Optimizing over the efficient set of a multi-objective optimization problem is among the difficult problems in global optimization because of its nonconvexity, even in the linear case. In this paper, we consider only properly efficient solutions which are characterized through weighted sum scalarization. We propose a numerical method to tackle this problem when the objective functions and the feasible set of the multi-objective optimization problem are convex. This algorithm penalizes progressively iterates that are not properly efficient and uses a sequence of convex nonlinear subproblems that can be solved efficiently. The proposed algorithm is shown to perform well on a set of standard problems from the literature, as it allows to obtain optimal solutions in all cases.

中文翻译：

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优化适当有效的凸多目标优化问题集

优化多目标优化问题的有效集是全局优化中的难题之一，因为它的非凸性，即使在线性情况下也是如此。在本文中，我们只考虑通过加权和标量化来表征的适当有效的解决方案。当多目标优化问题的目标函数和可行集是凸的时，我们提出了一种数值方法来解决这个问题。该算法逐渐惩罚效率不高的迭代，并使用一系列可以有效解决的凸非线性子问题。所提出的算法在文献中的一组标准问题上表现良好，因为它允许在所有情况下获得最佳解决方案。