ABSTRACT

A realistic solution concept associated with a multiobjective optimization problem is that named Pareto (or efficient) solution, which is more difficult to be treated from a mathematical point of view than the notion of weak Pareto (or weakly efficient) solution. This work provides a complete description of the efficient solution set, when the objective functions are defined on the real line. This is motivated, besides theoretical aspects, also by a numerical point of view, since most algorithms in scalar minimization involve the solvability of a one-dimensional optimization problem to find the next iterate. It is expected that the same situation occurs in the multiobjective optimization problem. We first consider the case when all the objective functions are semistrictly quasiconvex, and afterwards we consider the same problem under quasiconvexity along with some additional assumptions. The latter allows us to deal with the general bicriteria optimization problem under quasiconvexity. Several examples showing the applicability of our results are presented, and an algorithm is proposed to compute the whole efficient solution set.

摘要

与多目标优化问题相关的一个现实的解决方案概念是所谓的帕累托（或有效）解决方案，从数学的角度来看，它比弱帕累托（或弱有效）解决方案的概念更难处理。当目标函数在实线上定义时，这项工作提供了有效解决方案集的完整描述。除了理论方面，这也是出于数值的考虑，因为标量最小化中的大多数算法都涉及一维优化问题的可解性以找到下一个迭代。预计多目标优化问题也会出现同样的情况。我们首先考虑所有目标函数都是半严格拟凸的情况，然后我们在拟凸性下考虑相同的问题以及一些额外的假设。后者允许我们处理拟凸性下的一般双准则优化问题。几个例子展示了我们的结果的适用性，并提出了一种算法来计算整个有效的解决方案集。