Bayesian optimization for mixed-variable, multi-objective problems

Abstract

Optimizing multiple, non-preferential objectives for mixed variable, expensive black-box problems is important in many areas of engineering and science. The expensive, noisy, black-box nature of these problems makes them ideal candidates for Bayesian optimization (BO). mixed variable and multi-objective problems, however, are a challenge due to BO’s underlying smooth Gaussian process surrogate model. Current multi-objective BO algorithms cannot deal with mixed variable problems. We present MixMOBO, the first mixed variable, multi-objective Bayesian optimization framework for such problems. Using MixMOBO, optimal Pareto-fronts for multi-objective, mixed variable design spaces can be found efficiently while ensuring diverse solutions. The method is sufficiently flexible to incorporate different kernels and acquisition functions, including those that were developed for mixed variable or multi-objective problems by other authors. We also present HedgeMO, a modified Hedge strategy that uses a portfolio of acquisition functions for multi-objective problems. We present a new acquisition function, SMC. Our results show that MixMOBO performs well against other mixed variable algorithms on synthetic problems. We apply MixMOBO to the real-world design of an architected material and show that our optimal design, which was experimentally fabricated and validated, has a normalized strain energy density $10^4$ times greater than existing structures.

Appendix A: Benchmark Test Functions

In this section, we define the benchmark test functions, all of which are set to be maximized during our optimizations.

1.1 Contamination Problem

The contamination problem was introduced by Hu et al. (2011) and is used to test categorical variables with binary categories. The problem aims to maximize the reward function for applying a preventative measure to stop contamination in a food supply chain with D stages. At each $i^{th}$ stage, where $i\in [1,D]$, decontamination efforts can be applied. However, this effort comes at a cost c and will decrease the contamination by a random rate $\Gamma _i$. If no prevention effort is taken, the contamination spreads with a rate of $\Omega _i$. At each stage i, the fraction of contaminated food is given by the recursive relation:

$$\begin{aligned} Z_i=\Omega _i(1-w_i)(1-Z_{i-1}) + (1-\Sigma _i w_i) Z_{i-1} \end{aligned}$$

(A1)

here $w_i\in {0,1}$ and is the decision variable to determine if preventative measures are taken at $i^{th}$ stage or not. The goal is to decide which stages i action should be taken to make sure $Z_i$ does not exceed an upper limit $U_i$. $\Omega _i$ and $\Sigma _i$ are determined by a uniform distribution. We consider the problem setup with Langrangian relaxation (Baptista and Poloczek 2018):

$$\begin{aligned} f({\vec {w}})= -\sum _{i=1}^D\left( cw_i+\frac{\rho }{T}\sum _{k=1}^T1_{\{Z_k>U_i\}}\right) - \lambda ||\vec {w}||_1 \end{aligned}$$

(A2)

Here violation of $Z_k<U_i$ is penalized by $\rho =1$ and summing the contaminated stages if the limit is violated and our total stages or dimensions are $D=21$. The cost c is set to be 0.2 and $Z_1=0.01$. As in the setup for Baptista and Poloczek (2018), we use $T=100$ stages, $U_i=0.1$, $\lambda =0.01$ and $\epsilon =0.05$.

1.2 Encrypted Amalgamated

Analytic test functions generally cannot mimic mixed variables. To map the continuous output of a function into N discrete ordinal or categorical variables, the continuous range of the test function’s output is first discretized into N discrete subranges by selecting $(N-1)$ break points, often equally spaced, within the bounds of the range. Then, the continuous output variable is assigned the integer round-off value of the subrange defined by its surrounding pair of break points. If necessary, the domain of the test function’s output is first mapped into a larger domain so that each subrange has a unique integer value. To mimic ordinal variables, we are done, but for categorical variables, a random vector for each categorical variable is then generated which scrambles or ‘encrypts’ the indices of these values, thus creating random landscapes as is the case for categorical variables with a latent space. The optimization algorithm only sees the encrypted space and the random vector is only used when evaluating the black-box function.

We also define a new test function that we call the Amalgamated function, a piece-wise function formed from commonly used analytical test functions with different features (for more details on these functions we refer to Tušar et al. (2019)). The Amalgamated function is non-convex and anisotropic, unlike conventional test functions where isotropy can be exploited.

For $i=1...n$, $k=$mod$(i-1,7)$:

$$\begin{aligned} f(\vec {w})=\sum _{i=1}^{D}g(w_i) \end{aligned}$$

(A3)

where

$$\begin{aligned} g(w_i)= \left\{ \begin{array}{ll} sin(w_i) &{} \text {if}\, k=0, \ w_i\in (0,\pi )\\ -\frac{w_i^4-16w_i^2+5w_i}{2} &{} \text {if}\, k=1, \ w_i\in (-5,5)\\ -(w_i^2) &{} \text {if}\, k=2, \ w_i\in (-10,10)\\ -[10+w_i^2-10cos(2\pi w_i)] &{} \text {if}\, k=3, \ w_i\in (-5,5)\\ -[100(w_i-w_{i-1}^2)^2+(1-w_i)^2] \quad \qquad &{} \text {if}\, k=4, \ w_i\in (-2,2)\\ abs(cos(w_i)) &{} \text {if}\, k=5, \ w_i\in (-\pi /2,\pi /2)\\ - w_i &{} \text {if}\, k=6, \ w_i\in (-30,30) \end{array} \right. \end{aligned}$$

(A4)

To create the Encrypted Amalgamated function, for categorical and ordinal variables, equally spaced points are taken within the bounds defined above. For our current work, we use a $D=13$ with 8 categorical and 3 ordinal variables with 5 states each, and 2 continuous variables.

1.3 NK Landscapes

NK Landscapes were introduced by Kauffman and Levin (1987) as a way of creating optimization problems with categorical variables. N describes the number of genes or number of dimensions D and K is the number of epistatic links of each gene to other genes, which describes the ‘ruggedness’ of the landscape. A large number of random landscapes can be created for given N and K values. The global optimum of a generated landscape for experimentation can only be computed through complete enumeration. The landscape cost for any vector is calculated as an average of each component cost. Each component cost is based on the random values generated for the categories, not only by its own alleles, but also by the alleles in the other genes connected through the random epistasis matrix, with K probability or ruggedness. A $K=1$ ruggedness translates to a fully connected genome.

The NK Landscapes from Kauffman and Levin (1987) were formulated only for binary variables. They were extended by Li et al. (2006) for multi-categorical problems, which is the formulation we use. Details of the NK Landscape test-functions we use can be found in Li et al. (2006). For the current study, we use $N=8$ with 4 categories each and ruggedness $K=0.2$.

1.4 Rastringin

Rastringin function is a commonly used non-convex optimization function (Tušar et al. 2019) with a large number of local optima. It is defined as:

$$\begin{aligned} f(\vec {w})= -[10+w_i^2-10cos(2\pi w_i)], \ w_i\in (-5,5) \end{aligned}$$

(A5)

We use $D=9$ for testing with 6 ordinal with 5 discrete states and 3 continuous variables. The ordinal variables are equally spaced within the bounds.

1.5 Encrypted Syblinski-Tang

We use the Syblinski-Tang function (Tušar et al. 2019), an isotropic non-convex function. The function is considered difficult to optimize because many search algorithms get ‘stuck’ at a local optimum. For use with categorical variables, we encrypt it as described previously. The Syblinski-Tang function, in terms of input vector ${\vec {w}}$, is defined as:

$$\begin{aligned} f(\vec {w})= -\frac{\sum _{i=1}^{D}w_i^4-16w_i^2+5w_i}{2}, \ w_i\in (-5,2.5) \end{aligned}$$

(A6)

For the current study, this function was tested with $D=10$ categorical variables and 5 categories for each variable.

1.6 Encrypted ZDT6

ZDT benchmarks are a suite of multi-objective problems, suggested by Zitzler et al. (2000), and most commonly used for testing such problems. We use ZDT6, which is non-convex and non-uniform in its parameter space. We again modify the function by encrypting it to work with categorical problems. ZDT6 is defined as:

$$\begin{aligned} \ \begin{aligned} f_1(\vec {w})&=exp(-4w_1)sin^6(6 \pi w_1)-1 \\ f_2(\vec {w})&=-g(\vec {w})\left[ 1-(f_1(\vec {w})/g(\vec {w}))^{2}\right] \\ g(\vec {w})&=1+9\left[ \left( \sum _{i=2}^Dw_i\right) /(n-1)\right] ^{1/4} \end{aligned} \end{aligned}$$

(A7)

Here $w_1 \in [0,1]$ and $w_i =0$ for $i = 2,\dots ,D$. The function was tested for $D=10$ with 5 categories each. We note that to evaluate the performance of MixMOBO, we compared it against the NSGA-II variant Deb et al. (2002) that can deal with mixed variables (by running ZDT4 in a mixed variable setting and ZDT6 with categorical variables). No encryption is necessary for GAs. GAs required, on average, $10^2$ more function calls compared to MixMOBO.