Covariance matrix adaptation pareto archived evolution strategy with hypervolume-sorted adaptive grid algorithm

Abstract

Abstract. Real-world problems often involve the optimisation of multiple conflicting objectives. These problems, referred to as multi-objective optimisation problems, are especially challenging when more than three objectives are considered simultaneously. This paper proposes an algorithm to address this class of problems. The proposed algorithm is an evolutionary algorithm based on an evolution strategy framework, and more specifically, on the Covariance Matrix Adaptation Pareto Archived Evolution Strategy (CMA-PAES). A novel selection mechanism is introduced and integrated within the framework. This selection mechanism makes use of an adaptive grid to perform a local approximation of the hypervolume indicator which is then used as a selection criterion. The proposed implementation, named Covariance Matrix Adaptation Pareto Archived Evolution Strategy with Hypervolume-sorted Adaptive Grid Algorithm (CMA-PAES-HAGA), overcomes the limitation of CMA-PAES in handling more than two objectives and displays a remarkably good performance on a scalable test suite in five, seven, and ten-objective problems. The performance of CMA-PAES-HAGA has been compared with that of a competition winning meta-heuristic, representing the state-of-the-art in this sub-field of multi-objective optimisation.The proposed algorithm has been tested in a seven-objective real-world application, i.e. the design of an aircraft lateral control system. In this optimisation problem, CMA-PAES-HAGA greatly outperformed its competitors.

Progressive preference articulation for decision making in multi-objective optimisation problems

“…This paper proposes a novel progressive preference articulation mechanism for multi-objective optimisation problems. Furthermore, this mechanism is incorporated into an optimisation framework recently proposed in the literature, namely the Covariance Matrix Adaptation Pareto Archived Evolution Strategy with Hypervolume-sorted Adaptive Grid Algorithm (CMA-PAES-HAGA), see [50]. The resulting algorithm has been thoroughly tested and compared against modern algorithms which implicitly and explicitly incorporate preference articulation.…”

“…In accordance with the notation used in the field, the parent population size is indicated with µ while the offspring population size is indicated with λ , see [50]. As shown in the following subsections, the µ parent solutions generate λ offspring solutions at each generation, and µ solutions must then be selected from the N = µ + λ solutions in X.…”

“…The external framework embedding the WZ preference articulation operator is the CMA-PAES-HAGA framework proposed in [50]. This algorithmic framework uses the search exploration of the Pareto Adaptive Evolution Strategy (PAES) using a random perturbation aided by a Gaussian distribution, see [41], combined with Covariance Matrix Adaptation logic, see [28].…”

Estimation of Distribution Algorithm Using Correlation between Binary Elements: A New Binary-Code Metaheuristic

“…More details about this aircraft dynamic model can be found in [58][59][60]. To handle the constraints, the penalty function which was presented in [61] is used.…”

“…In this work, only the lateral/ directional motion control is considered. A state equation representing the dynamic motion of an aircraft is expressed as follows [57][58][59][60] where x = { , , , } , is the sideslip, a velocity in direction, is the yaw rate, rate of change of rotation about the -axis, is the roll rate, rate of change of rotation about the -axis, is the bank angle, rotation about the -axis, A is the kinetic energy matrix, B is Coriolis matrix, u = { } is the control vector, is the aileron deflection, and is the rudder deflection.…”