On the Effect of Connectedness for Biobjective Multiple and Long Path Problems

Recently, the property of connectedness has been claimed to give a strong motivation on the design of local search techniques for multiobjective combinatorial optimization (MOCO). Indeed, when connectedness holds, a basic Pareto local search, initialized with at least one non-dominated solution, allows to identify the efficient set exhaustively. However, this becomes quickly infeasible in practice as the number of efficient solutions typically grows exponentially with the instance size. As a consequence, we generally have to deal with a limited-size approximation, where a good sample set has to be found. In this paper, we propose the biobjective multiple and long path problems to show experimentally that, on the first problems, even if the efficient set is connected, a local search may be outperformed by a simple evolutionary algorithm in the sampling of the efficient set. At the opposite, on the second problems, a local search algorithm may successfully approximate a disconnected efficient set. Then, we argue that connectedness is not the single property to study for the design of local search heuristics for MOCO. This work opens new discussions on a proper definition of the multiobjective fitness landscape.Comment: Learning and Intelligent OptimizatioN Conference (LION 5), Rome : Italy (2011

Local and global order 3/2 convergence of a surrogate evolutionary algorithm

A Quasi-Monte-Carlo method based on the computation of a surrogate model of the fitness function is proposed, and its convergence at super-linear rate 3/2 is proved under rather mild assumptions on the fitness function – but assuming that the starting point lies within a small neighborhood of a global maximum. A memetic algorithm is then constructed, that performs both a random exploration of the search space and the exploitation of the best-so-far points using the previous surrogate local algorithm, coupled through selection. Under the same mild hypotheses, the global convergence of the memetic algorithm, at the same 3/2 rate, is proved

When non-elitism outperforms elitism for crossing fitness valleys

Crossing fitness valleys is one of the major obstacles to function optimization. In this paper we investigate how the structure of the fitness valley, namely its depth d and length ℓ, influence the runtime of different strategies for crossing these valleys. We present a runtime comparison between the (1+1) EA and two non-elitist nature-inspired algorithms, Strong Selection Weak Mutation (SSWM) and the Metropolis algorithm. While the (1+1) EA has to jump across the valley to a point of higher fitness because it does not accept decreasing moves, the non-elitist algorithms may cross the valley by accepting worsening moves. We show that while the runtime of the (1+1) EA algorithm depends critically on the length of the valley, the runtimes of the non-elitist algorithms depend crucially only on the depth of the valley. In particular, the expected runtime of both SSWM and Metropolis is polynomial in ℓ and exponential in d while the (1+1) EA is efficient only for valleys of small length. Moreover, we show that both SSWM and Metropolis can also efficiently optimize a rugged function consisting of consecutive valleys

On the Easiest and Hardest Fitness Functions

The hardness of fitness functions is an important research topic in the field of evolutionary computation. In theory, the study can help understanding the ability of evolutionary algorithms. In practice, the study may provide a guideline to the design of benchmarks. The aim of this paper is to answer the following research questions: Given a fitness function class, which functions are the easiest with respect to an evolutionary algorithm? Which are the hardest? How are these functions constructed? The paper provides theoretical answers to these questions. The easiest and hardest fitness functions are constructed for an elitist (1+1) evolutionary algorithm to maximise a class of fitness functions with the same optima. It is demonstrated that the unimodal functions are the easiest and deceptive functions are the hardest in terms of the time-fitness landscape. The paper also reveals that the easiest fitness function to one algorithm may become the hardest to another algorithm, and vice versa

A new approach to estimating the expected first hitting time of evolutionary algorithms

AbstractEvolutionary algorithms (EA) have been shown to be very effective in solving practical problems, yet many important theoretical issues of them are not clear. The expected first hitting time is one of the most important theoretical issues of evolutionary algorithms, since it implies the average computational time complexity. In this paper, we establish a bridge between the expected first hitting time and another important theoretical issue, i.e., convergence rate. Through this bridge, we propose a new general approach to estimating the expected first hitting time. Using this approach, we analyze EAs with different configurations, including three mutation operators, with/without population, a recombination operator and a time variant mutation operator, on a hard problem. The results show that the proposed approach is helpful for analyzing a broad range of evolutionary algorithms. Moreover, we give an explanation of what makes a problem hard to EAs, and based on the recognition, we prove the hardness of a general problem

How Mutation and Selection Solve Long Path Problems in Polynomial Expected Time

It is shown by means of Markov chain analysis that unimodal binary long path problems can be solved by mutation and elitist selection in a polynomially bounded number of trials on average

A Fitness Function Elimination Theory For Blackbox Optimization And Problem Class Learning

The modern view of optimization is that optimization algorithms are not designed in a vacuum, but can make use of information regarding the broad class of objective functions from which a problem instance is drawn. Using this knowledge, we want to design optimization algorithms that execute quickly (efficiency), solve the objective function with minimal samples (performance), and are applicable over a wide range of problems (abstraction). However, we present a new theory for blackbox optimization from which, we conclude that of these three desired characteristics, only two can be maximized by any algorithm. We put forward an alternate view of optimization where we use knowledge about the problem class and samples from the problem instance to identify which problem instances from the class are being solved. From this Elimination of Fitness Functions approach, an idealized optimization algorithm that minimizes sample counts over any problem class, given complete knowledge about the class, is designed. This theory allows us to learn more about the difficulty of various problems, and we are able to use it to develop problem complexity bounds. We present general methods to model this algorithm over a particular problem class and gain efficiency at the cost of specifically targeting that class. This is demonstrated over the Generalized Leading-Ones problem and a generalization called LO∗∗ , and efficient algorithms with optimal performance are derived and analyzed. We also iii tighten existing bounds for LO∗∗∗. Additionally, we present a probabilistic framework based on our Elimination of Fitness Functions approach that clarifies how one can ideally learn about the problem class we face from the objective functions. This problem learning increases the performance of an optimization algorithm at the cost of abstraction. In the context of this theory, we re-examine the blackbox framework as an algorithm design framework and suggest several improvements to existing methods, including incorporating problem learning, not being restricted to blackbox framework and building parametrized algorithms. We feel that this theory and our recommendations will help a practitioner make substantially better use of all that is available in typical practical optimization algorithm design scenarios