Proven Runtime Guarantees for How the MOEA/D Computes the Pareto Front From the Subproblem Solutions

Many real-world problems require the simultaneous optimization of different objectives. In this setting, known as multi-objective optimization, different solutions may not be comparable based on their objective values, as one solution can win over another solution in one objective, but lose in another objective. This results in a set of incomparable optimal objective values, commonly referred to as Pareto front. The aim in multi-objective optimization is to find the Pareto front of a problem, or a good approximation thereof.

Due to their population-based and heuristic nature, evolutionary algorithms lend themselves very well to multi-objective optimization, and they have been successfully applied for decades to a plethora of hard multi-objective optimization problems [32]. This strong interest has led to a variety of algorithms [24, 32], following different paradigms.

From the early days of theoretical analyses of evolutionary algorithms on, multi-objective evolutionary algorithms have been analyzed also via theoretical means [21, 17, 12, 22]. This area saw a boost of activity in the last two years, when the first mathematical runtime analysis of the NSGA-II [31] inspired many deep analyses of this important algorithm and variants such as the NSGA-III or SMS-EMOA [26, 1, 9, 4, 10, 11, 3, 2, 23, 27, 28, 29, 30, 20].

A substantially different, yet also very important algorithm is the decomposition-based multi-objective evolutionary algorithm (MOEA/D) [25]. It decomposes a multi-objective optimization problem $f$ into various single-objective subproblems of $f$. These subproblems are optimized in parallel. While doing so, the non-dominated solutions for $f$ are maintained in an archive.

Despite its popularity in empirical research and good performance in real-world problems [24, 32], the MOEA/D has not been extensively studied theoretically [19, 15, 14, 5]. In particular, it is currently not known how the basic MOEA/D using only standard mutation operators as variation operators finds Pareto optima that are not already an optimum of one of the subproblems (we refer to Section 2 for a detailed discussion of the previous works). We recall that the MOEA/D solves a number of single-objective subproblems essentially via single-objective approaches. Hence, if all Pareto optima of the original problem appear as optima of subproblems (we call these $g$-optima in the remainder), this is the setting regarded, e.g., in [19], then the multi-objective aspect of the problem vanishes and the only question is how efficiently the single-objective approaches solve the subproblems.

Theoretical analyses of the MOEA/D so far are scarce. Most results do not consider the MOEA/D with only standard mutation operators. And those that do make simplifying assumptions about the decomposition, using problem-specific knowledge. We also note that we could not find any proofs for how the MOEA/D finds its reference point (one of its parameters), which is important in the general scenarios. If the reference point is mentioned, it is immediately assumed to be best-possible.

Li et al. [19] conducted the first mathematical analysis of the MOEA/D. They study the runtime of the algorithm on the classic bi-objective OneMinMax ($\mathrm{OMM}$) [13] and LeadingOnesTrailingZeros ($\mathrm{LOTZ}$)¹¹1We note that the authors call $\mathrm{OMM}$ COCZ, and that they consider a version of $\mathrm{LOTZ}$, called LPTNO, that considers strings over $\{-1,1\}$ instead of binary strings, effectively replacing $0$s with $-1$s. This changes some values for certain parameters but effectively results in the same bounds, to the best of our knowledge. Hence, we use the names $\mathrm{OMM}$ and $\mathrm{LOTZ}$ throughout this section. benchmarks of size $n$, that is, the number of objective-function evaluations until the MOEA/D finds all Pareto optima of the problem. Both benchmarks have a Pareto front of size $n+1$. The authors consider a decomposition of each problem into $n+1$ subproblems—matching the size of the Pareto front—, and they assume that the MOEA/D uses standard bit mutation. The authors prove that if the subproblems are chosen such that the $n+1$ $g$-optima correspond one-to-one to the Pareto front of the problem, then the MOEA/D optimizes $\mathrm{OMM}$ in $O(n^{2}\log n)$ expected function evaluations, and $\mathrm{LOTZ}$ in $O(n^{3})$. We note that this requires a different subproblem structure for $\mathrm{OMM}$ and $\mathrm{LOTZ}$. Moreover, the authors show that if the MOEA/D uses the commonly used subproblem structure of $\mathrm{OMM}$ for $\mathrm{LOTZ}$, then the $g$-optima do not cover the entire Pareto front of $\mathrm{LOTZ}$, for sufficiently large $n$. In this case, the authors argue that optimization takes a long time, as the missing solutions need to be found, but these arguments are not made formal.

Huang and Zhou [14] analyze the MOEA/D when using contiguous hypermutations, a non-standard mutation operator stemming from artificial immune systems. The authors study two versions of contiguous hypermutation, and they consider the $\mathrm{OMM}$ and the $\mathrm{LOTZ}$ benchmarks as well as a deceptive bi-objective problem and one containing a plateau. Moreover, the authors consider a general decomposition of each problem into $N+1$ subproblems. This decomposition is always the same and corresponds to the commonly used one, which results in evenly spread $g$-optima on the Pareto front of $\mathrm{OMM}$, also analyzed by Li et al. [19] above. Huang and Zhou [14] prove that both MOEA/D variants optimize each of the four benchmarks in $O(Nn^{2}\log n)$ expected function evaluations. Moreover, they prove the same runtime bounds for the $4$-objective versions of $\mathrm{OMM}$ and $\mathrm{LOTZ}$. Overall, their results suggest that a choice of $N=O(1)$ is more beneficial, as the hypermutation operators are capable to cover larger distances than standard bit mutation and can thus find Pareto optima efficiently that do not correspond to $g$-optima.

Huang et al. [15] analyze a variant of the MOEA/D that employs standard bit mutation as well as one-point crossover at a rate of $0.5$. When performing crossover, one of the parents is the best solution of the current subproblem $g$ and the other one is chosen uniformly at random among the non-dominated solutions of the subproblems closest to $g$ with respect to the Euclidean distance in one of their parameters. The problem decomposition is always the commonly used one with equidistant $g$-optima, as in the two papers discussed above. The authors consider essentially the same four²²2The authors actually treat $\mathrm{LOTZ}$ and LPTNO as two functions, but the results are the same, which is why we count it as one problem. problems as Huang and Zhou [14] above, and they also consider a general number of $N+1$ subproblems. For $\mathrm{LOTZ}$ and the plateau function, the authors proved an expected runtime of $O(Nn^{2})$ function evaluations, and an $O(Nn\log n)$ bound for $\mathrm{OMM}$ and the deceptive function.

Very recently, Do et al. [5] analyzed the MOEA/D on the multi-objective minimum-weight-base problem, which is an abstraction of classical NP-hard combinatorial problems. Their MOEA/D variant uses a decomposition based on weight scalarization, different from the previous works above. The authors then prove that this variant finds an approximation of the Pareto front of the problem within expected fixed-parameter polynomial time.

We denote the natural numbers by $\mathbb{N}$, including $0$, and the real numbers by $\mathbb{R}$. For $a,b\in\mathbb{R}$, let $[a..b]=[a,b]\cap\mathbb{N}$ and $[a]=[1..a]$.

Let $n\in\mathbb{N}_{\geq 1}$. We consider bi-objective optimization problems, that is, functions $f\colon\{0,1\}^{n}\to\mathbb{R}^{2}$. We always assume that the dimension $n\in\mathbb{N}_{\geq 1}$ is given implicitly. When using big-O notation, it refers to asymptotics in this $n$. In this sense, an event occurs with high probability if its probability is at least $1-o(1)$.

We call a point $x\in\{0,1\}^{n}$ an individual and $f(x)$ the objective value of $x$. For all $i\in[n]$ and $j\in[2]$, we let $x_{i}$ denote the $i$-th component of $x$ and $f_{j}(x)$ the $j$-th component of $f(x)$. Moreover, let $|x|_{0}$ denote the number of $0$s of $x$, and let $|x|_{1}$ denote its number of $1$s.

For all $u,v\in\mathbb{R}^{2}$, we say that $v$ weakly dominates $u$ (written $v\succeq u$) if and only if for all $i\in[2]$ holds that $f_{i}(v)\geq f_{i}(u)$. We say that $v$ strictly dominates $u$ if and only if one of these inequalities is strict. We extend this notation to individuals, where a dominance holds if and only if it holds for their respective objective values.

We consider the maximization of bi-objective functions $f$, that is, we are interested in $\succeq$-maximal elements, called Pareto-optimal individuals. The set of all objective values that are not strictly dominated, that is, the set $F^{*}\coloneqq\{v\in\mathbb{R}^{2}\mid\not\exists x\in\{0,1\}^{n},f(x)\succ v\}$, is called the Pareto front of $f$.

We analyze the decomposition-based multi-objective evolutionary algorithm (MOEA/D; Algorithm 1) [25] for multi-objective optimization. The MOEA/D decomposes its objective function into a pre-specified number of single-objective subproblems. These subproblems are optimized in parallel. The MOEA/D maintains an archive of the $\succeq$-best solutions found so far, allowing it to find Pareto-optimal solutions for the original problem while only explicitly optimizing the subproblems.

More formally, given, besides others, a bi-objective optimization problem $f$ as well as a decomposition number $N\in[n]$ and a weight vector $w\in[0,1]^{N+1}$, the MOEA/D optimizes $f$ by decomposing $f$ into $N+1$ single-objective subproblems $\{g_{i}\colon\{0,1\}^{n}\times\mathbb{R}^{2}\to\mathbb{R}\}_{i\in[0..N]}$, weighted by $w$. Each of these subproblems is subject to minimization (of an error). The MOEA/D maintains a population of $N+1$ individuals $(x_{i})_{i\in[0..N]}\in(\{0,1\}^{n})^{N+1}$ as well as a reference point $z^{*}\in\mathbb{R}^{2}$ such that for each $i\in[0..N]$, individual $x_{i}$ is the currently best solution found for subproblem $i$ with respect to $z^{*}$. Ideally, the reference point is a point such that for all $j\in[2]$, value $z^{*}_{j}$ is optimal for objective $f_{j}$. To this end, the MOEA/D updates $z^{*}$ whenever it optimizes a subproblem. Moreover, the algorithm maintains a set $P\subseteq\mathbb{R}^{2}$ (the Pareto front) of non-dominated objective values.

We consider subproblems that measure the maximum distance to the reference point, known as Chebyshev approach. That is, for all $i\in[0..N]$, $x\in\{0,1\}^{n}$, and $z^{*}\in\mathbb{R}^{2}$, it holds that

When minimizing subproblem $i\in[0..N]$, the MOEA/D picks $x_{i}$ as parent and mutates it according to a given mutation operator. Afterward, it compares the offspring of the mutation to $x_{i}$ and selects the better one.³³3We note that the general MOEA/D allows to specify neighborhoods among the subproblems, which exchange non-dominated solutions among each other. In this article, we focus on no exchange.

We define the runtime of the MOEA/D on $f$ as the number of function evaluations of $f$ until the Pareto front $P$ of the algorithm is equal to the Pareto front $F^{*}$ of $f$ for the first time.

In this article, we consider the MOEA/D with different mutation operators.

We analyze the MOEA/D (Algorithm 1) on the OneMinMax ($\mathrm{OMM}$) function (of dimension $n\in\mathbb{N}_{\geq 1}$) with subproblems spread uniformly across the Pareto front of $\mathrm{OMM}$. To this end, we make the following assumptions about the input of the algorithm, which we refer to as the parametrization: We consider decomposition numbers $N\in[n]$, we define the weight vector as $w=(i\frac{n}{N})_{i\in[0..N]}$, and we consider the subproblems as defined in equation 1. In our calculations, we assume that $N$ divides $n$, as this avoids rounding, but we note that all results are equally valid if $N$ does not divide $n$, although the computations become more verbose, adding little merit. We note that, due to space restrictions, some of our proofs are in the appendix.

Our main results are the following, bounding the expected runtime for both standard bit mutation and power-law mutation.

Both runtime results present two terms, which stem from the two phases into which we separate our analysis. The first term in the results is an upper bound of the first phase, which is the number of function evaluations it takes the MOEA/D to optimize all subproblems. We call the solutions to these subproblems $g$-optima. Our bounds for either mutation operator for the first phase are the same (Corollary 3). The optimization mainly resembles performing $N+1$ times an optimization similar to that of the well-known $(1+1)$ evolutionary algorithm on the OneMax benchmark function. For $N=n$, our result for standard bit mutation recovers the result by Li et al. [19, Proposition $4$]. We go into detail about the analysis in Section 5.1.

The second phase starts immediately after the first phase and stops once the Pareto front of the MOEA/D covers the Pareto front of $\mathrm{OMM}$. During this analysis, we only assume that the MOEA/D found the $g$-optima so far. Thus, in the worst case, it still needs to find all other Pareto-optima of $\mathrm{OMM}$. To this end, each such optimum needs to be created via mutation directly from one of the $g$-optima, as the latter are not being replaced, due to them being optimal. Depending on the gap size, that is, the number of Pareto optima between two $g$-optima of neighboring subproblems, the mutation operator makes a big difference on the expected runtime bound. We analyze both mutation operators separately in Section 5.2.

Regarding Corollary 1, we see that the upper bound for standard bit mutation only handles values for $N\in[\frac{n}{2}..n]$ without any slowdown in comparison to the first phase. For smaller values, the upper bound is dominated by the second term and becomes super-polynomial once $N=o(n)$. In Theorem 5.1, we prove a lower bound for the second phase that shows that the expected runtime is also at least super-polynomial once $N=o(n)$. However, as this lower bound only applies to the second phase, it may be possible during the whole optimization of $\mathrm{OMM}$ that the algorithm finds most of the $\mathrm{OMM}$-Pareto optima already during the first phase, although we conjecture this not to happen for sufficiently small $N$.

For power-law mutation (Corollary 2), the bound for the second phase is independent of $N$ (Theorem 5.2). This shows that the power-law mutation picks up missing Pareto optima fairly quickly. In fact, even for an optimal value of $N=n$ for standard bit mutation, which results in a bound of $O(n^{2}\log n)$ for the second phase, the bound for power-law mutation is still smaller by a factor of $n^{\beta-2}$, which is less than $1$, as we assume that $\beta\in(1,2)$. Moreover, for the whole optimization, the upper bound for power-law mutation is best once $N=O(n^{\beta-1})$, that is, for smaller values of $N$. This is in contrast to the upper bound for standard bit mutation, which gets better for larger values of $N$.

Overall, our results suggest that standard bit mutation profits from having many subproblems, as creating initially skipped solutions may be hard to create once all subproblems are optimized. In contrast, power-law mutation is slowed down by optimizing many subproblems in parallel. Instead, it profits from optimizing fewer such subproblems and then creating initially skipped solutions.

In the following, we first analyze the first phase (Section 5.1) and then the second phase (Section 5.2).

We studied the impact of the decomposition number $N$ of the MOEA/D [25] on the classic multi-objective benchmark OneMinMax ($\mathrm{OMM}$) [13] theoretically. Our analyses considered subproblems that are evenly spread out on the Pareto front of $\mathrm{OMM}$. Especially, we studied the expected runtime when starting with all optima of the subproblems (the $g$-optima) and requiring to find the remaining Pareto optima of $\mathrm{OMM}$.

One of our theoretical results (Theorem 5.2) shows that using power-law mutation allows the MOEA/D to efficiently find the entire Pareto front of $\mathrm{OMM}$ even if it is initialized only with the $g$-optima. Interestingly, this bound is independent of the number of problems $N$ and thus the number of initially missing Pareto optima between two neighboring $g$-optima. Together with our general bound for finding all $g$-optima (Corollary 3), this shows that the MOEA/D with power-law mutation always optimizes $\mathrm{OMM}$ efficiently (Corollary 2). Depending on $N$, our total-runtime bound ranges from $O(n^{2}\log n)$ in the worst case to $O(n^{\beta}\log n)$ in the best case of $N=O(n^{\beta-1})$. This suggests that the MOEA/D is, in fact, slowed down when using many subproblems, despite large values of $N$ implying that the subproblems cover the Pareto front of $\mathrm{OMM}$ better than smaller values. The reason is that a large value of $N$ roughly translates to optimizing the same problem $N$ times. With the power-law mutation, it is better to optimize fewer and therefore more diverse subproblems and to then find the remaining Pareto optima efficiently via the power-law mutation.

For standard bit mutation, when starting with all $g$-optima, we show (Theorem 5.1) that the MOEA/D is not capable of efficiently finding all Pareto optima if $N$ is sufficiently small, as our lower bound is super-polynomial for $N=o(n)$. Nonetheless, for $N=\Theta(n)$, the expected runtime of the MOEA/D with standard bit mutation is polynomial in this part. This translates to an overall polynomial expected runtime (Corollary 1) for $N=\Theta(n)$ that it is even $O(n^{2}\log n)$ for $N\in[\frac{n}{2}..n]$, matching our worst-case bound with power-law.

Overall, our results suggest a clear benefit of power-law mutation over standard bit mutation of the MOEA/D in the setting we considered. Not only is the power-law variant faster, it is also far more robust to the choice of $N$ and thus to how the problem is decomposed.

For future work, it would be interesting to improve the upper bounds or prove matching lower bounds. A similar direction is to consider an exchange of best-so-far solutions among the subproblems. The classic MOEA/D supports such an exchange, which could potentially lead to finding the $g$-optima more quickly. Another promising direction is the study of different problem decompositions, for example, not-evenly spread subproblems or subproblem definitions different from equation 1. Last, we considered the $\mathrm{OMM}$ setting, with a stronger generalization some of our results (Theorems 5.1 and 5.2). However, it is not clear to what extent the benefit of power-law mutation carries over to problems with an entirely different structure, such as LeadingOnesTrailingZeros.