A Bayesian approach to constrained single- and multi-objective optimization

Abstract

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria—the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions—as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization.

Appendices

Appendix 1: On the bounded hyper-rectangles ${\mathbb {B}}_{\mathrm{o}}$ and ${{\mathbb {B}}}_{\mathrm{c}}$

We have assumed in Sect. 3 that ${\mathbb {B}}_{\mathrm{o}}$ and ${{\mathbb {B}}}_{\mathrm{c}}$ are bounded hyper-rectangles; that is, sets of the form

$$\begin{aligned} {\mathbb {B}}_{\mathrm{o}}&= \bigl \{ y \in {\mathbb {Y}}_{\mathrm{o}};\; y^{\mathrm{low}}_\mathrm{o}\le y \le y^{\mathrm{upp}}_\mathrm{o}\bigr \},\\ {{\mathbb {B}}}_{\mathrm{c}}&= \bigl \{ y \in {\mathbb {Y}}_{\mathrm{c}};\; y^{\mathrm{low}}_\mathrm{c}\le y \le y^{\mathrm{upp}}_\mathrm{c}\bigr \}, \end{aligned}$$

for some $y^{\mathrm{low}}_{\mathrm{o}}$, $y^{\mathrm{upp}}_{\mathrm{o}} \in {\mathbb {Y}}_{\mathrm{o}}$ and $y^{\mathrm{low}}_\mathrm{c}$, $y^{\mathrm{upp}}_\mathrm{c}\in {\mathbb {Y}}_{\mathrm{c}}$, with the additional assumption that $y^{\mathrm{low}}_{\mathrm{c}, j} < 0 < y^{\mathrm{upp}}_{\mathrm{c}, j}$ for all $j \le q$. Remember that upper bounds only where required in the unconstrained case discussed in Sect. 2.2. To shed some light on the role of these lower and upper bounds, let us now compute the improvement $I_1(X_1) = \left| H_1 \right|$ brought by a single evaluation.

If $X_1$ is not feasible, then

$$\begin{aligned} \left| H_1 \right| = \left| {\mathbb {B}}_{\mathrm{o}}\right| \;\cdot \; \prod _{j = 1}^q \left( y^{\mathrm{upp}}_{\mathrm{c},j} - y^{\mathrm{low}}_{\mathrm{c},j} \right) ^{\gamma _j} \left( y^{\mathrm{upp}}_{\mathrm{c},j} - \xi _{\mathrm{c},j}(X_1) \right) ^{1 - \gamma _j} \end{aligned}$$

(34)

where $\gamma _j = {\mathbbm {1}}_{\xi _{\mathrm{c},j}(X_1) \le 0}$. It is clear from the right-hand side of (34) that both ${\mathbb {B}}_{\mathrm{o}}$ and ${{\mathbb {B}}}_{\mathrm{c}}$ have to be bounded if we want $\left| H_1 \right| < +\infty$ for any $\gamma = \left( \gamma _1,, \ldots ,\, \gamma _q \right) \in \{ 0, 1 \}^q$. Note, however, that only the volume of ${\mathbb {B}}_{\mathrm{o}}$ actually matters in this expression, not the actual values of $y^{\mathrm{low}}_\mathrm{o}$ and $y^{\mathrm{upp}}_\mathrm{o}$. Equation (34) also reveals that the improvement is a discontinuous function of the observations: indeed, the jth term in the product jumps from $y^{\mathrm{upp}}_{\mathrm{c},j}$ to $y^{\mathrm{upp}}_{\mathrm{c},j} - y^{\mathrm{low}}_{\mathrm{c},j} > y^{\mathrm{upp}}_{\mathrm{c},j}$ when $\xi _{\mathrm{c},j}(X_1)$ goes from $0^+$ to 0. The increment $- y^{\mathrm{low}}_{\mathrm{c},j}$ can be thought of as a reward associated to finding a point which is feasible with respect to the jth constraint.

The value of $\left| H_1 \right|$ when $X_1$ is feasible is

$$\begin{aligned} \left| H_1 \right| = \left| {\mathbb {B}}_{\mathrm{o}}\right| \,\cdot \, \left( \left| {{\mathbb {B}}}_{\mathrm{c}}\right| - \left| {{\mathbb {B}}}_{\mathrm{c}}^{-}\right| \right) \;+\; \prod _{j \le p} \left( \min \left( \xi _{\mathrm{o},j}(X_1), y^{\mathrm{upp}}_{\mathrm{o},j} \right) - \max \left( \xi _{\mathrm{o},j}(X_1), y^{\mathrm{low}}_{\mathrm{o},j} \right) \right) \,\cdot \, \left| {{\mathbb {B}}}_{\mathrm{c}}^{-}\right| , \end{aligned}$$

(35)

where $\left| {{\mathbb {B}}}_{\mathrm{c}}^{-}\right| = \prod _{j=1}^q \left| y^{\mathrm{low}}_{\mathrm{c},j} \right|$ is the volume of the feasible subset of ${{\mathbb {B}}}_{\mathrm{c}}$, ${{\mathbb {B}}}_{\mathrm{c}}^{-}= {{\mathbb {B}}}_{\mathrm{c}}\cap ] -\infty ; 0 ]^q$. The first term in the right-hand side of (35) is the improvement associated to the domination of the entire unfeasible subset of ${\mathbb {B}}= {\mathbb {B}}_{\mathrm{o}}\times {{\mathbb {B}}}_{\mathrm{c}}$; the second term measures the improvement in the space of objective values.

Appendix 2: An adaptive procedure to set ${\mathbb {B}}_{\mathrm{o}}$ and ${{\mathbb {B}}}_{\mathrm{c}}$

This section describes the adaptive numerical procedure that is used, in our numerical experiments, to define the hyper-rectangles ${\mathbb {B}}_{\mathrm{o}}$ and ${{\mathbb {B}}}_{\mathrm{c}}$. As said in Sect. 3.3, these hyper-rectangles are defined using estimates of the range of the objective and constraint functions, respectively. To this end, we will use the available evaluations results, together with posterior quantiles provided by our Gaussian process models on the set of candidate points ${\mathcal {X}}_n$ (defined in Sect. 4.2).

More precisely, assume that n evaluation results $\xi (X_i)$, $1 \le i \le n$, are available. Then, we define the corners of ${\mathbb {B}}_{\mathrm{o}}$ by

$$\begin{aligned} \left\{ \begin{array}{lcl} y^{\mathrm{low}}_{\mathrm{o},i,n} &{}=&{} \min \left( \min _{i \le n} \xi _{\mathrm{o},i}(X_i),\; \min _{x \in {\mathcal {X}}_n} \widehat{\xi }_{\mathrm{o},\,i,\,n}(x)-\lambda _\mathrm{o}\sigma _{\mathrm{o},\,i,\,n}(x) \right) , \\ y^{\mathrm{upp}}_{\mathrm{o},i,n} &{}=&{} \max \left( \max _{i \le n} \xi _{\mathrm{o},i}(X_i),\; \max _{x \in {\mathcal {X}}_n} \widehat{\xi }_{\mathrm{o},\,i,\,n}(x)+\lambda _\mathrm{o}\sigma _{\mathrm{o},\,i,\,n}(x) \right) , \end{array} \right. \end{aligned}$$

(36)

for $1 \le i \le p$, and the corners of ${{\mathbb {B}}}_{\mathrm{c}}$ by

$$\begin{aligned} \left\{ \begin{array}{lcl} y^{\mathrm{low}}_{\mathrm{c},j,n} &{}=&{} \min \left( 0,\; \min _{i \le n} \xi _{\mathrm{c},j}(X_i),\; \min _{x \in {\mathcal {X}}_n} \widehat{\xi }_{\mathrm{c},\,j,\,n}(x) - \lambda _\mathrm{c}\sigma _{\mathrm{c},\,j,\,n}(x) \right) , \\ y^{\mathrm{upp}}_{\mathrm{c},j,n} &{}=&{} \max \left( 0,\; \max _{i \le n} \xi _{\mathrm{c},j}(X_i),\; \max _{x \in {\mathcal {X}}_n} \widehat{\xi }_{\mathrm{c},\,j,\,n}(x) + \lambda _\mathrm{c}\sigma _{\mathrm{c},\,j,\,n}(x) \right) , \end{array} \right. \end{aligned}$$

(37)

for $1 \le j \le q$, where $\lambda _\mathrm{o}$ and $\lambda _\mathrm{c}$ are positive numbers.

Table 11 Number of evaluations to find a first feasible point for the COBYLA, active-set, interior-point and SQP local optimization algorithms

Table 12 Number of evaluations to reach the target for the COBYLA, active-set, interior-point and SQP local optimization algorithms

Table 13 Number of evaluations to find a first feasible point for the COBRA-Local, COBRA-Global and Extended-ConstrLMSRBF optimization algorithms

Table 14 Number of evaluations to reach the target for the COBRA-Local, COBRA-Global and Extended-ConstrLMSRBF optimization algorithms

Appendix 3: Mono-objective benchmark result tables

In Sect. 5.3, only the best results for both the “Local” and the “Regis” groups of algorithms were shown. In Appendix 3, we present the full results. Tables 11 and 12, and Tables 13 and 14 present respectively the results obtained with the local optimization algorithms and the results obtained by Regis [61] on the single-objective benchmark test problems (see Table 1). Tables 11 and 12 show the performances for finding feasible solutions and for reaching the targets specified in Table 1 for the COBYLA, Active-Set, Interior-Point and SQP algorithms. Similarly, Tables 13 and 14 show the performances for finding feasible solutions and for reaching the targets for the COBRA-Local, COBRA-Global and Extended-ConstrLMSRBF algorithms of Regis [61].

Appendix 4: Modified g3mod, g10 and PVD4 test problems

We detail here the modified formulations of the g3mod, g10 and PVD4 problems that were used in Sect. 5.3 to overcome the modeling problems with BMOO. Our modifications are shown in boldface. The rationale of the modifications is to smooth local jumps.

• modified-g3mod problem

  $$\begin{aligned} \left\{ \begin{array}{lcl} f(x) &{}=&{} -\text {plog}((\sqrt{d})^d{\prod }_{i=1}^d x_i)^{\mathbf{0.1}}\\ c(x) &{}=&{} ({\sum }_{i=1}^d x_i^2) - 1 \end{array} \right. \end{aligned}$$

• modified-g10 problem

  $$\begin{aligned} \left\{ \begin{array}{lcl} f(x) &{}=&{} x_1 + x_2 + x_3\\ c_1(x) &{}=&{} 0.0025(x_4+x_6) - 1\\ c_2(x) &{}=&{} 0.0025(x_5+x_7-x_4) - 1\\ c_3(x) &{}=&{} 0.01(x_8-x_5) - 1\\ c_4(x) &{}=&{} \text {plog}(100x_1 - x_1x_6 + 833.33252x_4 - 83333.333)^{\mathbf{7}}\\ c_5(x) &{}=&{} \text {plog}(x_2x_4 - x_2x_7 -1250x_4 + 1250x_5)^{\mathbf{7}}\\ c_6(x) &{}=&{} \text {plog}(x_3x_5 - x_3x_8 -2500x_5 + 1250000)^{\mathbf{7}} \end{array} \right. \end{aligned}$$

• modified-PVD4 problem

  $$\begin{aligned} \left\{ \begin{array}{lcl} f(x) &{}=&{} 0.6224x_1x_3x_4 + 1.7781x_2x_3^2 + 3.1661x_1^2x_4 + 19.84x_1^2x_3\\ c_1(x) &{}=&{} -x_1 + 0.0193x_3\\ c_2(x) &{}=&{} -x_2 + 0.00954x_3\\ c_3(x) &{}=&{} \text {plog}(-\pi x_3^2x_4 - 4/3\pi x_3^3 + 1296000)^{\mathbf{7}} \end{array} \right. \end{aligned}$$

Note that the above defined problems make use of the plog function defined below (see Regis [61]).

$$\begin{aligned} \text {plog}(x) = \left\{ \begin{array}{ll} \log (1+x) &{}\quad \text {if } x \ge 0\\ -\log (1-x) &{}\quad \text {otherwise} \end{array} \right. \end{aligned}$$