Theoretical Analysis And Empirical Validation Of The Conical Area Evolutionary Algorithm For Bi-objective Optimization

There exist a large number of multi-objective optimization problems(MOPs)in many management and engineering fields.In general a MOP has many Pareto optimal solutions and the population-based evolutionary algorithms(EAs)can find these multiple solutions simultaneously in one single run.Although most of the multi-objective evolutionary algorithms(MOEAs)are Pareto dominance-based algorithms,several recent decompositionbased multi-objective evolutionary algorithms such as the multi-objective algorithm based on decomposition(MOEA/D)are becoming more and more successful and popular.Particularly,a conical area evolutionary algorithm(CAEA),which employs an efficient cone decomposition approach,was developed for the improvement in efficiency of runtime and population diversity of decomposition-based algorithms for the bi-objective optimization.The cone decomposition approach in CAEA divides the bi-objective space in many conical sub-regions and uses the conical area indicator as its scalar objectives to solve every scalar sub-problem in the corresponding conical sub-region.The global Pareto optimality of cone decomposition was proved.It means that the optimum of any given conical sub-problem in cone decomposition is Pareto optimal in the entire bi-objective space in existence of continuous frontier segment within related sub-region.But there is still an open question about if the optimal solutions of all ? conical sub-problems can approximately converge as a whole to the entire real Pareto frontier.In this thesis,we theoretically prove and empirically validate that cone decomposition method has the good ability of approximate ?-distribution convergence for solving the biobjective problems.On the basis of global Pareto optimality of cone decomposition,we first prove by the monotonicity and additivity properties of hyper-volume,the squeeze theorem and the L'Hospital's rule for limit that the set of optima of all ? sub-problems in cone decomposition can converge to optimal ?-distribution,a finite set of ? solutions maximizing hyper-volume,for large enough ? in existence of continuous frontier.Meanwhile,the original CAEA is further improved to better approximate the optimum solutions of conical subproblems.We perform some experiments to verify the above property on the 5 bi-objective MOP benchmark problems.Experimental results show that solutions of conical sub-problems gained through the improved CAEA named as CAEA-II achieve better qualities of the frontiers in terms of hyper-volume error metric,than the solutions through the other four famous multi-objective algorithms that are NSGA-II(non-dominated sorting genetic algorithm II),NSGA-III(non-dominated sorting genetic algorithm III),MOEA/D,MOEA/DDE(MOEA/D with differential evolution).