When optimizing a multi-objective function, it is interesting to find the points where the best trade-off is reached between the different objectives, i.e. to find the points where no improvement...Show moreWhen optimizing a multi-objective function, it is interesting to find the points where the best trade-off is reached between the different objectives, i.e. to find the points where no improvement in one of the objective functions can be made without degrading some of the other objective values. To find these optimal points, we suggest a new algorithm that uses the expected hypervolume improvement of a point. Subsequently this algorithm is considered for the single objective case in one dimension and compared to the existing algorithm for this single objective problem called Shubert’s Algorithm. Moreover, some problems with the existing proof of the convergence rate of Shubert’s Algorithm are resolved. Finally, the bi-objective case is investigated and a method is constructed to calculate the expected hypervolume improvement in a point in an explicit way.Show less
We propose and discuss by means of examples different methods of finding an Efficient Set and a Pareto Frontier in multi-objective optimization problems. Among those methods are both the...Show moreWe propose and discuss by means of examples different methods of finding an Efficient Set and a Pareto Frontier in multi-objective optimization problems. Among those methods are both the probabilistic and the deterministic search e.g. ‘the gradient method‘, Karlin theorem, Karush-Kuhn-Tucker condition. We get the Efficient Set in the explicit form and then we find the Pareto Frontier in parametric or explicit form. The key part of the thesis is a method for finding the Efficient Set and the Pareto Frontier for the implicitly defined objective functions. We consider first the general case and later focus on the special case of two equations of the form: f1(x1, ..., xn)−φ1(y1, y2) = 0, f2(x1, ..., xn)−φ2(y1, y2) = 0 that define two implicitly defined objective functions y1 = y1(x1, ..., xn), y2 = y2(x1, ..., xn). Next we study the Pareto optimization problem to minimize y1 and y2. The solution of this problem is divided into two parts. In the first part we solve the problem of finding the Pareto Efficient Set for functions that are given explicitly. In the second part we discuss and propose methods for finding the Pareto Frontier and Efficient Set for implicitly given objective functions. The results of this work were presented at 28th European Conference on Operational Research "EURO 2016" in Poznań, Poland on July 3-6, 2016, Emmerich M., Sklyar M. "Computing Pareto Fronts of Implicitly Defined Functions", Conference Handbook p. 20.Show less
