Multiobjective optimization with evolutionary algorithms
Michael Lahanas, Kostas Karouzakis, Natasa Milickovic, Stavroula Giannouli and Dimos Baltas
Evolutionary or genetic algorithms have been used in a variety of optimization problems. We have a special case for multiobjective optimization evolutionary algorithms, MOEAs. Goal of multiobjective optimization is the determination of a representative set of efficient solutions, i.e. a representative set of the Pareto front.
We used the first generation MOEAs, such as NSGA, NPGA and NRGA for dose optimization problems in brachytherapy. Later we used the more powerful elitistic algorithms SPEA and NSGA-II.
Many studies which compared the efficiency of various MOEAs involved functions with 2-10 decision variables and 2-3 objectives. We compare two multiobjective evolutionary algorithms, with deterministic gradient based optimization methods for the dose optimization problem in high-dose rate (HDR) brachytherapy or intensity modulated radiotherapy (IMRT). The optimization considers up to 300 parameters in brachytherapy and up to 10000 and more in IMRT. The objectives are expressed in terms of statistical parameters, from dose distributions. These parameters are approximated from dose values from a small number of points. For these objectives it is known that the deterministic algorithms converge to the global Pareto front. The evolutionary algorithms produce only local Pareto-optimal fronts.
The performance of the multiobjective evolutionary algorithms is improved if a small part of the population is initialized with solutions from deterministic algorithms. These solutions are called also supporting solutions. An explanation is that only a very small part of the search space is close to the global Pareto front.
Figure 1 Pareto front obtained by Pareto based MOEAs with and without initialization by deterministic algorithms
We estimated the performance of the algorithms in some cases in terms of probability compared to a random optimum search method. For this we estimated by random search the visiting probability of random configurations in objective space. We have found in some cases that this visiting probability distribution can be approximated by a multidimensional-gaussian distribution. By fitting the parameters of this distribution we have extrapolated the visiting probability of efficient solutions.
Figure 2 Distributions of random solutions in objective space
Figure 3 Approximation of this distribution by a two-dimensional Gaussian distribution
Figure 4 Extrapolation of probability to obtain a solution within an interval in the objective space. The distribution of random solutions, the MO algorithm SPEA and a gradient based deterministic algorithm is shown.
We have found that evolutionary algorithms are able to find solutions much beyond the capabilities of random search. Still in comparison with dedicated deterministic gradient based algorithms this performance is small, especially for multidimensional problems. Gradient based algorithms use the information from derivatives to drive the searcher close to the global minimum in a region of objective space of extreme low visiting probability.
Our current interest is to compare Pareto ranking based MOEAs with local search based and other heuristic approaches such as multiobjective simulated annealing.
We have found that the crossover operator determines the efficiency of MOEAs to a large extend. For multiobjective optimization problems with more than 2 objectives we have found that current algorithms require an accumulation of non-dominated solutions. Together with a good initialization of some solutions the performance of evolutionary algorithm increases significantly.
Still there are problems such as in intensity modulated beam radiotherapy where algorithms like NSGA-II still fail to find good solutions in comparison to deterministic algorithms. Here NSGA-IIc the controlled elitist version with supported solutions provides significant better results than NSGA-II, showing that lateral diversity is important.
Application of MOEAs in radiotherapy treatment planning systems
We have developed a toolkit WinOpt-HDR for the test of multiobjective evolutionary algorithms for HDR brachytherapy inverse planning. The algorithms have been further improved resulting in a commercial treatment planning system.
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M. Lahanas, D. Baltas and N. Zamboglou, Anatomy-based three-dimensional dose optimization in brachytherapy using multiobjective genetic algorithms, Med. Phys. 26 1904-1918, 1999
M. Lahanas, D. Baltas and N. Zamboglou. A Hybrid Evolutionary Multiobjective Algorithm for Anatomy-Based Dose Optimization Algorithm in HDR Brachytherapy, Phys. Med. Biol. 48 399-415, 2003
M. Lahanas, K. Karouzakis, S. Giannouli, R. F. Mould and D. Baltas. Inverse Planning in Brachytherapy: Radium to High Dose Rate 192 Iridium Afterloading, to be published as invited review in Nowotwory Journal of Oncology 2003
M. Lahanas, D. Baltas and E. Schreibmann. Application of Multiobjective Evolutionary Algorithms for radiotherapy, submitted to Applied Soft Computing 2003
Eduard Schreibmann, Michael Lahanas, Lei Xing and Dimos Baltas, Multiobjective evolutionary optimization of number of beams, their orientation and weights for IMRT, Phys. Med. Biol. 49 747-770, 2004
M. Lahanas, Application of Multiobjective Evolutionary Optimization Algorithms in Medicine, in Application of Multiobjective Evolutionary Optimization Algorithms, edited by C. Coello Coello et al, World Scientific 2004
M. Lahanas, N. Milickovic, D. Baltas and N. Zamboglou, Application of Multiobjective Evolutionary Algorithms for Dose Optimization Problems in Brachytherapy, in Proceedings of the first international conference, EMO 2001, Zurich, Switzerland, edited by E. Zitzler, K. Deb, L. Thiele, C. A. Coello Coello, D. Corne, Lecture Notes in Computer Science Vol. 1993, Springer, 574-587, 2001
N. Milickovic, M. Lahanas, D. Baltas and N. Zamboglou, Comparison of Evolutionary and Deterministic Multiobjective Algorithms for Dose Optimization in Brachytherapy, in Proceedings of the first international conference, EMO 2001, Zurich, Switzerland, edited by E.Zitzler, K. Deb, L. Thiele, C. A. Coello Coello, D. Corne, Lecture Notes in Computer Science Vol. 1993, Springer 167-180, 2001.
EUROGEN 2001 - Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems
M. Lahanas, N. Milickovic, M. Papagiannopoulou, K. Karouzakis, D. Baltas and N. Zamboglou, Application of a Hybrid NSGA-II Multiobjective Algorithm for Anatomy based Dose Optimization in Brachytherapy, "EUROGEN 2001 - Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems" Athens, Greece 19-21 September 2001.
N. Milickovic, M. Lahanas, M. Papagiannopoulou, K. Karouzakis, D. Baltas and N. Zamboglou, Application of Multiobjective Genetic Algorithms in Anatomy based Dose Optimization in Brachytherapy and its Comparison with Deterministic Algorithms, "EUROGEN 2001 - Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems" Athens, Greece 19-21 September 2001.
M. Lahanas, E. Schreibmann, N. Milickovic and D. Baltas. Intensity Modulated Beam Radiation Therapy Dose Optimization with Multiobjective Evolutionary Algorithms, Lecture Notes in Computer Science 2632, Springer, EMO 2003, pp 648-661
23rd Annual International Conference of the IEEE - Engineering in Medicine and Biology 2001
N. Milickovic, M. Lahanas, M. Papagiannopoulou, K. Karouzakis, D. Baltas and N. Zamboglou, Application of Multiobjective Genetic Algorithms in Anatomy Based Dose Optimization in Brachytherapy and its Comparison with Deterministic Algorithms, Proceedings of the 23rd IEEE EMBS Int. Conference, 23rd Annual International Conference of the IEEE - Engineering in Medicine and Biology Society, Istanbul, Turkey, 25-28 October, 2001.
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