Multiobjective optimization and evolutionary algorithms

What is Multiobjective optimization

Multiobjective optimization (MO), also known as multicritera optimization is an optimization of a vector function $\mathbf{f}(\mathbf{x})$, where $\mathbf{x}$ is a vector of parameters to the vector function $\mathbf{f}$, usually being a minimisation of components of $\mathbf{x}$ with respect to Pareto efficiency.

Pareto efficiency, also known as Pareto optimality, is a best choice of criteria, such that no criterion must be worsened in order to better some another criterion. $\mathbf{x}$ is Pareto optimal, iff does not exist $\mathbf{y}:y_i < x_i$.

What are Evolutionary Algorithms

Evolutionary Algorithms (EAs) are problem solving principles from nature applied to the metaheuristic search. Using mutation, crossover, and selection operators, stochastic search is developed to evolve better individuals in quest for (a) globally best individual(s).

When applying EAs to MO, multiple pareto optimal solutions are in the quest. One of the possible EAs for exploring the MO search space is the Differential Evolution (e.g. jDE, 'rand/1/bin': $v_i=x_i+F(y_i-z_i)$). The most known Genetic Algorithm (GA) for the MO is the Non-dominated Sorting Genetic Algorithm - II (NSGA-II) by Deb.

Performance Assessment of MOEAs

To assess an MOEA, test function suites are used, which comprise of several multiobjective test functions to solve.

When test function results are collected, performance metrics are used to assess the quality of an MOEA with respect to each performance metric. If the preference of decision maker is same as with performance metric, then we can say that an MOEA has better quality (with respect to this preference).

There exist several performance metrics (see Knowles, Thiele, Zitzler: TIK-Report No. 214, 2006), such as:

• Dominance Ranking,
• Quality Indicators
  • Hypervolume Indicator $I_H$,
  • Epsilon Indicator $I_{\epsilon}$,
  • R Indicator $I_R$,
• Empirical Attainment Function.

Our Papers

(pdf) A. Zamuda, J. Brest. Tree Model Reconstruction Innovization Using Multi-objective Differential Evolution. 2012 IEEE World Congress on Computational Intelligence (IEEE WCCI 2012), Brisbane, Australia, 2012, pp. 575-582.

BibTeX

(pdf) A. Zamuda, J. Brest, B. Boskovic, V. Zumer. Differential Evolution with Self-adaptation and Local Search for Constrained Multiobjective Optimization. IEEE Congress on Evolutionary Computation (CEC) 2009, 2009, pp. 195-202.

BibTeX

(pdf) A. Zamuda, J. Brest, B. Boskovic, V. Zumer. Differential Evolution for Multiobjective Optimization with Self Adaptation. The 2007 IEEE Congress on Evolutionary Computation CEC2007, 2007, pp. 3617-3624.

BibTeX

A. Zamuda. Differential Evolution for Parameterized Procedural Woody Plant Models Reconstruction: PhD thesis. Faculty of Electrical Engineering and Computer Science, 2012.

A. Zamuda. Samoprilagajanje krmilnih parametrov pri algoritmu diferencialne evolucije za večkriterijsko optimizacijo: MSc thesis. Faculty of Electrical Engineering and Computer Science, 2008.

(pdf) A. Zamuda, J. Brest, B. Bošković, V. Žumer. Študija samoprilagajanja krmilnih parametrov pri algoritmu DEMOwSA. Elektrotehniški vestnik, 2008, vol. 75, no. 4, pp. 223-228.

BibTeX

(pdf) A. Zamuda, J. Brest, B. Bošković, V. Žumer. Diferencialna evolucija za večkriterijsko optimizacijo s samoprilagajanjem in z lokalnim preiskovanjem SQP. Zbornik sedemnajste mednarodne Elektrotehniške in računalniške konference ERK, 2008, pp. 103-106.

BibTeX

(pdf) J. Brest, A. Zamuda, B. Bošković, V. Žumer. Večkriterijska optimizacija: primerjava algoritmov MOjDE in DEMO. Zbornik šestnajste mednarodne Elektrotehniške in računalniške konference ERK, 2007, pp. 85-88.

BibTeX

(pdf) A. Zamuda, J. Brest, B. Bošković, V. Žumer. Večkriterijska optimizacija: eksperimentalni rezultati algoritmov MOjDE in DEMO. Zbornik šestnajste mednarodne Elektrotehniške in računalniške konference ERK, 2007, pp. 89-92.

BibTeX

Journals

• IEEE Transactions on Evolutionary Computation (Online - IEEE Xplore)
• Evolutionary Computation (The MIT Press)
• Journal of Global Optimization
• Soft Computing (online content)
• Applied Intelligence (online content)

Magazines:

• IEEE Computational Intelligence Magazine
• IEEE Internet Computing Magazine (online content)
• Electrotechnical Review (online content)

Conferences

• IEEE Congress on Evolutionary Computation (CEC2007)
  • Special Section on Performance Assessment of Multi-Objective Optimization Algorithms (Test codes)
• Genetic and Evolutionary Computation Conference (GECCO-2007)
• Parallel Problem Solving From Nature (PPSN)
• Evolutionary Multi-Criterion Optimization (EMO2007)

External links

• Prof. Kalyanmoy Deb's Homepage @ Kanpur Genetic Algorithms Laboratory
• Carlos A. Coello Coello's Publications
• Walking Fish Group
• Tea Tušar's DEMO program