# American Institute of Mathematical Sciences

April  2014, 10(2): 397-412. doi: 10.3934/jimo.2014.10.397

## A ladder method for linear semi-infinite programming

 1 School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476, Melbourne, Victoria 3001, Australia, Australia

Received  November 2012 Revised  July 2013 Published  October 2013

This paper presents a new method for linear semi-infinite programming. With the introduction of the so-called generalized ladder point, a ladder method for linear semi-infinite programming is developed. This work includes the generalization of the inclusive cone version of the fundamental theorem of linear programming and the extension of a linear programming ladder algorithm. The extended ladder algorithm finds a generalized ladder point optimal solution of the linear semi-infinite programming problem, which is approximated by a sequence of ladder points. Simple convergence properties are provided. The algorithm is tested by solving a number of linear semi-infinite programming examples. These numerical results indicate that the algorithm is very efficient when compared with other methods.
Citation: Yanqun Liu, Ming-Fang Ding. A ladder method for linear semi-infinite programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 397-412. doi: 10.3934/jimo.2014.10.397
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##### References:
 [1] Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial & Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851 [2] Zhi Guo Feng, Kok Lay Teo, Volker Rehbock. A smoothing approach for semi-infinite programming with projected Newton-type algorithm. Journal of Industrial & Management Optimization, 2009, 5 (1) : 141-151. doi: 10.3934/jimo.2009.5.141 [3] Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 [4] Yanqun Liu. An exterior point linear programming method based on inclusive normal cones. Journal of Industrial & Management Optimization, 2010, 6 (4) : 825-846. doi: 10.3934/jimo.2010.6.825 [5] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [6] Jinchuan Zhou, Naihua Xiu, Jein-Shan Chen. Solution properties and error bounds for semi-infinite complementarity problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 99-115. doi: 10.3934/jimo.2013.9.99 [7] Burcu Özçam, Hao Cheng. A discretization based smoothing method for solving semi-infinite variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 219-233. doi: 10.3934/jimo.2005.1.219 [8] Yi Zhang, Yong Jiang, Liwei Zhang, Jiangzhong Zhang. A perturbation approach for an inverse linear second-order cone programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 171-189. doi: 10.3934/jimo.2013.9.171 [9] Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 965-976. doi: 10.3934/jimo.2014.10.965 [10] Rafael del Rio, Mikhail Kudryavtsev, Luis O. Silva. Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems. Inverse Problems & Imaging, 2012, 6 (4) : 599-621. doi: 10.3934/ipi.2012.6.599 [11] Igor Chueshov. Remark on an elastic plate interacting with a gas in a semi-infinite tube: Periodic solutions. Evolution Equations & Control Theory, 2016, 5 (4) : 561-566. doi: 10.3934/eect.2016019 [12] Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317 [13] Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022 [14] Azhar Ali Zafar, Khurram Shabbir, Asim Naseem, Muhammad Waqas Ashraf. MHD natural convection boundary-layer flow over a semi-infinite heated plate with arbitrary inclination. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 1007-1015. doi: 10.3934/dcdss.2020059 [15] Xiaodong Fan, Tian Qin. Stability analysis for generalized semi-infinite optimization problems under functional perturbations. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1221-1233. doi: 10.3934/jimo.2018201 [16] Andrew E.B. Lim, John B. Moore. A path following algorithm for infinite quadratic programming on a Hilbert space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 653-670. doi: 10.3934/dcds.1998.4.653 [17] Rong Hu, Ya-Ping Fang. A parametric simplex algorithm for biobjective piecewise linear programming problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 573-586. doi: 10.3934/jimo.2016032 [18] Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457 [19] 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 [20] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3821-3838. doi: 10.3934/dcdsb.2017192

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