# Differential Evolution with Estimation of Distribution for Worst-Case Scenario Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Definition of Classical Optimization Problem

#### 2.2. Definition of Worst Case Scenario Optimization Problem

## 3. Algorithm Method

#### 3.1. Differential Evolution (DE)

- 1.
- Initialization: A population of $NPop$ individuals is randomly initialized. Each individual is represented by a D dimensional parameter vector, ${X}_{i,g}=({x}_{i,g}^{1},{x}_{i,g}^{2},...,{x}_{i,g}^{D})$ where $i=1,2,...,nPop$, $g=1,2,...MaxGen$, where $MaxGen$ is the maximum number of generations. Each vector component is subject to upper and lower bounds ${X}_{min}$ and ${X}_{max}$. The initial values of the ith individual are generated as:$${X}_{i}={X}_{min}+rand(0,1)\ast ({X}_{max}-{X}_{min})$$
- 2.
- Mutation: The new individual is generated by adding the weighted difference vector between two randomly selected population members to a third member. This process is expressed as:$${V}_{i,G}={X}_{r1,G}+F\ast ({X}_{r2,G}-{X}_{r3,G})$$V is the mutant vector, X is an individual, $r1,r2,r3$ are randomly chosen integers within the range of $[1,NPop]$ and $r1,r2,r3\ne i$, G corresponds to the current generation, F is the scale factor, usually a positive real number between 0.2 and 0.8. F controls the rate at which the population evolves.
- 3.
- Crossover: After mutation, the binomial crossover operation is applied. The mutant individual ${V}_{i,G}$ is recombined with the parent vector ${X}_{i,G}$, in order to generate the offspring ${U}_{i,G}$. The vectors of the offspring are inherited from ${X}_{i,G}$ or ${V}_{i,G}$ depending on a parameter called crossover probability, ${C}_{r}\in [0,1]$ as follows:$${U}_{i,G}=\left\{\begin{array}{cc}{V}_{i,G},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}rand\phantom{\rule{4.pt}{0ex}}\le \phantom{\rule{4.pt}{0ex}}{C}_{r}\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}t=random\left(i\right).\hfill \\ {X}_{i,G},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$
- 4.
- Selection: The selection operation is a competition between each individual ${X}_{i,G}$ and its offspring ${U}_{i,G}$ and defines which individual will prevail in the next generation. The winner is the one with the best fitness value. The operation is expressed by the following equation:$${X}_{i,G+1}=\left\{\begin{array}{cc}{U}_{i,G},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}f\left({U}_{i,G}\right)\phantom{\rule{4.pt}{0ex}}\le \phantom{\rule{4.pt}{0ex}}f\left({X}_{i,G}\right)\phantom{\rule{4.pt}{0ex}}.\hfill \\ {X}_{i,G},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$The above steps of mutation, crossover, and selection are repeated for each generation until a certain set of termination criteria has been met. Figure 2 shows the basic flowchart of the DE.

#### 3.2. Estimation of Distribution Algorithms (EDAs)

- 1.
- Initialization: A population is initialized randomly.
- 2.
- Selection: The most promising individuals $S\left(t\right)$ from the population $P\left(t\right)$, where t is the current generation, are selected.
- 3.
- Estimation of the probabilistic distribution: A probabilistic model $M\left(t\right)$ is built from $S\left(t\right)$.
- 4.
- Generate new individuals: New candidate solutions are generated by sampling from the $M\left(t\right)$.
- 5.
- Create new population: The new solutions are incorporated into $P\left(t\right)$, and go to the next generation. The procedure ends when the termination criteria are met.

#### 3.3. Proposed Algorithm

- 1.
- Initialization: A population of size $NPop$ is initialized according to the general DE procedure mentioned in the previous section, where the individuals are representing candidate solutions in the design space X.
- 2.
- Evaluation: To evaluate the fitness function, we need to solve the problem in the scenario space. For a fixed candidate UL solution ${X}_{i}$, the LL DE is executed. More detailed steps are given in the next paragraphs. The LL DE returns the solution corresponding to the worst-case scenario for the specific ${X}_{i}$. For each individual, the corresponding best ${Y}_{best}=argma{x}_{y\in Y}f({X}_{i},y)$ solutions are stored, meaning the solution y that for a fixed x maximizes the objective function.
- 3.
- Building: The individuals in the population $P\left(i\right)$ are sorted as the ascending of the UL fitness values. The best $nPop/2$ are selected. From the best $nPop/2$ individuals, we build the distribution to establish a probabilistic model ${M}_{G}$ for the LL solution. The d-dimensional multivariate normal densities to factorize the joint probability density function (pdf) are:$$F(x,\mu ,\Sigma )=\frac{1}{\sqrt{{\left|\Sigma \right|\left(2\pi \right)}^{d}}}{e}^{-1/2(x-\mu ){\Sigma}^{-1}{(x-\mu )}^{{}^{\prime}}}$$
- 4.
- Evolution: Evolve UL with the steps of the standard DE of mutation, crossover, producing an offspring ${U}_{i,G}$.
- 5.
- Selection: As mentioned above, the selection operation is a competition between each individual ${X}_{i,G}$ and its offspring ${U}_{i,G}$. The offspring will be evaluated in the scenario space and sent in LL only if $f({U}_{i,G},{Y}_{i,G})$≤$f({X}_{i,G},{Y}_{i,G})$, where ${Y}_{i,G}$ corresponds to the worst case vector of the parent individual ${X}_{i,G}$. In that way, a lot of unneeded LL optimization calls will be avoided, reducing FEs. If the offspring is evaluated in the scenario space, the selection procedure in Equation (6) is applied.
- 6.
- Termination criteria:
- Stop if the maximum number of function evaluations $MaxFEs$ is reached.
- Stop if the improvement of the best objective value of the last $MaxImpGen$ generations is below a specific number.
- Stop if the absolute difference of the best and the known true optimal objective value is below a specific number.

- 7.
- Output: the best worst case function value $f({x}^{*},{y}^{*})$, the solution corresponding to the best worst-case scenario ${x}^{*},{y}^{*}$

- 1.
- Setting: Set the parameters of the probability of crossover $CR$, the population size $nPop$, the mutation rate F, the sampling probability $\beta $.
- 2.
- Initialization: Sample $nPop$ individuals to initialize the population. If $\beta \le random(0,1)$, then the individual is sampled from the probabilistic model ${M}_{{G}^{UL}}$ built in the UL with the Equation (7). The model here is sampled with the $mvnrnd(mu,Sigma)$ built-in function of Matlab, which accepts a mean vector mu and covariance matrix sigma as input and returns a random vector chosen from the multivariate normal distribution with that mean and covariance [23]. Otherwise, it is uniformly sampled in the scenario space according to the Equation (3). Please note that for the first UL generation, $\beta $ is always 0, as no probabilistic model is built yet. For the following generations, $\beta $ can range from (0,1) number, where $\beta $ = 1 means that the population will be sampled only from the probabilistic model. This might lead the algorithm to be stuck in local optima and to converge prematurely. An example of an initial population generated with the aforementioned method with $\beta $ = 0.5 is shown in Figure 5. Magenta asterisk points represent the population generated by the probabilistic model ${M}_{{G}^{UL}}$ of the previous UL generation. Blue points are samples uniformly distributed in the search space. In Figure 6, the effect of the probabilistic model on the initial population of LL for ${f}_{8}$ during the optimization is shown. As the iterations increase, the LL members of the populations sampled from the probabilistic distribution reach the promising area that maximizes the function. In the zoomed subplot in each subfigure, one can see that all such members of the population are close to the global maximum, compared to the randomly distributed members.
- 3.
- Mutation, crossover, and selection as the standard DE.
- 4.
- Termination criteria:
- Stop if the maximum number of generations $MaxGen$ is reached.
- Stop if the absolute difference between the best and the known true optimal objective value is below a specific number.

- 5.
- Output: the maximum function value $f({x}^{*},{y}^{*})$, the solution corresponding to the worst-case scenario ${y}^{*}=argmax\left(f({x}^{*},y)\right)$.

## 4. Experimental Settings

#### 4.1. Test Functions

#### 4.2. Parameter Settings

## 5. Experimental Results and Discussion

#### 5.1. Effectiveness of the Probabilistic Sharing Mechanism

#### 5.2. Comparison with State-of-the-Art Method MMDE

#### 5.3. Engineering Application

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DE | differential evolution |

EDA | estimation of distribution algorithm |

UL | upper level |

LL | lower level |

BOP | bilevel optimization problem |

FEs | function evaluations |

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**Figure 1.**A general sketch of the min-max optimization problem as a bilevel problem, inspired by [16].

**Figure 5.**Balancing exploration and exploitation with the sharing distribution mechanism of the LL population. Magenta asterisk points represent the population generated by the distribution of the previous UL generations. Blue points are samples uniformly distributed in the search space. The idea behind this is to keep the “knowledge” already gained in previous generations while also giving the opportunity to the algorithm to search the whole search space. Here with $\beta =0.5$.

**Figure 6.**Effect of the probabilistic model on the initial population of LL for ${f}_{8}$. As the iterations increase, the LL members of the population that were produced from the probabilistic model reach the promising area that maximizes the function. The area they are concentrated is shown in the zoomed plots of each plot.

**Figure 7.**Three-dimensional mesh and contour plots of the symmetrical test function f8. Green dot corresponds to the known optimum. (

**a**) A 3D mesh of the symmetrical test function

**f**8. (

**b**) Contour plot of the symmetrical test function

**f**8.

**Figure 8.**3D mesh and contour plots of the asymmetrical test function

**f**9. Green dot corresponds to the known optimum. (

**a**) 3D mesh of the asymmetrical test function

**f**9. (

**b**) Contour plot of the asymmetrical test function

**f**9.

**Figure 9.**Barchart of the success rate (%) of each algorithmic instance and each test function. The red color corresponds to the instance where $\beta =0.0$, magenta $\beta =0.5$ and blue $\beta =0.8$.

**Figure 10.**Fitness accuracy convergence of the upper level of the median run for all the test functions and algorithm instances. The red color corresponds to the instance where $\beta =0.0$, magenta $\beta =0.5$ and blue $\beta =0.8$. Generations axes is in logarithmic scale.

**Figure 11.**Boxplots of MSE values of the proposed method and MMDE over 30 runs for the test functions ${f}_{8}$–${f}_{13}$.

Upper-Level | Lower-Level | |
---|---|---|

Population size | $max({n}_{x}+{n}_{y},5)\ast 2$ | $max({n}_{y},5)\ast 2$ |

Crossover | 0.9 | 0.9 |

Mutation | uniformly (0.2, 0.8) | uniformly (0.2, 0.8) |

Desired Accuracy | 1 × 10^{−5} | 1 × 10^{−5} |

Maximum Number of Generations | - | 10 |

Maximum Number of Function Evaluations | 5000 | - |

Maximum Number of Improvement Generations | 30 | - |

Least Improvement | 1 × 10^{−5} | - |

Problems | $\mathit{\beta}$ = 0 | $\mathit{\beta}$ = 0.5 | $\mathit{\beta}$ = 0.8 | |
---|---|---|---|---|

${f}_{1}$ | Mean | 3.45 × 10${}^{-1}$ | 2.77 × 10${}^{-5}$ | 2.29 × 10${}^{-5}$ |

Median | 9.49 × 10${}^{-2}$ | 3.33 × 10${}^{-\mathbf{5}}$ | 3.33 × 10${}^{-\mathbf{5}}$ | |

Std | 6.55 × 10${}^{-1}$ | 3.43 × 10${}^{-5}$ | 1.53 × 10${}^{-5}$ | |

p-value | ≤0.05 | NA | >0.05 | |

Median FEs | 20,115 | 28,300 | 46,535 | |

${f}_{2}$ | Mean | 1.11 × 10${}^{-1}$ | 1.25 × 10${}^{-3}$ | 1.28 × 10${}^{-4}$ |

Median | 4.96 × 10${}^{-2}$ | 5.53 × 10${}^{-\mathbf{6}}$ | 5.79 × 10${}^{-\mathbf{6}}$ | |

Std | 5.38 × 10${}^{-1}$ | 4.27 × 10${}^{-3}$ | 5.43 × 10${}^{-4}$ | |

p-value | ≤0.05 | NA | >0.05 | |

Median FEs | 20,665 | 16,180 | 17,140 | |

${f}_{3}$ | Mean | 1.64 × 10${}^{0}$ | 2.27 × 10${}^{-3}$ | 2.35 × 10${}^{-2}$ |

Median | 9.51 × 10${}^{-1}$ | 1.86 × 10${}^{-\mathbf{5}}$ | 2.47 × 10${}^{-\mathbf{5}}$ | |

Std | 2.29 × 10${}^{0}$ | 1.30 × 10${}^{-2}$ | 8.35 × 10${}^{-2}$ | |

p-value | ≤0.05 | NA | >0.05 | |

Median FEs | 27,535 | 39,830 | 46,785 | |

${f}_{4}$ | Mean | 3.49 × 10${}^{-1}$ | 2.93 × 10${}^{-5}$ | 1.64 × 10${}^{-3}$ |

Median | 2.27 × 10${}^{-1}$ | 2.03 × 10${}^{-\mathbf{5}}$ | 3.39 × 10${}^{-\mathbf{5}}$ | |

Std | 4.49 × 10${}^{-1}$ | 4.41 × 10${}^{-5}$ | 8.88 × 10${}^{-3}$ | |

p-value | ≤0.05 | NA | >0.05 | |

Median FEs | 19,940 | 26,478 | 40,516 | |

${f}_{5}$ | Mean | 6.23 × 10${}^{-2}$ | 9.95 × 10${}^{-4}$ | 8.43 × 10${}^{-6}$ |

Median | 2.63 × 10${}^{-2}$ | 2.99 × 10${}^{-\mathbf{4}}$ | 8.55 × 10${}^{-\mathbf{7}}$ | |

Std | 1.05 × 10${}^{-1}$ | 4.18 × 10${}^{-3}$ | 5.08 × 10${}^{-5}$ | |

p-value | ≤0.05 | >0.05 | NA | |

Median FEs | 38,694 | 78,444 | 97,506 | |

${f}_{6}$ | Mean | 2.16 × 10${}^{-1}$ | 1.96 × 10${}^{-3}$ | 1.19 × 10${}^{-2}$ |

Median | 1.62 × 10${}^{-1}$ | 7.86 × 10${}^{-\mathbf{6}}$ | 6.33 × 10${}^{-\mathbf{6}}$ | |

Std | 2.67 × 10${}^{-1}$ | 6.74 × 10${}^{-3}$ | 6.50 × 10${}^{-2}$ | |

p-value | ≤0.05 | >0.05 | NA | |

Median FEs | 55,740 | 69,798 | 77,356 | |

${f}_{7}$ | Mean | 5.57 × 10${}^{-1}$ | 7.90 × 10${}^{-2}$ | 7.90 × 10${}^{-2}$ |

Median | 4.76 × 10${}^{-1}$ | 7.90 × 10${}^{-\mathbf{2}}$ | 7.90 × 10${}^{-\mathbf{2}}$ | |

Std | 3.75 × 10${}^{-1}$ | 1.34 × 10${}^{-4}$ | 9.54 × 10${}^{-6}$ | |

p-value | ≤0.05 | NA | ≤0.05 | |

Median FEs | 143,580 | 360,460 | 541,940 | |

${f}_{8}$ | Mean | 9.27 × 10${}^{-6}$ | 3.03 × 10${}^{-6}$ | 3.34 × 10${}^{-6}$ |

Median | 6.16 × 10${}^{-6}$ | 1.17 × 10${}^{-\mathbf{6}}$ | 2.12 × 10${}^{-\mathbf{6}}$ | |

Std | 1.99 × 10${}^{-5}$ | 3.24 × 10${}^{-6}$ | 3.23 × 10${}^{-6}$ | |

p-value | ≤0.05 | NA | >0.05 | |

Median FEs | 9120 | 8150 | 8070 | |

${f}_{9}$ | Mean | 0.00 × 10${}^{0}$ | 0.00 × 10${}^{0}$ | 2.96 × 10${}^{-3}$ |

Median | 0.00 × 10${}^{0}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

Std | 0.00 × 10${}^{0}$ | 0.00 × 10${}^{0}$ | 1.62 × 10${}^{-2}$ | |

p-value | NaN | NA | >0.05 | |

Median FEs | 3435 | 3935 | 3715 | |

${f}_{10}$ | Mean | 7.54 × 10${}^{-6}$ | 8.03 × 10${}^{-8}$ | 8.74 × 10${}^{-4}$ |

Median | 2.86 × 10${}^{-\mathbf{7}}$ | 2.98 × 10${}^{-7}$ | 2.95 × 10${}^{-7}$ | |

Std | 3.60 × 10${}^{-5}$ | 4.96 × 10${}^{-7}$ | 4.79 × 10${}^{-3}$ | |

p-value | NA | >0.05 | >0.05 | |

FEs | 4435 | 3995 | 3880 | |

${f}_{11}$ | Mean | 5.11 × 10${}^{-3}$ | 3.62 × 10${}^{-3}$ | 1.43 × 10${}^{-2}$ |

Median | 1.79 × 10${}^{-\mathbf{3}}$ | 2.95 × 10${}^{-\mathbf{4}}$ | 1.14 × 10${}^{-2}$ | |

Std | 7.44 × 10${}^{-3}$ | 8.06 × 10${}^{-3}$ | 1.25 × 10${}^{-2}$ | |

p-value | >0.05 | NA | ≤0.05 | |

FEs | 21,965 | 30,480 | 33,485 | |

${f}_{12}$ | Mean | 3.72 × 10${}^{-1}$ | 5.21 × 10${}^{-1}$ | 7.05 × 10${}^{-1}$ |

Median | 2.25 × 10${}^{-\mathbf{1}}$ | 4.77 × 10${}^{-1}$ | 7.43 × 10${}^{-1}$ | |

Std | 5.98 × 10${}^{-1}$ | 5.23 × 10${}^{-1}$ | 1.42 × 10${}^{-1}$ | |

p-value | NA | ≤0.05 | ≤0.05 | |

Median FEs | 14,945 | 15,795 | 28,210 | |

${f}_{13}$ | Mean | 3.99 × 10${}^{-2}$ | 2.42 × 10${}^{-2}$ | 1.98 × 10${}^{-1}$ |

Median | 6.51 × 10${}^{-2}$ | 1.82 × 10${}^{-\mathbf{4}}$ | 1.07 × 10${}^{-\mathbf{5}}$ | |

Std | 7.29 × 10${}^{-1}$ | 1.21 × 10${}^{-1}$ | 1.04 × 10${}^{0}$ | |

p-value | ≤0.05 | >0.05 | NA | |

Median FEs | 22,430 | 56,880 | 61525 |

Problems | $\mathit{\beta}$ = 0.5 | MMDE | |
---|---|---|---|

${f}_{8}$ | Mean | 2.0234 × 10${}^{-5}$ | 2.6618 × 10${}^{-5}$ |

Median | 1.8269 × 10${}^{-\mathbf{7}}$ | 9.5487 × 10${}^{-6}$ | |

Std | 7.5060 × 10${}^{-5}$ | 6.1841 × 10${}^{-5}$ | |

p-value | NA | ≤0.05 | |

${f}_{9}$ | Mean | 2.5849 × 10${}^{-1}$ | 3.3719 × 10${}^{-3}$ |

Median | 0.0000 × 10${}^{\mathbf{0}}$ | 0.0000 × 10${}^{\mathbf{0}}$ | |

Std | 9.6251 × 10${}^{-1}$ | 1.1081 × 10${}^{-2}$ | |

p-value | NA | >0.05 | |

${f}_{10}$ | Mean | 1.0029 × 10${}^{0}$ | 5.1712 × 10${}^{-1}$ |

Median | 0.0000 × 10${}^{\mathbf{0}}$ | 0.0000 × 10${}^{\mathbf{0}}$ | |

Std | 2.8499 × 10${}^{0}$ | 2.4408 × 10${}^{0}$ | |

p-value | NA | ≤0.05 | |

${f}_{11}$ | Mean | 3.3428 × 10${}^{-1}$ | 7.9495 × 10${}^{-4}$ |

Median | 5.5485 × 10${}^{-2}$ | 8.8027 × 10${}^{-\mathbf{5}}$ | |

Std | 8.7588 × 10${}^{-1}$ | 1.4876 × 10${}^{-3}$ | |

p-value | ≤0.05 | NA | |

${f}_{12}$ | Mean | 8.1786 × 10${}^{-3}$ | 1.1339 × 10${}^{-5}$ |

Median | 9.6258 × 10${}^{-5}$ | 2.1344 × 10${}^{-\mathbf{6}}$ | |

Std | 1.9804 × 10${}^{-2}$ | 3.2300 × 10${}^{-5}$ | |

p-value | ≤0.05 | NA | |

${f}_{13}$ | Mean | 5.0537 × 10${}^{-2}$ | 5.5425 × 10${}^{-3}$ |

Median | 1.9716 × 10${}^{-2}$ | 2.7037 × 10${}^{-\mathbf{3}}$ | |

Std | 7.7093 × 10${}^{-2}$ | 7.7943 × 10${}^{-3}$ | |

p-value | ≤0.05 | NA |

**Table 4.**Statistical comparison with MMDE over 30 runs and ${10}^{4}$ FEs for the engineering application.

Problems | $\mathit{\beta}$ = 0.5 | MMDE | |
---|---|---|---|

$Acc\left({J}_{minmax}\right)$ | Mean | 1.7100 × 10${}^{-1}$ | 1.0472 × 10${}^{-1}$ |

Median | 6.4784 × 10${}^{-\mathbf{2}}$ | 9.9247 × 10${}^{-\mathbf{2}}$ | |

Std | 2.6084 × 10${}^{-1}$ | 6.0458 × 10${}^{-2}$ | |

p-value | NA | >0.05 | |

$MSE\left(x\right)$ | Mean | 1.4668 × 10${}^{-2}$ | 7.6533 × 10${}^{-4}$ |

Median | 6.4275 × 10${}^{-\mathbf{4}}$ | 3.6815 × 10${}^{-\mathbf{4}}$ | |

Std | 5.1909 × 10${}^{-2}$ | 7.6206 × 10${}^{-4}$ | |

p-value | >0.05 | NA |

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Antoniou, M.; Papa, G. Differential Evolution with Estimation of Distribution for Worst-Case Scenario Optimization. *Mathematics* **2021**, *9*, 2137.
https://doi.org/10.3390/math9172137

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Antoniou M, Papa G. Differential Evolution with Estimation of Distribution for Worst-Case Scenario Optimization. *Mathematics*. 2021; 9(17):2137.
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Antoniou, Margarita, and Gregor Papa. 2021. "Differential Evolution with Estimation of Distribution for Worst-Case Scenario Optimization" *Mathematics* 9, no. 17: 2137.
https://doi.org/10.3390/math9172137