首页 | 本学科首页   官方微博 | 高级检索  
     检索      


Optimizing a linear function over an efficient set
Authors:J G Ecker  J H Song
Institution:(1) Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York
Abstract:The problem (P) of optimizing a linear functiond T x over the efficient set for a multiple-objective linear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function directiond and the set of domination directionsD, ifd T pgrgE0 for all nonzero pgrisinD, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point 
$$\hat x$$
, if there is no adjacent efficient edge yielding an increase ind T x, then a cutting plane 
$$d^T x = d^T \hat x$$
is used to obtain a multiple-objective linear program ( 
$$\bar M$$
) with a reduced feasible set and an efficient set 
$$\bar E$$
. To find a better efficient point, we solve the problem (Ii) of maximizingc i T x over the reduced feasible set in ( 
$$\bar M$$
) sequentially fori. If there is a 
$$x^i  \in \bar E$$
that is an optimal solution of (Ii) for somei and 
$$d^T x^i  > d^T \hat x$$
, then we can choosex i as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point 
$$\hat x$$
is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.The authors would like to thank an anonymous referee, H. P. Benson, and P. L. Yu for numerous helpful comments on this paper.
Keywords:Multiple-objective linear programming  efficient sets  domination cones  nonconvex optimization
本文献已被 SpringerLink 等数据库收录!
设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号