# American Institute of Mathematical Sciences

January  2010, 6(1): 15-28. doi: 10.3934/jimo.2010.6.15

## A numerical approach to infinite-dimensional linear programming in $L_1$ spaces

 1 Department of Mathematical Analysis and Statistical Inference, Institute of Statistical Mathematics, Research Organization of Information and Systems, Tokyo 106-8569, Japan 2 Department of Mathematics, National Cheng Kung University, Tainan, Taiwan 3 Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845

Received  August 2008 Revised  July 2009 Published  November 2009

An infinite-dimensional linear programming formulated on $L_1$ spaces, problem (P), is studied in this paper. A related optimization problem, general capacity problem (GCAP), is also mentioned in this paper. But we find that the optimal solution does not exist in problem (P). Thus, we approach the optimal value for problem (P) via solving the problem (GCAP). A proposed algorithm is shown that we solve a sequence of semi-infinite subproblems to approach the optimal value of problem (P). The error bound for the difference between the optimal value for problem (P) and optimal value for semi-infinite subproblem is also given in this paper. Finally, numerical examples are implemented and compared with discretization method to show our computational efficiency.
Citation: Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial & Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15
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