# Strength Pareto Evolutionary Algorithm 2 (SPEA2) in MATLAB

URL: https://yarpiz.com/spea2-ypea122
Author: Mostapha Kalami Heris
Published: 2015-09-01
Updated: 2026-10-30
Topics: Multiobjective Optimization

> Implement SPEA2 in MATLAB with strength-based fitness, a nearest-neighbor density measure, an external archive, and multi-objective examples.

Strength Pareto Evolutionary Algorithm 2 (SPEA2) is an extended version of the SPEA multi-objective evolutionary optimization algorithm. Zitzler and Thiele proposed SPEA in 1999 ([paper](https://doi.org/10.1109/4235.797969)), and Zitzler, Laumanns and Thiele presented SPEA2 in 2001 in a technical report of ETH Zurich. SPEA2 uses a mechanism similar to k-Nearest Neighbor (kNN) and a specialized ranking system to sort the population. The next generation is selected from the combination of the current population and an archive of the best solutions found so far. SPEA2 is one of the best-known multi-objective optimization algorithms.

## The problem

SPEA2 minimizes several conflicting objectives $f_1, \dots, f_M$ at the same time. Solutions are compared with Pareto dominance: $x$ dominates $y$ ($x \prec y$) when $f_i(x) \le f_i(y)$ for every objective $i$ and $f_j(x) < f_j(y)$ for at least one objective $j$. The goal is a set of non-dominated solutions that lies close to the Pareto front and covers it well.

## How SPEA2 works

SPEA2 keeps a population $P$ of size $N$ and an archive $\bar{P}$ of size $\bar{N}$. In each generation, every member $i$ of $P \cup \bar{P}$ gets a fitness value in three steps.

The **strength** $S(i)$ is the number of members that $i$ dominates:

$$
S(i) = \left|\{\, j \in P \cup \bar{P} : i \prec j \,\}\right|
$$

The **raw fitness** $R(i)$ is the sum of the strengths of all members that dominate $i$. A non-dominated member has $R(i) = 0$, and a member dominated by strong members gets a large value:

$$
R(i) = \sum_{j \,:\, j \prec i} S(j)
$$

The **density** $D(i)$ comes from the distance $\sigma_i^k$ between member $i$ and its $k$-th nearest neighbor in objective space, with $k = \sqrt{N + \bar{N}}$ rounded to an integer. It is always below 1, so it only separates members with the same raw fitness:

$$
D(i) = \frac{1}{\sigma_i^k + 2}, \qquad F(i) = R(i) + D(i)
$$

Lower fitness is better. The next archive is filled as follows:

1. All non-dominated members ($R(i) = 0$) are copied to the archive.
2. If there are fewer than $\bar{N}$ of them, the best dominated members by fitness fill the rest.
3. If there are more than $\bar{N}$, the **truncation** step removes members one at a time. Each time, it removes the member that is closest to its neighbors, so the archive keeps a good spread along the front.

Parents are then chosen from the archive by binary tournament on $F$, and crossover and mutation create the new population.

## Implementation in MATLAB

This post shares an open-source implementation of SPEA2 in MATLAB. The code is structured, so if you know MATLAB, you should find it easy to use in your research projects.

The `SPEA2` folder holds the main script `spea2.m`, a one line entry point `main.m`, and the helper functions `Dominates.m`, `BinaryTournamentSelection.m`, `Crossover.m`, `Mutate.m` and `PlotCosts.m`. The default problem is `ZDT.m`, the ZDT1 test function with two objectives, and `MOP2.m` is a second test problem.

To run the code, open the `SPEA2` folder in MATLAB and run `main.m`. In each iteration the program prints the number of members on the approximate Pareto front and plots them. At the end it prints the minimum, maximum, range, standard deviation and mean of each objective over the front.

The code uses `unifrnd`, `randsample` and `pdist2`, functions of the Statistics and Machine Learning Toolbox, so it probably needs that toolbox. The distances between members are computed with `pdist2` and the standardized Euclidean metric. The archive-truncation code is a simplified implementation and does not recompute the full neighbor-distance ordering after every removal, so it should not be treated as an exact reproduction of the report.

### Parameters in the code

The parameters are set at the top of `spea2.m`:

| Parameter | Meaning | Value |
| --- | --- | ---: |
| `nVar` | Number of decision variables | 30 |
| `VarMin`, `VarMax` | Lower and upper bounds of the variables | 0, 1 |
| `MaxIt` | Maximum number of iterations | 200 |
| `nPop` | Population size | 50 |
| `nArchive` | Archive size | 50 |
| `K` | kNN parameter | `round(sqrt(nPop+nArchive))` |
| `pCrossover` | Share of the new population created by crossover | 0.7 |
| `crossover_params.gamma` | Range extension of the arithmetic crossover | 0.1 |
| `mutation_params.h` | Gaussian mutation step, as a share of the variable range | 0.2 |

The rest of the new population (a share of $1 - 0.7$) is created by mutation. To solve your own problem, write a cost function that takes a column vector and returns a column vector of objective values, and set `CostFunction`, `nVar`, `VarMin` and `VarMax` in `spea2.m`.

## Check the result

Check dominance among the reported objective vectors, confirm decision-variable bounds, and inspect the spread across repeated runs. An archive can include dominated members while it is being filled, so distinguish its contents from the final reported non-dominated subset. Standardized distances need meaningful variation in each objective, and an objective with zero variance needs explicit handling. Use a consistent evaluation budget when comparing with NSGA-II or other methods.

## Related posts

[NSGA-II](https://yarpiz.com/nsga2-ypea120), [PESA-II](https://yarpiz.com/pesa2-ypea123), [MOEA/D](https://yarpiz.com/moead-ypea124) and [Multi-Objective PSO](https://yarpiz.com/mopso-ypea121) are other multi-objective algorithms on Yarpiz, implemented in the same style.

## How to cite

Mostapha Kalami Heris, Strength Pareto Evolutionary Algorithm 2 (SPEA2) in MATLAB (YPEA122), Yarpiz, 2015. https://yarpiz.com/spea2-ypea122