Decomposition Evolutionary Algorithms for Noisy Multiobjective Optimization

Multi-objective problems are a category of optimization problem that contain more than one objective function and these objective functions must be optimized simultaneously. Should the objective functions be conflicting, then a set of solutions instead of a single solution is required. This set is known as Pareto optimal. Multi-objective optimization problems arise in many real world applications where several competing objectives must be evaluated and optimal solutions found for them, in the presence of trade offs among conflicting objectives. Maximizing returns while minimizing the risk of stock market investments, or maximizing performance whilst minimizing fuel consumption and hazardous gas emission when buying a car are typical examples of real world multi-objective optimization problems. In this case a number of optimal solutions can be found, known as non-dominated or Pareto optimal solutions. Pareto optimal solutions are reached when it is impossible to improve one objective without making the others worse. Classical ways to address this problem used direct or gradient based methods that rendered them insufficient or computationally expensive for large scale or combinatorial problems. Other difficulties attended the classical methods, such as problem knowledge, which may not be available, or sensitivity to some problem features. For example, finding solutions on the entire Pareto optimal set can only be guaranteed for convex problems. Classical methods for generating the Pareto front set aggregate the objectives into a single or parametrized function before search. Thus, several runs and parameter settings are performed to achieve a set of solutions that approximate the Pareto optimals. Subsequently new methods have been developed, based on computer experiments with meta-heuristic algorithms. Most of these meta-heuristics implement some sort of stochastic search method, amongst which the 'Evolutionary Algorithm' is garnering much attention. It possesses several characteristics that make it a desirable method for confronting multi-objective problems. As a result, a number of studies in recent decades have developed or modified the MOEA for different purposes. This algorithm works with a population of solutions which are capable of searching for multiple Pareto optimal solutions in a single run. At the same time, only the fittest individuals in each generation are offered the chance for reproduction and representation in the next generation. The fitness assignment function is the guiding system of MOEA. Fitness value represents the strength of an individual. Unfortunately, many real world applications bring with them a certain degree of noise due to natural disasters, inefficient models, signal distortion or uncertain information. This noise affects the performance of the algorithm's fitness function and disrupts the optimization process. This thesis explores and targets the effect of this disruptive noise on the performance of the MOEA. In this thesis, we study the noisy MOP and modify the MOEA/D to improve its performance in noisy environments. To achieve this, we will combine the basic MOEA/D with the 'Ordinal Optimization' technique to handle uncertainties. The major contributions of this thesis are as follows. First, MOEA/D is tested in a noisy environment with different levels of noise, to give us a deeper understanding of where the basic algorithm fails to handle the noise. Then, we extend the basic MOEA/D to improve its noise handling by employing the ordinal optimization technique. This creates MOEA/D+OO, which will outperform MOEA/D in terms of diversity and convergence in noisy environments. It is tested against benchmark problems of varying levels of noise. Finally, to test the real world application of MOEA/D+OO, we solve a noisy portfolio optimization with the proposed algorithm. The portfolio optimization problem is a classic one in finance that has investors wanting to maximize a portfolio's return while minimizing risk of investment. The latter is measured by standard deviation of the portfolio's rate of return. These two objectives clearly make it a multi-objective problem.