Multi-objective Optimization In Computational I...
We have provided here evidence, for the first time, that cells within both T cell and neutrophil populations exhibit a continuum of inherent directionalities and translational speeds. Further, we have shown that cells do not simultaneously perform very fast translational and turn movements. We have developed a novel framework to fit statistical distributions to cell translation and turn speeds whilst accounting for experimental bias. Thereafter, the manner in which these two components of motility combine to impact overall spatial exploration is analysed through a novel coupling of 3D agent-based simulation with multi-objective optimization. This latter framework for the first time calibrates and assesses putative motility models through simultaneous consideration of several motility metrics, accounting for trade-offs in performance against each. These frameworks provide the means to robustly analyse and accurately reproduce cellular motility patterns, as they explicitly reflect the constraints of in vivo data.
Multi-objective optimization in computational i...
The advent of two-photon in vivo cellular imaging techniques facilitates detailed examination of cellular motility and interaction. The resultant data permits identification of cellular motility strategies, which can be incorporated into broader immune simulations to understand the development and potential manipulation of the immune response. Determining which motility model best fits a biological dataset requires simultaneous consideration of several metrics of motility; three dimensional motility is too intricate a phenomenon to be fully specified in only one metric. Here we have evaluated the capacity of six random walk models, including Brownian motion, Lévy walk and four correlated random walks, to reproduce the motility dynamics of lymph node T cells and neutrophil datasets under inflammatory conditions. Our evaluation is made possible through the development of a novel simulation calibration methodology, where multi-objective optimization identifies parameter values that provide optimal trade-offs for a given model against several metrics of motility.
Previous lymphocyte modeling efforts have incorporated explicit cellular arrest phases between periods of fixed speed, straight-line motility [15, 26]. Our in vivo datasets do not record cells as being stationary, or moving in straight lines (S1A and S1B Fig). As such, we have explored CRW models that explicitly capture distributions of translational and turn speeds. Other work has focused on modeling lymphocytes as point-processes confined to the lymph node reticular network , explicitly modeling cellular morphology [25, 28], and conceptualizing cell trajectories as features of environmental obstacles . The possibility of calibrating the configuration of an environment by proxy of the resultant cellular motility is intriguing. Our multi-objective optimization framework is independent of the motility paradigm and could be more broadly applied in these contexts.
We opted to employ three objectives in our multi-objective approach, based on the pooled translational speeds of all cells across all time points into a single distribution, similarly for turn speeds, and track meandering indices. We consider this the minimum required to accurately specify motility, capturing how cells move translationally through space, how subsequent trajectories are correlated, and how these two aspects integrate to define overall spatial coverage. Multi-objective optimisation can accommodate more objectives, and hence additional motility metrics could be incorporated (or substituted). In particular, we believe there is merit in studying how recent, more sophisticated motility metrics might be incorporated into our framework [20, 21]. It is practical, rather than technical, considerations that limit the number of objectives one can use: in our experience the number of Pareto front members tends to increase with each additional objective, and more objectives constitute a more complex problem which can require greater computational effort to solve to a similar extent (e.g., as measured through objective KS values). Convention in multi-objective optimisation dictates that one choose objectives which are not correlated with one another; to do so increases the complexity of the optimisation problem whilst providing little benefit in capturing better quality solutions. Candidates for additional objectives might include the median track translational or turn speed distributions, however we note that for our favoured motility model, the inverse heterogeneous CRW (IHeteroCRW), these characteristics are well captured despite not being explicit criteria in model calibration (S17B, S17C, S23B and S23C Figs).
This work discusses robustness assessment during multi-objective optimization with a Multi-Objective Evolutionary Algorithm (MOEA) using a combination of two types of robustness measures. Expectation quantifies simultaneously fitness and robustness, while variance assesses the deviation of the original fitness in the neighborhood of the solution. Possible equations for each type are assessed via application to several benchmark problems and the selection of the most adequate is carried out. Diverse combinations of expectation and variance measures are then linked to a specific MOEA proposed by the authors, their selection being done on the basis of the results produced for various multi-objective benchmark problems. Finally, the combination preferred plus the same MOEA are used successfully to obtain the fittest and most robust Pareto optimal frontiers for a few more complex multi-criteria optimization problems.
The covariancematrix adaptation evolution strategy (CMA-ES) is one of themost powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.
GOMORS is a parallel response surface-assisted evolutionary algorithm approach to multi-objective optimization that is designed to obtain good non-dominated solutions to black box problems with relatively few objective function evaluations. GOMORS uses Radial Basic Functions to iteratively compute surrogate response surfaces as an approximation of the computationally expensive objective function. A multi objective search utilizing evolution, local search, multi method search and non-dominated sorting is done on the surrogate radial basis function surface because it is inexpensive to compute. A balance between exploration, exploitation and diversification is obtained through a novel procedure that simultaneously selects evaluation points within an algorithm iteration through different metrics including Approximate Hypervolume Improvement, Maximizing minimum domain distance, Maximizing minimum objective space distance, and surrogate-assisted local search, which can be computed in parallel. The results are compared to ParEGO (a kriging surrogate method solving many weighted single objective optimizations) and the widely used NSGA-II. The results indicate that GOMORS outperforms ParEGO and NSGA-II on problems tested. For example, on a groundwater PDE problem, GOMORS outperforms ParEGO with 100, 200 and 400 evaluations for a 6 dimensional problem, a 12 dimensional problem and a 24 dimensional problem. For a fixed number of evaluations, the differences in performance between GOMORS and ParEGO become larger as the number of dimensions increase. As the number of evaluations increase, the differences between GOMORS and ParEGO become smaller. Both surrogate-based methods are much better than NSGA-II for all cases considered.
Many authors (for instance, Deb et al.  and Zhang et al. ) have successfully employed evolutionary strategies for solving multi-objective optimization problems. Even with the improvement over traditional methods, these algorithms require, typically, many objective function evaluations which can be infeasible for computationally expensive problems. Added challenges to multi-objective optimization of expensive functions arise with increase in dimensionality of the decision variables and objectives.
The use of iterative response surface modeling or function approximation techniques inside an optimization algorithm can be highly beneficial in reducing time for computing objectives for multi-objective optimization of such problems. Since the aim of efficient multi-objective optimization is to identify good solutions within a limited number of expensive function evaluations, approximating techniques can be incorporated into the optimization process to reduce computational costs. Gutmann  introduced the idea of using radial basis functions (RBF)  for addressing single objective optimization of computationally expensive problems. Jin et al.  appears to be the first journal paper to combine a non quadratic response surface with a single objective evolutionary algorithm by using neural net approximations. Regis and Shoemaker  were the first to use Radial Basis Functions (not a neural net) to improve the efficiency of an evolutionary algorithm with limited numbers of evaluations. Later they introduced a non-evolutionary algorithm Stochastic-RBF  , which is a very effective radial basis function-based method for single objective optimization of expensive global optimization problems. These methods have been extended to include parallelism ; high-dimensional problems ; constraints ; local optimization [49, 50]; integer problems [30, 31] and other extensions [39, 41]. Kriging-based methods have also been explored for addressing single objective optimization problems [9, 16, 19]. Jones et al.  introduced Efficient Global Optimization (EGO), which is an algorithm for single objective global optimization within a limited budget of evaluations. 041b061a72