When to Truncate the Archive? On the Effect of the Truncation Frequency in Multi-Objective Optimisation

Using an archive to store nondominated solutions found during the search of a multi-objective evolutionary algorithm (MOEA) is a useful practice. However, as nondominated solutions of a multi-objective optimisation problem can be enormous or infinitely many, it is desirable to provide the decision-maker with only a small, representative portion of all the nondominated solutions in the archive, thus entailing a truncation operation. Then, an important issue is when to truncate the archive. This can be done once a new solution generated, a batch of new solutions generated, or even using an unbounded archive to keep all nondominated solutions generated and truncate it later. Intuitively, the last approach may lead to a better result since we have all the information in hand before performing the truncation. In this paper, we study this issue and investigate the effect of the timing of truncating the archive. We apply well-established truncation criteria that are commonly used in the population maintenance procedure of MOEAs (e.g., crowding distance, hypervolume indicator, and decomposition). We show that, interestingly, truncating the archive once a new solution generated tends to be the best, whereas considering an unbounded archive is often the worst. We analyse and discuss this phenomenon. Our results highlight the importance of developing effective subset selection techniques (rather than employing the population maintenance methods in MOEAs) when using a large archive.

Multi-objective Pseudo Boolean Functions in Runtime Analysis: A Review

Recently, there has been growing interest within the theoretical community in analytically studying multi-objective evolutionary algorithms. This runtime analysis-focused research can help formally understand algorithm behaviour, explain empirical observations, and provide theoretical insights to support algorithm development and exploration. However, the test problems commonly used in the theoretical analysis are predominantly limited to problems with heavy ``artificial'' characteristics (e.g., symmetric objectives and linear Pareto fronts), which may not be able to well represent realistic scenarios. In this paper, we survey commonly used multi-objective functions in the theory domain and systematically review their features, limitations and implications to practical use. Moreover, we present several new functions with more realistic features, such as local optimality and nonlinearity of the Pareto front, through simply mixing and matching classical single-objective functions in the area (e.g., LeadingOnes, Jump and RoyalRoad). We hope these functions can enrich the existing test problem suites, and strengthen the connection between theoretic and practical research.

A Weight Adaptation Trigger Mechanism in Decomposition-based Evolutionary Multi-Objective Optimisation

Decomposition-based multi-objective evolutionary algorithms (MOEAs) are widely used for solving multi-objective optimisation problems. However, their effectiveness depends on the consistency between the problems Pareto front shape and the weight distribution. Decomposition-based MOEAs, with uniformly distributed weights (in a simplex), perform well on problems with a regular (simplex-like) Pareto front, but not on those with an irregular Pareto front. Previous studies have focused on adapting the weights to approximate the irregular Pareto front during the evolutionary process. However, these adaptations can actually harm the performance on the regular Pareto front via changing the weights during the search process that are eventually the best fit for the Pareto front. In this paper, we propose an algorithm called the weight adaptation trigger mechanism for decomposition-based MOEAs (ATM-MOEA/D) to tackle this issue. ATM-MOEA/D uses an archive to gradually approximate the shape of the Pareto front during the search. When the algorithm detects evolution stagnation (meaning the population no longer improves significantly), it compares the distribution of the population with that of the archive to distinguish between regular and irregular Pareto fronts. Only when an irregular Pareto front is identified, the weights are adapted. Our experimental results show that the proposed algorithm not only performs generally better than seven state-of-the-art weight-adapting methods on irregular Pareto fronts but also is able to achieve the same results as fixed-weight methods like MOEA/D on regular Pareto fronts.

Stochastic Population Update Provably Needs An Archive in Evolutionary Multi-objective Optimization

Evolutionary algorithms (EAs) have been widely applied to multi-objective optimization, due to their nature of population-based search. Population update, a key component in multi-objective EAs (MOEAs), is usually performed in a greedy, deterministic manner. However, recent studies have questioned this practice and shown that stochastic population update (SPU), which allows inferior solutions have a chance to be preserved, can help MOEAs jump out of local optima more easily. While introducing randomness in the population update process boosts the exploration of MOEAs, there is a drawback that the population may not always preserve the very best solutions found, thus entailing a large population. Intuitively, a possible solution to this issue is to introduce an archive that stores the best solutions ever found. In this paper, we theoretically show that using an archive can allow a small population and accelerate the search of SPU-based MOEAs substantially. Specifically, we analyze the expected running time of two well-established MOEAs, SMS-EMOA and NSGA-II, with SPU for solving a commonly studied bi-objective problem OneJumpZeroJump, and prove that using an archive can bring (even exponential) speedups. The comparison between SMS-EMOA and NSGA-II also suggests that the $(\mu+\mu)$ update mode may be more suitable for SPU than the $(\mu+1)$ update mode. Furthermore, our derived running time bounds for using SPU alone are significantly tighter than previously known ones. Our theoretical findings are also empirically validated on a well-known practical problem, the multi-objective traveling salesperson problem. We hope this work may provide theoretical support to explore different ideas of designing algorithms in evolutionary multi-objective optimization.

Distilled Lifelong Self-Adaptation for Configurable Systems

Modern configurable systems provide tremendous opportunities for engineering future intelligent software systems. A key difficulty thereof is how to effectively self-adapt the configuration of a running system such that its performance (e.g., runtime and throughput) can be optimized under time-varying workloads. This unfortunately remains unaddressed in existing approaches as they either overlook the available past knowledge or rely on static exploitation of past knowledge without reasoning the usefulness of information when planning for self-adaptation. In this paper, we tackle this challenging problem by proposing DLiSA, a framework that self-adapts configurable systems. DLiSA comes with two properties: firstly, it supports lifelong planning, and thereby the planning process runs continuously throughout the lifetime of the system, allowing dynamic exploitation of the accumulated knowledge for rapid adaptation. Secondly, the planning for a newly emerged workload is boosted via distilled knowledge seeding, in which the knowledge is dynamically purified such that only useful past configurations are seeded when necessary, mitigating misleading information. Extensive experiments suggest that the proposed DLiSA significantly outperforms state-of-the-art approaches, demonstrating a performance improvement of up to 229% and a resource acceleration of up to 2.22x on generating promising adaptation configurations. All data and sources can be found at our repository: https://github.com/ideas-labo/dlisa.

A First Running Time Analysis of the Strength Pareto Evolutionary Algorithm 2 (SPEA2)

Evolutionary algorithms (EAs) have emerged as a predominant approach for addressing multi-objective optimization problems. However, the theoretical foundation of multi-objective EAs (MOEAs), particularly the fundamental aspects like running time analysis, remains largely underexplored. Existing theoretical studies mainly focus on basic MOEAs, with little attention given to practical MOEAs. In this paper, we present a running time analysis of strength Pareto evolutionary algorithm 2 (SPEA2) for the first time. Specifically, we prove that the expected running time of SPEA2 for solving three commonly used multi-objective problems, i.e., $m$OneMinMax, $m$LeadingOnesTrailingZeroes, and $m$-OneJumpZeroJump, is $O(\mu n\cdot \min\{m\log n, n\})$, $O(\mu n^2)$, and $O(\mu n^k \cdot \min\{mn, 3^{m/2}\})$, respectively. Here $m$ denotes the number of objectives, and the population size $\mu$ is required to be at least $(2n/m+1)^{m/2}$, $(2n/m+1)^{m-1}$ and $(2n/m-2k+3)^{m/2}$, respectively. The proofs are accomplished through general theorems which are also applicable for analyzing the expected running time of other MOEAs on these problems, and thus can be helpful for future theoretical analysis of MOEAs.

An Archive Can Bring Provable Speed-ups in Multi-Objective Evolutionary Algorithms

In the area of multi-objective evolutionary algorithms (MOEAs), there is a trend of using an archive to store non-dominated solutions generated during the search. This is because 1) MOEAs may easily end up with the final population containing inferior solutions that are dominated by other solutions discarded during the search process and 2) the population that has a commensurable size of the problem's Pareto front is often not practical. In this paper, we theoretically show, for the first time, that using an archive can guarantee speed-ups for MOEAs. Specifically, we prove that for two well-established MOEAs (NSGA-II and SMS-EMOA) on two commonly studied problems (OneMinMax and LeadingOnesTrailingZeroes), using an archive brings a polynomial acceleration on the expected running time. The reason is that with an archive, the size of the population can reduce to a small constant; there is no need for the population to keep all the Pareto optimal solutions found. This contrasts existing theoretical studies for MOEAs where a population with a commensurable size of the problem's Pareto front is needed. The findings in this paper not only provide a theoretical confirmation for an increasingly popular practice in the design of MOEAs, but can also be beneficial to the theory community towards studying more practical MOEAs.

Maintaining Diversity Provably Helps in Evolutionary Multimodal Optimization

In the real world, there exist a class of optimization problems that multiple (local) optimal solutions in the solution space correspond to a single point in the objective space. In this paper, we theoretically show that for such multimodal problems, a simple method that considers the diversity of solutions in the solution space can benefit the search in evolutionary algorithms (EAs). Specifically, we prove that the proposed method, working with crossover, can help enhance the exploration, leading to polynomial or even exponential acceleration on the expected running time. This result is derived by rigorous running time analysis in both single-objective and multi-objective scenarios, including $(\mu+1)$-GA solving the widely studied single-objective problem, Jump, and NSGA-II and SMS-EMOA (two well-established multi-objective EAs) solving the widely studied bi-objective problem, OneJumpZeroJump. Experiments are also conducted to validate the theoretical results. We hope that our results may encourage the exploration of diversity maintenance in the solution space for multi-objective optimization, where existing EAs usually only consider the diversity in the objective space and can easily be trapped in local optima.

Adapting Multi-objectivized Software Configuration Tuning

When tuning software configuration for better performance (e.g., latency or throughput), an important issue that many optimizers face is the presence of local optimum traps, compounded by a highly rugged configuration landscape and expensive measurements. To mitigate these issues, a recent effort has shifted to focus on the level of optimization model (called meta multi-objectivization or MMO) instead of designing better optimizers as in traditional methods. This is done by using an auxiliary performance objective, together with the target performance objective, to help the search jump out of local optima. While effective, MMO needs a fixed weight to balance the two objectives-a parameter that has been found to be crucial as there is a large deviation of the performance between the best and the other settings. However, given the variety of configurable software systems, the "sweet spot" of the weight can vary dramatically in different cases and it is not possible to find the right setting without time-consuming trial and error. In this paper, we seek to overcome this significant shortcoming of MMO by proposing a weight adaptation method, dubbed AdMMO. Our key idea is to adaptively adjust the weight at the right time during tuning, such that a good proportion of the nondominated configurations can be maintained. Moreover, we design a partial duplicate retention mechanism to handle the issue of too many duplicate configurations without losing the rich information provided by the "good" duplicates. Experiments on several real-world systems, objectives, and budgets show that, for 71% of the cases, AdMMO is considerably superior to MMO and a wide range of state-of-the-art optimizers while achieving generally better efficiency with the best speedup between 2.2x and 20x.

Stochastic Population Update Can Provably Be Helpful in Multi-Objective Evolutionary Algorithms

Evolutionary algorithms (EAs) have been widely and successfully applied to solve multi-objective optimization problems, due to their nature of population-based search. Population update is a key component in multi-objective EAs (MOEAs), and it is performed in a greedy, deterministic manner. That is, the next-generation population is formed by selecting the first population-size ranked solutions (based on some selection criteria, e.g., non-dominated sorting, crowdedness and indicators) from the collections of the current population and newly-generated solutions. In this paper, we question this practice. We analytically present that introducing randomness into the population update procedure in MOEAs can be beneficial for the search. More specifically, we prove that the expected running time of a well-established MOEA (SMS-EMOA) for solving a commonly studied bi-objective problem, OneJumpZeroJump, can be exponentially decreased if replacing its deterministic population update mechanism by a stochastic one. Empirical studies also verify the effectiveness of the proposed stochastic population update method. This work is an attempt to challenge a common practice for the population update in MOEAs. Its positive results, which might hold more generally, should encourage the exploration of developing new MOEAs in the area.