arXiv subjects

Andre Opris

Publications and source records attributed to Andre Opris.

At least 19 recordsLinked to original sources

arXiv2026

Provable Speedups From Dynamic Population Sizes in Evolutionary Algorithms for Multiobjective Optimization

This paper investigates the role of dynamic population sizes in evolutionary multi-objective optimization. Although such approaches are widely used in practice, their benefits remain poorly understood, and rigorous runtime analyses explaining when and why they help are still scarce. To address this, we introduce the bi-objective problem class CLIMB and analyze the runtime of GSEMO and the widely used NSGA-II on this problem. Our results show that allowing a dynamic population size for NSGA-II can lead to a moderate improvement, yielding a speedup of order $\Omega(\sqrt{n}/\log n)$. In particular, we prove that GSEMO and NSGA-II-DYN, a version of NSGA-II with dynamic population sizes we propose in this paper, can find the Pareto front of CLIMB in expected $O(n \log n)$ fitness evaluations, whereas NSGA-II with a fixed population size requires $\Omega(n^{1.5})$ fitness evaluations in expectation. To the best of our knowledge, this is the first rigorous runtime analysis in multi-objective optimization demonstrating a super-constant speedup of GSEMO over NSGA-II. Our analysis builds on concepts from single-objective optimization, like the evolution of population diversity over time, and employs the well-known family-three method to prove the lower bound.

cs.NE↗

arXiv2026

Runtime Analysis of Cartesian Genetic Programming in Evolving Boolean Functions

Cartesian Genetic Programming (CGP) is among the practical and popular forms of Genetic Programming as it uses a graph-based representation of programs. This paper presents a first runtime analysis of CGP in evolving Boolean functions using complete training sets. We prove an asymptotic bound $O(n D^5)$ for the expected number of fitness evaluations of CGP to construct a conjunction of $n$ inputs using at most $D \geq n-1$ binary gates, a minimal function set, and even with a strict survival selection. When the non-strict selection is used, the bound is improved to $O(n D^4)$. Our analysis reveals interesting characteristics of CGP induced search, which have been only observed empirically. In particular, enabling the acceptance of equally good solutions, including those with connected gates non-contributing to fitness, can lead to a speedup, and consequently a better asymptotic time bound. In contrast to conjunctions, we also prove a negative result which shows that CGP requires exponential time to evolve an exclusive disjunction. Experiments evolving conjunctions complement our theoretical findings. The use of incomplete training sets is found to further reduce the average number of fitness evaluations while maintaining a good level of generalisation.

cs.NE↗

arXiv2026

SPEA2$^+$: Improved Density Estimation in SPEA2 with Provable Runtime Guarantees

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is a popular and prominent evolutionary algorithm for solving multi-objective optimisation problems. Despite its popularity, theoretical analyses of SPEA2 have only appeared recently. Moreover, these analyses focus exclusively on how SPEA2 handles non-dominated solutions and disregard the algorithmic components responsible for handling dominated solutions. We conduct a first runtime analysis of SPEA2 for which these components are analysed. We prove that, unlike other prominent algorithms, including NSGA-II, NSGA-III and SMS-EMOA under the same setting of constant population size and duplicate elimination, SPEA2 is unable to cover the Pareto front of the OneTrapZeroTrap benchmark efficiently. Our results indicate that using k-th nearest-neighbour distance in the fitness assignment provides an insufficient signal to maintain diversity among dominated individuals. To address this issue, we propose an improved variant, SPEA2$^+$, that considers all pairwise distances. The new algorithm achieves the same performance guarantees as the other prominent algorithms on OneTrapZeroTrap, while matching the performance of the original SPEA2 on simpler problems. Experimental results complement our theoretical findings.

cs.NE↗

arXiv2026

On the Impact of Crossover in Many-Objective Optimization: A Runtime Analysis of NSGA-III

In recent years, a theoretical understanding has rapidly advanced regarding how popular multi-objective evolutionary algorithms (MOEAs) can optimize many-objective problems. However, the benefits of using crossover in many-objective optimization are theoretically not understood, except for specifically designed benchmark functions tuned to particular crossover operators, and still lag significantly behind its practical use. In this paper, we build upon this line of research and present a theoretical runtime analysis of the widely used NSGA-III algorithm on the classical $m$-objective $m$-OneJumpZeroJump function ($m$-OJZJ for short). Our results demonstrate that NSGA-III with crossover optimizes $m$-OJZJ asymptotically faster than NSGA-III without crossover for any number $m$ of objectives for huge parameter regimes. We complement our analysis by providing a lower runtime bound on $4$-OJZJ when crossover is turned off.

cs.NE↗

arXiv2026

Parent Selection Mechanisms in Elitist Crossover-Based Algorithms

Parent selection methods are widely used in evolutionary computation to accelerate the optimization process, yet their theoretical benefits are still poorly understood. In this paper, we address this gap by proposing a parent selection strategy for the $(\mu+1)$ genetic algorithm (GA) that prioritizes the selection of maximally distant parents for crossover. We show that, with an appropriately chosen population size, the resulting algorithm solves the Jump$_k$ problem in $O(k4^kn\log(n))$ expected time. This bound is significantly smaller than the best known bound of $O(n\mu\log(\mu)+n\log(n)+n^{k-1})$ for any $(\mu+1)$~GA using no explicit diversity-preserving mechanism and a constant crossover probability. To establish this result, we introduce a novel diversity metric that captures both the maximum distance between pairs of individuals in the population and the number of pairs achieving this distance. The main novelty of our analysis is that it relies on crossover as a mechanism for creating and maintaining diversity throughout the run, rather than using crossover only in the final step to combine already diversified individuals. The insights provided by our analysis contribute to a deeper theoretical understanding of the role of crossover in the population dynamics of genetic algorithms.

cs.NE↗

arXiv2025

Towards a Rigorous Understanding of the Population Dynamics of the NSGA-III: Tight Runtime Bounds

Evolutionary algorithms are widely used for solving multi-objective optimization problems. A prominent example is NSGA-III, which is particularly well suited for solving problems involving more than three objectives, distinguishing it from the classical NSGA-II. Despite its empirical success, the theoretical understanding of NSGA III remains very limited, especially with respect to runtime analysis. A central open problem concerns its population dynamics, which involve controlling the maximum number of individuals sharing the same fitness value during the exploration process. In this paper, we make a significant step towards such an understanding by proving tight runtime bounds for NSGA-III on the bi-objective OneMinMax ($2$-OMM) problem. Firstly, we prove that NSGA-III requires $\Omega(n^2 \log(n) / \mu)$ generations in expectation to optimize $2$-OMM assuming the population size $\mu$ satisfies $n+1 \leq \mu =O(\log(n)^c(n+1))$ where $n$ denotes the problem size and $c<1$ is a constant. Apart from~\cite{opris2025multimodal}, this is the first proven lower runtime bound for NSGA-III on a classical benchmark problem. Complementing this, we secondly improve the best known upper bound of NSGA-III on the $m$-objective OneMinMax problem ($m$-OMM) of $O(n \log(n))$ generations by a factor of $\mu /(2n/m + 1)^{m/2}$ for a constant number $m$ of objectives and population size $(2n/m + 1)^{m/2} \leq \mu \in O(\sqrt{\log(n)} (2n/m + 1)^{m/2})$. This yields tight runtime bounds in the case $m = 2$, and the surprising result that NSGA-III beats NSGA-II by a factor of $\mu/n$ in the expected runtime.

cs.NE↗

arXiv2025

A First Runtime Analysis of the PAES-25: An Enhanced Variant of the Pareto Archived Evolution Strategy

This paper presents a first mathematical runtime analysis of PAES-25, an enhanced version of the original Pareto Archived Evolution Strategy (PAES) coming from the study of telecommunication problems over two decades ago to understand the dynamics of local search of MOEAs on many-objective fitness landscapes. We derive tight expected runtime bounds of PAES-25 with one-bit mutation on $m$-LOTZ until the entire Pareto front is found: $\Theta(n^3)$ iterations if $m=2$, $\Theta(n^3 \log^2(n))$ iterations if $m=4$ and $\Theta(n(2n/m)^{m/2} \log(n/m))$ iterations if $m>4$ where $n$ is the problem size and $m$ the number of objectives. To the best of our knowledge, these are the first known tight runtime bounds for an MOEA outperforming the best known upper bound of $O(n^{m+1})$ for (G)SEMO on $m$-LOTZ when $m$ is at least $4$. We also show that archivers, such as the Adaptive Grid Archiver (AGA), Hypervolume Archiver (HVA) or Multi-Level Grid Archiver (MGA), help to distribute the set of solutions across the Pareto front of $m$-LOTZ efficiently. We also show that PAES-25 with standard bit mutation optimizes the bi-objective LOTZ benchmark in expected $O(n^4)$ iterations, and we discuss its limitations on other benchmarks such as OMM or COCZ.

cs.NE↗

arXiv2025

Runtime Analyses of NSGA-III on Many-Objective Problems: Provable Exponential Speedup via Stochastic Population Update

NSGA-III is a prominent algorithm in evolutionary many-objective optimization. It is particularly well suited for optimizing problems with more than three objectives, distinguishing it from the classical NSGA-II. However, theoretical understanding of when and why NSGA-III performs well is still at an early stage. In this paper, we contribute to closing this gap by conducting rigorous runtime analyses on the classical many-objective benchmark problems $d$-\textsc{LeadingOnesTrailingZeros} ($d$-LOTZ), $d$-\textsc{CountingOnesCountingZeros} ($d$-COCZ), $d$-\textsc{OneMinMax} ($d$-OMM), and $d$-\textsc{OneJumpZeroJump} ($d$-OJZJ) for arbitrary numbers of objectives $d$. In particular, we improve upon previous results when the population size is asymptotically larger than the size of the Pareto front. Notably, in the bi-objective case, the derived upper runtime bounds are asymptotically tighter than those known for NSGA-II. For the problems $2$-OMM and $2$-OJZJ, NSGA-III even outperforms NSGA-II in terms of expected runtime for suitable population sizes $\mu$. Further, we show that a stochastic population update mechanism provably yields an exponential speedup in the expected runtime on many-objective multimodal problems such as $d$-OJZJ, as well as on the function $d$-\textsc{RRMO}, a many-objective variant of the Real-Royal-Road function, for certain parameter regimes. To complement our analysis, we also establish tight runtime bounds for NSGA-III on $2$-\textsc{OJZJ} and $4$-\textsc{OJZJ}. In particular, the result for $4$-OJZJ provides, to the best of our knowledge, the first lower bound for NSGA-III on a classical benchmark problem with more than two objectives. Deriving these bounds requires a substantially deeper analysis of the population dynamics of NSGA-III than has been achieved in previous work.

cs.NE↗

arXiv2025

Tight Runtime Guarantees From Understanding the Population Dynamics of the GSEMO Multi-Objective Evolutionary Algorithm

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order $\Omega(n^2 \log n)$, for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time $O(n^2)$, improving over the previous estimate of $O(n^2 \log n)$ for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first $\Omega(n^{k+1})$ lower bound for the OJZJ benchmark in the case that the gap parameter is $k \in \{2,3\}$. We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.

cs.NE↗

arXiv2024

Many Objective Problems Where Crossover is Provably Essential

This article addresses theory in evolutionary many-objective optimization and focuses on the role of crossover operators. The advantages of using crossover are hardly understood and rigorous runtime analyses with crossover are lagging far behind its use in practice, specifically in the case of more than two objectives. We present two many-objective problems $RR_{\text{RO}}$ and $uRR_{\text{RO}}$ together with a theoretical runtime analysis of the GSEMO and the widely used NSGA-III algorithm to demonstrate that one point crossover on $RR_{\text{RO}}$, as well as uniform crossover on $uRR_{\text{RO}}$, can yield an exponential speedup in the runtime. In particular, when the number of objectives is constant, this algorithms can find the Pareto set of both problems in expected polynomial time when using crossover while without crossover they require exponential time to even find a single Pareto-optimal point. For both problems, we also demonstrate a significant performance gap in certain superconstant parameter regimes for the number of objectives. To the best of our knowledge, this is one of the first rigorous runtime analysis in many-objective optimization which demonstrates an exponential performance gap when using crossover for more than two objectives. Additionally, it is the first runtime analysis involving crossover in many-objective optimization where the number of objectives is not necessarily constant.

cs.NE↗

arXiv2024

Theoretical Analysis of Quality Diversity Algorithms for a Classical Path Planning Problem

Quality diversity (QD) algorithms have shown to provide sets of high quality solutions for challenging problems in robotics, games, and combinatorial optimisation. So far, theoretical foundational explaining their good behaviour in practice lack far behind their practical success. We contribute to the theoretical understanding of these algorithms and study the behaviour of QD algorithms for a classical planning problem seeking several solutions. We study the all-pairs-shortest-paths (APSP) problem which gives a natural formulation of the behavioural space based on all pairs of nodes of the given input graph that can be used by Map-Elites QD algorithms. Our results show that Map-Elites QD algorithms are able to compute a shortest path for each pair of nodes efficiently in parallel. Furthermore, we examine parent selection techniques for crossover that exhibit significant speed ups compared to the standard QD approach.

cs.AI↗

arXiv2024

Illustrating the Efficiency of Popular Evolutionary Multi-Objective Algorithms Using Runtime Analysis

Runtime analysis has recently been applied to popular evolutionary multi-objective (EMO) algorithms like NSGA-II in order to establish a rigorous theoretical foundation. However, most analyses showed that these algorithms have the same performance guarantee as the simple (G)SEMO algorithm. To our knowledge, there are no runtime analyses showing an advantage of a popular EMO algorithm over the simple algorithm for deterministic problems. We propose such a problem and use it to showcase the superiority of popular EMO algorithms over (G)SEMO: OneTrapZeroTrap is a straightforward generalization of the well-known Trap function to two objectives. We prove that, while GSEMO requires at least $n^n$ expected fitness evaluations to optimise OneTrapZeroTrap, popular EMO algorithms NSGA-II, NSGA-III and SMS-EMOA, all enhanced with a mild diversity mechanism of avoiding genotype duplication, only require $O(n \log n)$ expected fitness evaluations. Our analysis reveals the importance of the key components in each of these sophisticated algorithms and contributes to a better understanding of their capabilities.

cs.NE↗

arXiv2024

Runtime Analyses of NSGA-III on Many-Objective Problems

NSGA-II and NSGA-III are two of the most popular evolutionary multi-objective algorithms used in practice. While NSGA-II is used for few objectives such as 2 and 3, NSGA-III is designed to deal with a larger number of objectives. In a recent breakthrough, Wietheger and Doerr (IJCAI 2023) gave the first runtime analysis for NSGA-III on the 3-objective OneMinMax problem, showing that this state-of-the-art algorithm can be analyzed rigorously. We advance this new line of research by presenting the first runtime analyses of NSGA-III on the popular many-objective benchmark problems mLOTZ, mOMM, and mCOCZ, for an arbitrary constant number $m$ of objectives. Our analysis provides ways to set the important parameters of the algorithm: the number of reference points and the population size, so that a good performance can be guaranteed. We show how these parameters should be scaled with the problem dimension, the number of objectives and the fitness range. To our knowledge, these are the first runtime analyses for NSGA-III for more than 3 objectives.

cs.NE↗

arXiv2024

Achieving Tight $O(4^k)$ Runtime Bounds on Jump$_k$ by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity

The JUMP$_k$ benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of $O(\text{poly}(n) + 4^k/p_c)$ for the ($\mu$+1) Genetic Algorithm ($(\mu+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic $p_c$; the best known runtime bound, in terms of function evaluations, for $p_c = \Omega(1)$ is $O((n/\chi)^{k-1})$, $\chi$ a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the $(\mu+1)$ GA on JUMP$_k$. The $(\mu+1)$-$\lambda_c$-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover $\lambda_c$ times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the $(\mu+1)$-$\lambda_c$-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of $O(\mu n \log(\mu) + 4^k)$ function evaluations for a range of $k$ under the mild assumptions $p_c = O(1/k)$ and $\mu \in \Omega(kn)$. For all constant $k$, the restriction is satisfied for some $p_c = \Omega(1)$ and it implies that the expected runtime for all constant $k$ and an appropriate $\mu = \Theta(kn)$ is bounded by $O(n^2 \log n)$, irrespective of $k$. For larger $k$, the expected time of the $(\mu+1)$-$\lambda_c$-GA is $\Theta(4^k)$, which is tight for a large class of unbiased black-box algorithms and faster than the original $(\mu+1)$ GA by a factor of $\Omega(1/p_c)$. We also show that our analysis can be extended to other unitation functions such as JUMP$_{k, \delta}$ and HURDLE.

cs.NE↗

arXiv2023

Analysing the Robustness of NSGA-II under Noise

Runtime analysis has produced many results on the efficiency of simple evolutionary algorithms like the (1+1) EA, and its analogue called GSEMO in evolutionary multiobjective optimisation (EMO). Recently, the first runtime analyses of the famous and highly cited EMO algorithm NSGA-II have emerged, demonstrating that practical algorithms with thousands of applications can be rigorously analysed. However, these results only show that NSGA-II has the same performance guarantees as GSEMO and it is unclear how and when NSGA-II can outperform GSEMO. We study this question in noisy optimisation and consider a noise model that adds large amounts of posterior noise to all objectives with some constant probability $p$ per evaluation. We show that GSEMO fails badly on every noisy fitness function as it tends to remove large parts of the population indiscriminately. In contrast, NSGA-II is able to handle the noise efficiently on \textsc{LeadingOnesTrailingZeroes} when $p<1/2$, as the algorithm is able to preserve useful search points even in the presence of noise. We identify a phase transition at $p=1/2$ where the expected time to cover the Pareto front changes from polynomial to exponential. To our knowledge, this is the first proof that NSGA-II can outperform GSEMO and the first runtime analysis of NSGA-II in noisy optimisation.

cs.NE↗

arXiv2023

Analysing Equilibrium States for Population Diversity

Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how population diversity of $(\mu+1)$ algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not.

cs.NE↗

arXiv2023

Crossover Can Guarantee Exponential Speed-Ups in Evolutionary Multi-Objective Optimisation

Evolutionary algorithms are popular algorithms for multiobjective optimisation (also called Pareto optimisation) as they use a population to store trade-offs between different objectives. Despite their popularity, the theoretical foundation of multiobjective evolutionary optimisation (EMO) is still in its early development. Fundamental questions such as the benefits of the crossover operator are still not fully understood. We provide a theoretical analysis of the well-known EMO algorithms GSEMO and NSGA-II to showcase the possible advantages of crossover: we propose classes of "royal road" functions on which these algorithms cover the whole Pareto front in expected polynomial time if crossover is being used. But when disabling crossover, they require exponential time in expectation to cover the Pareto front. The latter even holds for a large class of black-box algorithms using any elitist selection and any unbiased mutation operator. Moreover, even the expected time to create a single Pareto-optimal search point is exponential. We provide two different function classes, one tailored for one-point crossover and another one tailored for uniform crossover, and we show that immune-inspired hypermutations cannot avoid exponential optimisation times. Our work shows the first example of an exponential performance gap through the use of crossover for the widely used NSGA-II algorithm and contributes to a deeper understanding of its limitations and capabilities.

cs.NE↗

arXiv2022

Restricted Log-Exp-Analytic Power Functions

A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$h: \mathbb{R} \to \mathbb{R}, x \mapsto \left\{\begin{array}{ll} x^r, & x > 0, \\ 0, & \textnormal{ else, } \end{array}\right.$$ for $r \in \mathbb{R}$ is given. Consequently we obtain a parametric version of Tamm's theorem for this class of functions which is indeed a full generalisation of the parametric version of Tamm's theorem for $\mathbb{R}_{\textnormal{an}}^{\mathbb{R}}$-definable functions.

math.LO↗