7), we have b = β.

The KKT conditions for this problem result (Luenberger and Ye 2008)

$$ \nabla_{\mathbf{d}} c + \boldsymbol{\lambda}^{T} \nabla_{\mathbf{d}} \mathbf{g} = \mathbf{0}. $$

(62)

By the chain rule, we have

$$ \nabla_{\mathbf{b}} c = \nabla_{\mathbf{d}} c \nabla_{\mathbf{b}} {\mathbf{d}} + \frac{d c}{d \mathbf{b}}, $$

(63)

where dc/db is the explicit ordinary derivative of c according to b. Assuming active constraints we also have the following:

$$ \nabla_{\mathbf{b}} \mathbf{g} = \nabla_{\mathbf{d}} \mathbf{g} \nabla_{\mathbf{b}} {\mathbf{d}} = \mathbf{I}. $$

(64)

Thus, sensitivity of the Lagrangian results (Luenberger and Ye 2008)

$$ \nabla_{\mathbf{b}} c + \boldsymbol{\lambda}^{T} \nabla_{\mathbf{b}} \mathbf{g} = \nabla_{\mathbf{d}} c \nabla_{\mathbf{b}} {\mathbf{d}} + \frac{d c}{d \mathbf{b}} + \boldsymbol{\lambda}^{T} \nabla_{\mathbf{d}} \mathbf{g} \nabla_{\mathbf{b}} {\mathbf{d}}, $$

(65)

that can be written as follows:

$$ \nabla_{\mathbf{b}} c + \boldsymbol{\lambda}^{T} \mathbf{I} = \left( \nabla_{\mathbf{d}} c + \boldsymbol{\lambda}^{T} \nabla_{\mathbf{d}} \mathbf{g} \right) \nabla_{\mathbf{b}} {\mathbf{d}} + \frac{d c}{d \mathbf{b}}. $$

(66)

From (62), we get

$$ \nabla_{\mathbf{b}} c + \boldsymbol{\lambda}^{T} = \frac{d c}{d \mathbf{b}} $$

(67)

and finally,

$$ \nabla_{\mathbf{b}} c = \frac{d c}{d \mathbf{b}} - \boldsymbol{\lambda}^{T}. $$

(68)

Note that this result is basically the same as that presented in literature (Luenberger and Ye 2008), with a correction dc/db, accounting for the explicit dependence of the objective function on b. Further mathematical requirements are as discussed in details by Luenberger and Ye (2008).

Appendix B: Sensitivity for the PMA

In order to employ the results from Appendix A for the PMA (12)–(13), we first write a single constraint of the PMA as follows:

$$ g_{i}(\mathbf{d},\mathbf{x}_{i},\beta_{i}) \leq b_{i}. $$

(69)

For active and inactive constraints, we have, respectively,

$$ \frac{\partial g_{i}}{\partial b_{j}} = \left\lbrace \begin{array}{ll} 1, & \textrm{ if } i = j \textrm{ and } g_{i} = b_{i}\\ 0, & \textrm{ otherwise } \end{array} \right.. $$

(70)

If the reliability constraint is inactive, then ∂g_i/∂β_i = 0. Otherwise,

$$ \frac{\partial g_{i}}{\partial \beta_{i}} = \frac{\partial g_{i}}{\partial b_{i}} \frac{\partial b_{i}}{\partial \beta_{i}}. $$

(71)

Consequently,

$$ \frac{\partial g_{i}}{\partial \beta_{j}} = \left\lbrace \begin{array}{ll} \partial b_{i}/\partial \beta_{i}, & \textrm{ if } i = j \textrm{ and } g_{i} = b_{i}\\ 0, & \textrm{ otherwise }\\ \end{array} \right. $$

(72)

From the sensitivity results presented in Appendix A, we have

$$ \nabla_{\mathbf{b}} c = \frac{d c}{d \mathbf{b}} - \boldsymbol{\lambda}^{T}, $$

(73)

where λ are Lagrange multipliers related to the PMA constraints. By multiplying the above sensitivity by ∇_βb, we get the following:

$$ \nabla_{\mathbf{b}} c \nabla_{\boldsymbol{\beta}} \mathbf{b} = \left( \frac{d c}{d \mathbf{b}} - \boldsymbol{\lambda}^{T} \right) \nabla_{\boldsymbol{\beta}} \mathbf{b} $$

(74)

and, from (72),

$$ \nabla_{\boldsymbol{\beta}} c = \frac{d c}{d \boldsymbol{\beta}} - \boldsymbol{\lambda}^{T} \nabla_{\boldsymbol{\beta}} \mathbf{g}. $$

(75)

From this result, we conclude that in the PMA, the sensitivity expressions from Appendix A must be corrected by ∇_βg. We thus require the derivatives of the constraints g_i according to β_i.

Considering a single constraint g = g_i, the first-order optimality condition for the inverse reliability analysis problem of (14)–(15) is known to be (Wu et al. 1990; Yi et al. 2008)

$$ \mathbf{u} = -\beta \frac{\nabla_{\mathbf{u}} g}{\Vert \nabla_{\mathbf{u}} g \Vert}. $$

(76)

We thus conclude that,

$$ \frac{\partial \mathbf{u}}{\partial \beta} = -\frac{\nabla_{\mathbf{u}} g}{\Vert \nabla_{\mathbf{u}} g \Vert} = \frac{1}{\beta} \mathbf{u}. $$

(77)

and

$$ \frac{\partial \mathbf{x}}{\partial \beta} = \nabla_{\mathbf{u}} \mathbf{x} \frac{\partial \mathbf{u}}{\partial \beta}, $$

(78)

where ∇_ux is evaluated from the transformation of (17). Note that ∇_ux can be evaluated numerically with little computational effort, if necessary. We finally have the following:

$$ \frac{\partial g}{\partial \beta} = \left( \nabla_{\mathbf{x}} g \right)^{T} \frac{\partial \mathbf{x}}{\partial \beta} = \frac{1}{\beta} \left( \nabla_{\mathbf{x}} g \right)^{T} \left( \nabla_{\mathbf{u}} \mathbf{x} \right) \mathbf{u}. $$

(79)

If X is a vector of independent normal random variables, we have as follows:

$$ \mathbf{x} = \boldsymbol{\mu} + \boldsymbol{\sigma} \mathbf{u}, $$

(80)

where μ is the vector of expected values and σ is a diagonal matrix of standard deviations of the random variables. In this case, we get the following:

$$ \nabla_{\mathbf{u}} \mathbf{x} = \boldsymbol{\sigma}, $$

(81)

$$ \mathbf{u} = \boldsymbol{\sigma}^{-1}(\mathbf{x} - \boldsymbol{\mu}). $$

(82)

and (79) results the following:

$$ \frac{\partial g}{\partial \beta} = \frac{1}{\beta} \left( \nabla_{\mathbf{x}} g \right)^{T} (\mathbf{x} - \boldsymbol{\mu}). $$

(83)