On characterizing the "knee" of the Pareto curve based on Normal-Boundary Intersection
On characterizing the "knee" of the Pareto curve based on Normal-Boundary Intersection
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DOI：
10.1007/s001580050111
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发表时间：
1999-10-01
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影响因子：
--
通讯作者：
Das, I
中科院分区：
文献类型：
--
作者：
Das, I
本文讨论的问题是产生一个帕累托最优点，保证是在一个“理想”的一部分，帕累托集在一个给定的多准则优化问题。最近开发的正常边界相交技术的基础上的帕累托集的参数化是用来制定一个子问题，其解决方案产生的点的“最大凸起”，通常被称为“膝盖的帕累托曲线”。这使得能够通过解决一个非线性规划问题来识别帕累托集合的“好区域”，从而绕过生成许多帕累托点的需要。此外，这种表示扩展了“膝盖”的概念，用于具有两个以上目标的问题。进一步证明了该拐点对于多目标函数的尺度是不变的，但该拐点的产生需要每个目标函数在其最小值（支付矩阵）处的值.该文件的特点的情况下，近似的函数值，包括支付矩阵将足以产生一个很好的近似膝盖。数值结果说明了这一点。此外，加权和最小化问题的基础上开发的支付矩阵中的信息，通过求解可以得到的膝盖。
This paper deals with the issue of generating one Pareto optimal point that is guaranteed to be in a "desirable" part of the Pareto set in a given multicriteria optimization problem. A parameterization of the Pareto set based on the recently developed normal-boundary intersection technique is used to formulate a subproblem, the solution of which yields the point of "maximum bulge", often referred to as the "knee of the Pareto curve". This enables the identification of the "good region" of the Pareto set by solving one nonlinear programming problem, thereby bypassing the need to generate many Pareto points. Further, this representation extends the concept of the "knee" for problems with more than two objectives. Tt is further proved that this knee is invariant with respect to the scales of the multiple objective functions.The generation of this knee however requires the value of each objective function at the minimizer of every objective function (the pay-off matrix). The paper characterizes situations when approximations to the function values comprising the pay-off matrix would suffice in generating a good approximation to the knee. Numerical results are provided to illustrate this point. Further, a weighted sum minimization problem is developed based on the information in the pay-off matrix, by solving which the knee can be obtained.