Abstract-We propose a surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive optimization problems with more than three objectives. The proposed algorithm is based on a recently developed evolutionary algorithm for many-objective optimization that relies on a set of adaptive reference vectors for selection. The proposed surrogateassisted evolutionary algorithm uses Kriging to approximate each objective function to reduce the computational cost. In managing the Kriging models, the algorithm focuses on the balance of diversity and convergence by making use of the uncertainty information in the approximated objective values given by the Kriging models, the distribution of the reference vectors as well as the location of the individuals. In addition, we design a strategy for choosing data for training the Kriging model to limit the computation time without impairing the approximation accuracy. Empirical results on comparing the new algorithm with the state-of-the-art surrogate-assisted evolutionary algorithms on a number of benchmark problems demonstrate the competitiveness of the proposed algorithm.

Evolutionary algorithms are widely used for solving multiobjective optimization problems but are often criticized because of a large number of function evaluations needed. Approximations, especially function approximations, also referred to as surrogates or metamodels are commonly used in the literature to reduce the computation time. This paper presents a survey of 45 different recent algorithms proposed in the literature between 2008-2016 to handle computationally expensive multiobjective optimization problems. Several algorithms are discussed based on what kind of an approximation such as problem, function or fitness approximation they use. Most emphasis is given to function approximation based algorithms. We also compare these algorithms based on different criteria such as metamodelling technique and evolutionary algorithm used, type and dimensions of the problem solved, handling constraints, training time and the type of evolution control. Furthermore, we identify and discuss some promising elements and major issues among algorithms in the literature related to using an approximation and numerical settings used. In addition, we discuss selecting an algorithm to solve a given computationally expensive mmultiobjective optimization problem based on the dimensions in both objective and decision spaces and the computation budget available.

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We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one's preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.

A survey on handling computationally expensive multiobjective optimization problems using surrogates: non-nature inspired methods Tabatabaei, Mohammad; Hakanen, Jussi; Hartikainen, Markus; Miettinen, Kaisa; Sindhya, Karthik Tabatabaei, M., Hakanen, J., Hartikainen, M., Miettinen, K., & Sindhya, K. (2015). A survey on handling computationally expensive multiobjective optimization problems using surrogates: non-nature inspired methods. Structural and Multidisciplinary Optimization, 52 (1) AbstractComputationally expensive multiobjective optimization problems arise, e.g. in many engineering applications, where several conflicting objectives are to be optimized simultaneously while satisfying constraints. In many cases, the lack of explicit mathematical formulas of the objectives and constraints may necessitate conducting computationally expensive and time-consuming experiments and/or simulations. As another challenge, these problems may have either convex or nonconvex or even disconnected Pareto frontier consisting of Pareto optimal solutions. Because of the existence of many such solutions, typically, a decision maker is required to select the most preferred one. In order to deal with the high computational cost, surrogate-based methods are commonly used in the literature. This paper surveys surrogate-based methods proposed in the literature, where the methods are independent of the underlying optimization algorithm and mitigate the computational burden to capture different types of Pareto frontiers. The methods considered are classified, discussed and then compared. These methods are divided into two frameworks: the sequential and the adaptive frameworks. Based on the comparison, we recommend the adaptive framework to tackle the aforementioned challenges.

Complexity in solving real-world multicriteria optimization problems often stems from the fact that complex, expensive, and/or time-consuming simulation tools or physical experiments are used to evaluate solutions to a problem. In such settings, it is common to use efficient computational models, often known as surrogates or metamodels, to approximate the outcome (objective or constraint function value) of a simulation or physical experiment. The presence of multiple objective functions poses an additional layer of complexity for surrogate-assisted optimization.For example, complexities may relate to the appropriate selection of metamodels for the individual objective functions, extensive training time of surrogate models, or the optimal use of many-core computers to approximate efficiently multiple objectives simultaneously. Thinking out of the box, complexity can also be shifted from approximating the individual objective functions to approximating the entire Pareto front. This leads to further complexities, namely, how to validate statistically and apply the techniques developed to real-world problems. In this paper, we discuss emerging complexity-related topics in surrogate-assisted multicriteria optimization that may not be prevalent in nonsurrogate-assisted single-objective optimization. These complexities are motivated using several real-world problems in which the authors were involved. We then discuss several promising future research directions and prospective solutions to tackle emerging complexities in surrogate-assisted multicriteria optimization. Finally, we provide insights from an industrial point of view into how surrogate-assisted multicriteria optimization techniques can be developed and applied within a collaborative business environment to tackle real-world problems.

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