High-Dose-Rate (HDR) brachytherapy (BT) treatment planning involves determining an appropriate schedule of a radiation source moving through a patient's body such that target volumes are irradiated with the planning-aim dose as much as possible while healthy tissues (i.e., organs at risk) should not be irradiated more than certain thresholds. Such movement of a radiation source can be defined by so-called dwell times at hundreds of potential dwell positions, which must be configured to satisfy a clinical protocol of multiple different treatment criteria within a strictly-limited time frame of not more than one hour. In this article, we propose a bi-objective optimization model that intuitively encapsulates in two objectives the complicated high-dimensional multi-criteria nature of the BT treatment planning problem. The resulting Pareto-optimal fronts exhibit possible trade-offs between the coverage of target volumes and the sparing of organs at risk, thereby intuitively facilitating the decision-making process of treatment planners when creating a clinically-acceptable plan. We employ real medical data for conducting experiments and benchmark four different Multi-Objective Evolutionary Algorithms (MOEAs) on solving our problem: the Non-dominated Sorting Genetic Algorithm II (NSGA-II), the Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), the Multi-objective Adapted Maximum-Likelihood Gaussian Model Iterated Density-Estimation Evolutionary Algorithm (MAMaLGaM), and the recently-introduced Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA). The variation operator that is specific to MO-RV-GOMEA enables performing partial evaluations to efficiently calculate objective values of offspring solutions without incurring the cost of fully recomputing the radiation dose distributions for new treatment plans. Experimental results show that MO-RV-GOMEA is the best performing MOEA that effectively exploits dependencies between decision variables to efficiently solve the multi-objective BT treatment planning problem.

The recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA), with a lean, but sufficient, linkage model and an efficient variation operator, has been shown to be a robust and efficient methodology for solving single objective (SO) optimization problems with superior performance compared to classic genetic algorithms (GAs) and estimation-of-distribution algorithms (EDAs). In this paper, we bring the strengths of GOMEAs to the multiobjective (MO) optimization realm. To this end, we modify the linkage learning procedure and the variation operator of GOMEAs to better suit the need of finding the whole Paretooptimal front rather than a single best solution. Based on state-of-the-art studies on MOEAs, we further pinpoint and incorporate two other essential components for a scalable MO optimizer. First, the use of an elitist archive is beneficial for keeping track of non-dominated solutions when the main population size is limited. Second, clustering can be crucial if different parts of the Pareto-optimal front need to be handled differently. By combining these elements, we construct a multi-objective GOMEA (MO-GOMEA). Experimental results on various MO optimization problems confirm the capability and scalability of our MO-GOMEA that compare favorably with those of the well-known GA NSGA-II and the more recently introduced EDA mohBOA.

e recently introduced Multi-Objective Gene-pool Optimal Mixing Evolutionary Algorithm (MO-GOMEA) exhibits excellent scalability in solving a wide range of challenging discrete multi-objective optimization problems. In this paper, we address scalability issues in solving multi-objective optimization problems with continuous variables by introducing the Multi-Objective Real-Valued GOMEA (MO-RV-GOMEA), which combines MO-GOMEA with aspects of the multi-objective estimation-of-distribution algorithm known as MAMaLGaM. MO-RV-GOMEA exploits linkage structure in optimization problems by performing distribution estimation, adaptation, and sampling as well as solution mixing based on an explicitlyde ned linkage model. Such a linkage model can be de ned a priori when some problem-speci c knowledge is available, or it can be learned from the population. e scalability of MO-RV-GOMEA using di erent linkage models is compared to the state-of-the-art multi-objective evolutionary algorithms NSGA-II and MAMaLGaM on a wide range of benchmark problems. MO-RV-GOMEA is found to retain the excellent scalability of MO-GOMEA through the successful exploitation of linkage structure, scaling substantially be er than NSGA-II and MAMaLGaM. is scalability is even further improved when partial evaluations are possible, achieving strongly sub-linear scalability in terms of the number of evaluations.

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The Multi-objective Gene-pool Optimal Mixing Evolutionary Algorithm (MO-GOMEA) has been shown to be a promising solver for multi-objective combinatorial optimization problems, obtaining an excellent scalability on both standard benchmarks and real-world applications. To attain optimal performance, MO-GOMEA requires its two parameters, namely the population size and the number of clusters, to be set properly with respect to the problem instance at hand, which is a non-trivial task for any EA practitioner. In this article, we present a new version of MO-GOMEA in combination with the so-called Interleaved Multi-start Scheme (IMS) for the multi-objective domain that eliminates the manual setting of these two parameters. The new MO-GOMEA is then evaluated on multiple benchmark problems in comparison with two well-known multi-objective evolutionary algorithms (MOEAs): Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D). Experiments suggest that MO-GOMEA with the IMS is an easy-to-use MOEA that retains the excellent performance of the original MO-GOMEA.