Computing the Set of Approximate Solutions

In the previous chapters, we have investigated archivers that used the concept of \(\epsilon \)-dominance to discretize the Pareto set/front of a given MOP. As a consequence, even the limit archives for certain archivers could contain solutions that are not optimal, but only nearly optimal. This fact, however, does not represent a problem at least from the practical point of view, as in many cases we are satisfied with such nearly optimal solutions. More precisely, for a given \(\epsilon \) (which is of course again problem dependent) we may not mind if the performances of a given product (measured by the objectives \(f_i\)) differ by the entries of \(\epsilon \). As an example, consider the design of a vehicle where one objective, say \(f_1\), is its maximal speed. In this case, we may not care if the maximal speed of this object differs by \(\epsilon _1 = 0.5\) (km/h). That is, if \(x^*\) is vector in decision variable space for those \(f_1\) takes its maximum, we will hence accept all vectors x with \(f_1(x) - f_1(x^*) \le \epsilon _1\). So far, we have used nearly optimal (or \(\epsilon \)-approximate) solutions to obtain a finite size representation of the Pareto sets/fronts. In this chapter, we will go one step further and investigate the approximation of the entire set of approximate solutions of a given multi-objective optimization problem. The reason for this is that this entire set—which we will call \(P_{Q,\epsilon }\) in the sequel—might be of potential interest for the decision maker. One reason for this is that if two points x and y are similar in objective space, i.e., \(F(x)\approx F(y)\), respectively \( \Vert F(x)-F(y)\Vert \le \epsilon \) for a given vector \(\epsilon \), this does not have to hold for the entries of x or y. In fact, x and y could be very distinct from each other. Hence, if both x and y are nearly optimal solutions, both could represent different realizations of a given project which practically yield the same performance.