An adaptive bivariate decomposition method for interval optimization problems with multiple uncertain parameters

Abstract

A novel interval optimization method based on an adaptive bivariate decomposition algorithm is developed to solve the engineering uncertain optimization problems with multiple interval parameters. Unlike the traditional highly time-consuming nested optimization approach, the interval perturbation method-based interval optimization avoids tedious inner optimization, nonetheless, it faces huge computational challenges in the optimization issues with large uncertainties and requires derivative information that may be unavailable for complex engineering systems. To overcome these shortcomings, an adaptive bivariate decomposition method (ABDM) is proposed to compute the interval ranges of the uncertain function. In the optimization, the objective function and constraints are decomposed by ABDM into a sum of several one- and two-dimensional subsystems. The extrema of the subsystems are approximately calculated through subinterval analysis, and an adaptive convergence strategy is applied to guarantee the accuracy of the obtained bounds. Based on the interval order relation and the reliability-based possibility degree of interval, the interval uncertain optimization model is converted into a deterministic optimization one. To effectively solve the deterministic optimization model, a robust optimization solver known as lightning attachment procedure optimization is employed in the optimization algorithm. Finally, a numerical example and two engineering applications illustrate the accuracy and effectiveness of the method for uncertain optimization issues with multiple interval parameters. Moreover, the new method does not need to determine the derivative information of the uncertain objective and constraints, and thus applies to different engineering optimization issues.

Appendices

Appendix

Interval perturbation method-based interval uncertain optimization

The interval perturbation method utilizes Taylor expansion to evaluate the interval bounds of Eqs. (3) and (9). By applying the first-order Taylor expansion at the interval midpoint vector U^C, the lower and upper bounds of the uncertain objective function can be written as:

$$\begin{gathered} f^{{\text{L}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right) - \sum\limits_{j = 1}^{n} {\left| {\frac{{\partial f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)}}{{\partial U_{j} }}\Delta U_{j} } \right|} \hfill \\ f^{{\text{R}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right) + \sum\limits_{j = 1}^{n} {\left| {\frac{{\partial f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)}}{{\partial U_{j} }}\Delta U_{j} } \right|} \hfill \\ \end{gathered}$$

(37)

where $f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)$ and ${{\partial f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)} \mathord{\left/ {\vphantom {{\partial f\left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)} {\partial U_{j} }}} \right. \kern-\nulldelimiterspace} {\partial U_{j} }}$ are the specific value and the first-order partial derivative of the uncertain objective function at U^C, respectively; $\Delta U_{j} = U_{j} - U_{j}^{{\text{C}}}$. Likewise, the interval values of the jth constraint in Eq. (9) can be expressed as:

$$\begin{gathered} g_{j}^{{\text{L}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = g_{j} \left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right) - \sum\limits_{j = 1}^{n} {\left| {\frac{{\partial g_{j} \left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)}}{{\partial U_{j} }}\Delta U_{j} } \right|} \hfill \\ g_{j}^{{\text{R}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = g_{j} \left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right) + \sum\limits_{j = 1}^{n} {\left| {\frac{{\partial g_{j} \left( {{\varvec{X}},{\varvec{U}}^{{\text{C}}} } \right)}}{{\partial U_{j} }}\Delta U_{j} } \right|} \hfill \\ \end{gathered}$$

(38)

Like the proposed method, IPM-based IUO avoids time-consuming inner optimization but has two inherent shortcomings. First, its precision is high only when the nonlinearity of the function is weak or the uncertainties of interval parameter is small. Second, it requires the first-order derivative information, which is usually too complex to be derived for most practical engineering problems.

Vertex method-based interval uncertain optimization

The vertex method assumes that the interval bounds of the structural response can be reached at the vertex of the uncertain region. Hence, by substituting the ends of the interval parameters into Eq. (3), the lower and upper bounds can be formulated as:

$$\begin{gathered} f^{{\text{L}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = \min \left\{ {f\left( {{\varvec{X}},{\varvec{U}}_{V1} } \right),f\left( {{\varvec{X}},{\varvec{U}}_{V2} } \right), \ldots ,f\left( {{\varvec{X}},{\varvec{U}}_{{V2^{q} }} } \right)} \right\} \hfill \\ f^{{\text{R}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = \max \left\{ {f\left( {{\varvec{X}},{\varvec{U}}_{V1} } \right),f\left( {{\varvec{X}},{\varvec{U}}_{V2} } \right), \ldots ,f\left( {{\varvec{X}},{\varvec{U}}_{{V2^{q} }} } \right)} \right\} \hfill \\ \end{gathered}$$

(39)

where

$$\begin{gathered} {\varvec{U}}_{V1} \;{ = }\;\left[ {U_{1}^{{\text{L}}} ,U_{2}^{{\text{L}}} , \ldots ,U_{q}^{{\text{L}}} } \right] \hfill \\ {\varvec{U}}_{V2} \;{ = }\left[ {U_{1}^{{\text{L}}} ,U_{2}^{{\text{L}}} , \ldots ,U_{q}^{{\text{R}}} } \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ {\varvec{U}}_{{V2^{q} }} { = }\left[ {U_{1}^{{\text{R}}} ,U_{2}^{{\text{R}}} , \ldots ,U_{q}^{{\text{R}}} } \right] \hfill \\ \end{gathered}$$

(40)

Similarly, the upper and lower bounds of the jth uncertain constraint can be determined as:

$$\begin{gathered} g_{j}^{{\text{L}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = \min \left\{ {g_{j} \left( {{\varvec{X}},{\varvec{U}}_{V1} } \right),g_{j} \left( {{\varvec{X}},{\varvec{U}}_{V2} } \right), \ldots ,g_{j} \left( {{\varvec{X}},{\varvec{U}}_{{V2^{q} }} } \right)} \right\} \hfill \\ g_{j}^{{\text{R}}} \left( {{\varvec{X}},{\varvec{U}}} \right) = \max \left\{ {g_{j} \left( {{\varvec{X}},{\varvec{U}}_{V1} } \right),g_{j} \left( {{\varvec{X}},{\varvec{U}}_{V2} } \right), \ldots ,g_{j} \left( {{\varvec{X}},{\varvec{U}}_{{V2^{q} }} } \right)} \right\} \hfill \\ \end{gathered}$$

(41)

Clearly, ${\varvec{U}}_{V1} ,{\varvec{U}}_{V2} , \ldots ,\user2{U}_{{V2^{q} }}$ means that 2^q vertexes require to compute when there are q uncertain parameters. The calculation cost of VM-based IUO will be extremely expensive for the uncertain optimization with multiple uncertain parameters.