Saku Kukkonen

Real-parameter optimization with differential evolution

This study reports how the differential evolution (DE) algorithm performed on the test bed developed for the CEC05 contest for real parameter optimization. The test bed includes 25 scalable functions, many of which are both non-separable and highly multi-modal. Results include DEs performance on the 10 and 30-dimensional versions of each function

GDE3: the third evolution step of generalized differential evolution

A developed version of generalized differential evolution, GDE3, is proposed. GDE3 is an extension of differential evolution (DE) for global optimization with an arbitrary number of objectives and constraints. In the case of a problem with a single objective and without constraints GDE3 falls back to the original DE. GDE3 improves earlier GDE versions in the case of multi-objective problems by giving a better distributed solution. Performance of GDE3 is demonstrated with a set of test problems and the results are compared with other methods

Constrained Real-Parameter Optimization with Generalized Differential Evolution

This paper presents results for the CEC 2006 Special Session on Constrained Real-Parameter Optimization where the Generalized Differential Evolution (GDE) has been used to solve given test problems. The given problems consist of 24 problems having one objective function and one or more in-/equality constraints. Almost all the problems were solvable in a given maximum number of solution candidate evaluations. The paper also shows how GDE actually needs lower number of function evaluations than usually required.

Improved Pruning of Non-Dominated Solutions Based on Crowding Distance for Bi-Objective Optimization Problems

In this paper an algorithm for pruning a set of non-dominated solutions is proposed. The algorithm is based on the crowding distance calculation used in the elitist non-dominated sorting genetic algorithm (NSGA-II). The time complexity class of the new algorithm is estimated and in most cases it is the same as for the original pruning algorithm. Numerical results also support this estimate. For used bi-objective test problems, the proposed pruning algorithm is demonstrated to provide better distribution compared to the original pruning algorithm of NSGA-II. However, with tri-objective test problems there is no improvement and this study reveals that crowding distance does not estimate crowdedness well in this case and presumably also in cases of more objectives.

Multi-objective test problems, linkages, and evolutionary methodologies

Existing test problems for multi-objective optimization are criticized for not having adequate linkages among variables. In most problems, the Pareto-optimal solutions correspond to a fixed value of certain variables and diversity of solutions comes mainly from a random variation of certain other variables. In this paper, we introduce explicit linkages among variables so as to develop difficult two and multi-objective test problems along the lines of ZDT and DTLZ problems. On a number of such test problems, this paper compares the performance of a number of EMO methodologies having (i) variable-wise versus vector-wise recombination operators and (ii) spatial versus unidirectional recombination operators. Interesting and useful conclusions on the use of above operators are made from the study.

Ranking-Dominance and Many-Objective Optimization

An alternative relation to Pareto-dominance is studied. The relation is based on ranking a set of solutions according to each separate objective and an aggregation function to calculate a scalar fitness value for each solution. The relation is called as ranking-dominance and it tries to tackle the curse of dimensionality commonly observed in multi-objective optimization. Ranking-dominance can be used to sort a set of solutions even for a large number of objectives when the Pareto-dominance relation cannot distinguish solutions from one another anymore. This permits the search to advance even with a large number of objectives. Experimental results indicate that in some cases the selection based on ranking-dominance is able to advance the search towards the Pareto-front better than the selection based on Pareto-dominance. However, in some cases it is also possible that the search does not proceed into direction of the Pareto-front because the ranking-dominance relation permits deterioration of individual objectives. The results also show that when the number of objectives increases, the selection based on just Pareto-dominance without diversity maintenance is able to advance the search better than with diversity maintenance. Therefore, diversity maintenance connives at difficulties solving problems with a high number of objectives.

A fast and effective method for pruning of non-dominated solutions in many-objective problems

Diversity maintenance of solutions is an essential part in multi-objective optimization. Existing techniques are suboptimal either in the sense of obtained distribution or execution time. This paper proposes an effective and relatively fast method for pruning a set of non-dominated solutions. The proposed method is based on a crowding estimation technique using nearest neighbors of solutions in Euclidean sense, and a technique for finding these nearest neighbors quickly. The method is experimentally evaluated, and results indicate a good trade-off between the obtained distribution and execution time. Distribution is good also in many-objective problems, when number of objectives is more than two.

An Extension of Generalized Differential Evolution for Multi-objective Optimization with Constraints

In this paper an extension of Generalized Differential Evolution for constrained multi-objective (Pareto-)optimization is proposed. The proposed extension adds a mechanism for maintaining extent and distribution of the obtained non-dominated solutions approximating a Pareto front. The proposed extension is tested with a set of five benchmark multi-objective test problems and results are numerically compared to known global Pareto fronts and to results obtained with the elitist Non-Dominated Sorting Genetic Algorithm and Generalized Differential Evolution. Results show that the extension improves extent and distribution of solutions of Generalized Differential Evolution.

Performance assessment of Generalized Differential Evolution 3 with a given set of constrained multi-objective test problems

This paper presents results for the CEC 2009 Special Session on “Performance Assessment of Constrained / Bound Constrained Multi-Objective Optimization Algorithms” when Generalized Differential Evolution 3 has been used to solve a given set of test problems. The set consist of 23 problems having two, three, or five objectives. Problems have different properties in the sense of separability, modality, and geometry of the Pareto-front. The most of the problems are unconstrained, but 10 problems have one or two constraints. According to the numerical results with an inverted generational distance, Generalized Differential Evolution 3 performed well with all the problems except with one five objective problem. It was noticed that a low crossover control parameter value provides the best average results according to the metric.

Color features for quality control in ceramic tile industry

We study visual quality control in the ceramics industry. In tile manufacturing, it is important that in each set of tiles, every single tile looks similar. Currently, the estimation is usually done by human vision. Our goal is to design a machine vision system that can estimate the sufficient similarity, or same appearance, to the human eye. Our main approach is to use accurate spectral representation of color, and com- pare spectral features to the RGB color features. A laboratory system for color measurements is built. Experimentations with five classes of brown tiles are presented and discussed. In addition to the k-nearest neighbor (k-NN) classifier, a neural network called the self-organizing map (SOM) is used to provide understanding of the spectral features. Every single spectrum in each tile of a training set is used as input to a 2-D SOM. The SOM is analyzed to understand how spectra are clustered. As a result, tiles are classified using a trained 2-D SOM. It is also of interest to know whether the order of spectral colors can be determined. In our approach, all spectra are clustered in a 1-D SOM, and each pixel (spectrum) is presented by pseudocolors according to the trained nodes. Finally, the results are compared to experiments with human vision.