Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution
Finding the optimum solution in engineering and science is a common problem where one wishes to get the objective under certain constraints.This situation is also a typical issue in manufacturing industries where maximum profit and minimum cost are a common objective under certain constraints on the...
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Finding the optimum solution in engineering and science is a common problem where one wishes to get the objective under certain constraints.This situation is also a typical issue in manufacturing industries where maximum profit and minimum cost are a common objective under certain constraints on the available resources.One approach to solve optimization is to use formulation problem in linear form and subjects to linear constraints,the problem can be deliberated as linear programming problem.The linear constraints can be in a form of a matrix.There are limited researches that discuss the effect of the properties of matrix constraint to the solution.In fact,the matrix constraint has significant influence to the existent of the optimal solution to the optimization problem.This research focused on the investigation of characteristics of non-symmetric indefinite square matrices of linear programming problems which represent the constraints of linear programming problems.The non-symmetric indefinite square matrices are generated randomly by the MATLAB simulation software and its indefinite properties are verified through the principal minor test,quadratic form test and eigenvalues test.The solutions of the primal and dual linear programming problem are simulated and discussed.Optimization software,LINGO,is used to validate the solutions to assure the reliability of the simulated solutions in the MATLAB software.Based on the simulation results,some of the non-symmetric indefinite random matrices found duality gap and those matrices could not provide optimal solution to the problem.Whereas,some indefinite matrices with certain characteristics could achieve optimal solution and no duality gap presented.An indefinite random matrix with all positive off-diagonal entries and the determinant of leading principal minors with positive sign at odd orders and negative sign at even orders surely deliver the optimal solution to the linear programming problems.This research may contribute to the advancement of linear programming solution particularly when the constraints form an indefinite matrix. |
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Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution |
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Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution |
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Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution |
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Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution |
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properties of indefinite matrix constraints for linear programming in optimal solution |
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my-utem-ep.233352022-03-15T15:37:29Z Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution 2018 Sam, Mei Lee T Technology (General) Finding the optimum solution in engineering and science is a common problem where one wishes to get the objective under certain constraints.This situation is also a typical issue in manufacturing industries where maximum profit and minimum cost are a common objective under certain constraints on the available resources.One approach to solve optimization is to use formulation problem in linear form and subjects to linear constraints,the problem can be deliberated as linear programming problem.The linear constraints can be in a form of a matrix.There are limited researches that discuss the effect of the properties of matrix constraint to the solution.In fact,the matrix constraint has significant influence to the existent of the optimal solution to the optimization problem.This research focused on the investigation of characteristics of non-symmetric indefinite square matrices of linear programming problems which represent the constraints of linear programming problems.The non-symmetric indefinite square matrices are generated randomly by the MATLAB simulation software and its indefinite properties are verified through the principal minor test,quadratic form test and eigenvalues test.The solutions of the primal and dual linear programming problem are simulated and discussed.Optimization software,LINGO,is used to validate the solutions to assure the reliability of the simulated solutions in the MATLAB software.Based on the simulation results,some of the non-symmetric indefinite random matrices found duality gap and those matrices could not provide optimal solution to the problem.Whereas,some indefinite matrices with certain characteristics could achieve optimal solution and no duality gap presented.An indefinite random matrix with all positive off-diagonal entries and the determinant of leading principal minors with positive sign at odd orders and negative sign at even orders surely deliver the optimal solution to the linear programming problems.This research may contribute to the advancement of linear programming solution particularly when the constraints form an indefinite matrix. 2018 Thesis http://eprints.utem.edu.my/id/eprint/23335/ http://eprints.utem.edu.my/id/eprint/23335/1/Properties%20Of%20Indefinite%20Matrix%20Constraints%20For%20Linear%20Programming%20In%20Optimal%20Solution.pdf text en public http://eprints.utem.edu.my/id/eprint/23335/2/Properties%20Of%20Indefinite%20Matrix%20Constraints%20For%20Linear%20Programming%20In%20Optimal%20Solution.pdf text en validuser https://plh.utem.edu.my/cgi-bin/koha/opac-detail.pl?biblionumber=112662 mphil masters UTeM Faculty Of Manufacturing Engineering 1. Abbott, B., 2006. Definite and Indefinite. Encyclopedia of Language and Linguistics, 3, pp.392-399. 2. Allahviranloo, T., Hosseinzadeh Lotfi, F., Khorasani Kiasari, M. and Khezerloo, M., 2013. On the Fuzzy Solution of LR Fuzzy Linear Systems. Applied Mathematical Modelling, 37(3), pp. 1170-1176. 3. Al-Najjar, C. and Malakooti, B., 2011. Hybrid-LP: Finding Advanced Starting Points for Simplex and Pivoting LP Methods. Computer & Operations Research, 38, pp. 427-434. 4. Amiri, N.M. and Nasseri, S.H., 2007. Duality Results and a Dual Simplex Method for Linear Programming Problems with Trapezoidal Fuzzy Variables. Fuzzy Sets and Systems, 158, pp. 1961-1978. 5. Anthony, M. and Harvey, M., 2012. Linear Algebra: Concepts and Method, USA, New York: Cambridge University Press. 6. Arjmandzadeh, Z., Safi, M. and Nazemi, A., 2017. A New Neural Network Model for Solving Random Internal Linear Programming Problems. Neural Networks, 89, pp. 11-18. 7. Bai Z.Z., Ng, M.K. and Wang, Z.Q., 2009. Constraint Preconditioners for Symmetric Indefinite Matrices, SIAM Journal on Matrix Analysis and Applications, 31(2), pp. 410-433. 8. Balbas, A., Jimenez, P. and Heras, A., 1999. Duality Theory and Slackness Conditions in Multi-objective Linear Programming. Computers and Mathematics with Applications, 37, pp. 101-109. 9. Barnes, E., Chen, V., Gopalakrishnan, B. and Johnson, E.L., 2002. A Least-squares Primal-dual Algorithm for Solving Linear Programming Problems. Operations Research Letters, 30(5), pp. 289-294. 10. Bazaraa, M.S., Jarvis, J.J. and Sherali, H.D., 2010. Linear Programming and Network Flows, 4th ed., Canada: John Wiley & Sons. 11. Behera, D. and Chakraverty, S., 2014. Solving Fuzzy Complex System of Linear Equations. Information Sciences, 277, pp. 154-162. 12. Bertsimas, D. and Tsitsiklis, J. N., 1997. Introduction to Linear Optimization, Belmont, Massachusetts: Athena Scientific. 13. Bhatti, M.A., 2000. Practical Optimization Methods: With Mathematica ® Applications, New York: Springer. 14. Boley, D., 2013. Local Linear Convergence of the Alternating Direction Method of Multipliers on Quadratic or Linear Programs. SIAM Journal on Optimization, 23(4), pp. 2183-2207. 15. Bonnabel, S., Collard, A. and Sepulchre, R., 2013. Rank-preserving Geometric Means of Positive Semidefinite Matrices. Linear Algebra and its Applications, 438(8), pp. 3202-3216. 16. Bouras, A. and Syam, W.P., 2013. Hybrid Chaos Optimization and Affine Scaling Search Algorithm for Solving Linear Programming Problems. Applied Soft Computing, 13, pp. 2703-2710. 17. Burger, M., Notarstefano, G., Bullo, F. and Allgower, F., 2012. A Distributed Simplex Algorithm for Degenerate Linear Programs and Multi-agent Assignments. Automatica, 48(9), pp. 2298-2304. 18. Canto, R., Ricarte, B. and Urbano, A.M., 2010. Characterizations of Rectangular Totally and Strictly Totally Positive Matrices. Linear Algebra and Its Applications, 432, pp. 2623-2633. 19. Capitanescu, F. and Wehenkel, L., 2013. Experiments with the Interior-point Method for Solving Large Scale Optimal Power Flow Problems. Electric Power Systems Research, 95, pp. 276-283. 20. Carter, M., 2001. Foundations of Mathematical Economics, Cambridge, US: Massachusetts Institute of Technology. 21. Casacio, L., Lyra, C., Oliveira, A.R.L. and Castro, C.O., 2017. Improving the Preconditioning of Linear Systems from Interior Point Methods. Computers & Operations Research, 85, pp. 129-138. 22. Chabrillac, Y. and Crouzeix, J.P., 1984. Definiteness and Semidefiniteness of Quadratic Forms Revisited. Linear Algebra and Its Applications, 63, pp. 283-292. 23. Cheney, W. and Kincaid, D., 2012. Linear Algebra: Theory and Application, 2nd ed., United Kingdom: London. 24. Corley, H.W., 1984. Duality Theory for the Matrix Linear Programming Problem. Journal of Mathematical Analysis and Applications, 104, pp. 47-52. 25. Cui, J.L., Li, C.K. and Sze, N.S., 2017. Products of Positive Semidefinite Matrices. Linear Algebra and Its Applications, 528, pp. 17-24. 26. Dantzig, G.B., 1947. Maximization of a Linear Function of Variables Subject to Linear Inequalities, T.C. Koopmans (ed.): Activity Analysis of Production and Allocation”, New York-London 1951 (Wiley & Chapman-Hall), pp. 339-347. 27. Dantzig, G.B., 1963. Linear Programming and Extensions, New Jersey, USA: Princeton University Press. 28. Datta, K.B., 2011. Matrix and Linear Algebra: Aided with MATLAB, 2nd ed., New Delhi: PHI Learning Private Limited. 29. Di Pilla, L., Desogus, G., Mura, S., Ricciu, R. and Di Francesco, M., 2016. Optimizing the Distribution of Italian Building Energy Retrofit Incentives with Linear Programming. Energy and Buildings, 112, pp. 21-27. 30. Dhagat, V.B. and Tiwari, S., 2009. Some Results in Duality. International Journal of Contemporary Mathematical Sciences, 4(30), pp. 1493-1501. 31. Dowman K.D. and Wilson, J.M., 2002. Developments in Linear and Integer Programming. Journal of the Operational Research Society, 53(10), pp. 1065-1071. 32. Eiselt, H.A. and Sandblom, C.L., 2007. Linear Programming and Its Applications, Berlin: Springer. 33. Ezzati, R., Khorram, E. and Enayati, R., 2015. A New Algorithm to Solve Fully Fuzzy Linear Programming Problems Using the MOLP Problem. Applied Mathematical Modelling, 39(12), pp. 3183-3193. 34. Fares, D., Chedid, R., Karaki, S., Jabr, R., Panik, F., Gabele, H. and Huang, Y., 2014. Optimal Power Allocation for a FCHV Based on Linear Programming and PID Controller. International Journal of Hydrogen Energy, 39, pp. 21724-21738. 35. Feng, H.X., Da, W., Xi, L.F., Pan, E.S. and Xia, T.B., 2017. Solving the Integrated Cell Formation and Worker Assignment Problem Using Particle Swarm Optimization and Linear Programming. Computers & Industrial Engineering, 110, pp. 126-137. 36. Feng, J.W., Wu, D., Gao, W. and Li, G.Y., 2017. Uncertainty Analysis for Structures with Hybrid Random and Interval Parameters Using Mathematical Programming Approach. Applied Mathematical Modelling, 48, pp. 208-232. 37. Fiedler, M., 2015. Old and New about Positive Definite Matrices. Linear Algebra and Its Applications, 484, pp. 496-503. 38. Forti, M., 1994. On Global Asymptotic Stability of a Class of Nonlinear Systems Arising in Neural Network Theory. Journal of Differential Equations, 113, pp. 246-264. 39. Gao, P.W., 2013. The Phase 1 Simplex Algorithm on the Objective Hyperplane. Operational Research and Its Application in Engineering, Technology and Management 2013 (ISORA 2013), pp. 1-5. 40. Gao, Y. and Qin, Z.F., 2016. A Chance Constrained Programming Approach for Uncertain P-hub Center Location Problem. Computers & Industrial Engineering, 102, pp. 10-20. 41. Ghanbari, R., 2015. Solutions of Fuzzy LR Algebraic Linear Systems Using Linear Programs. Applied Mathematical Modelling, 39(17), pp. 5164-5173. 42. Gohberg, I., Lancaster, P. and Rodman, L., 2005. Indefinite Linear Algebra and Applications, Basel, Switzerland: Birkhauser Verlag. 43. Guler, O., 2010. Foundations of Optimization, New York: Springer. 44. Hannah, L.A., 2015. Stochastic Optimization. International Encyclopedia of the Social & Behavioral Sciences, 2, pp.473-481. 45. Hillier, S. and Lieberman, G., 2001. Introduction to Operations Research, 7th ed., New York: McGraw-Hill. 46. Huang, K.C. and Lu, H., 2015. A Linear Programming-based Method for the Network Revenue Management Problem of Air Cargo. Transportation Research Procedia, 7, pp. 459-473. 47. Huhtanen, M. and Seiskari, O., 2012. Computational Geometry of Positive Definiteness. Linear Algebra and its Applications, 437(7), pp. 1562-1578. 48. Hungerlander, P. and Anjos, M.F., 2015. A Semidefinite Optimization-based Approach for Global Optimization of Multi-row Facility Layout. European Journal of Operational Research, 245(1), pp. 46-61. 49. Illes, T. and Terlaky, T., 2002. Pivot Versus Interior Point Methods: Pros and Cons. European Journal of Operational Research, 140, pp. 170-190. 50. Jeffrey, A., 2010. Matrix Operations for Engineers and Scientists, UK: Springer. 51. Jeyakumar, V. and Li, G. Y., 2009. New Dual Constraint Qualifications Characterizing Zero Duality Gaps of Convex Programs and Semidefinite Programs. Nonlinear Analysis: Theory, Methods & Applications, 71(12), pp. e2239-e2249. 52. Johri, P. K., 1993. Implied Constraints and a Unified Theory of Duality in Linear and Nonlinear Programming. European Journal of Operational Research, 71(1), pp. 61-69. 53. Karmarkar, N., 1984. A New Polynomial-time Algorithm for Linear Programming. Combinatorica, 4(4), pp. 373-395. 54. Kim, K.S., Jung, L.H., Lee, S.C. and Moon, U.C., 2006. Security Constrained Economic Dispatch Using Interior Point Method. 2006 International Conference on Power System Technology, pp. 1-6. 55. Kim, M.K., Lee, Y.H. and Cho, G.M., 2009. An Adaptive-step Primal-dual Interior Point Algorithm for Linear Optimization. Nonlinear Analysis, 71, pp. 2305-2315. 56. Koch, R. and Nasrabadi, E., 2014. Flows over Time in Time-varying Networks: Optimality Conditions and Strong Duality. European Journal of Operational Research, 237, pp. 580-589. 57. Lavaei, J. and Low, S.H., 2012. Zero Duality Gap in Optimal Power Flow Problem. IEEE Transactions on Power Systems, 27(1), pp. 92-107. 58. Leu, S.Y. and Li, J.S., 2014. Shakedown Analysis of Framed Structures: Strong Duality and Primal-Dual Analysis. Procedia Engineering, 79, pp. 204-211. 59. Lim, L.L., Alpan, G. and Penz, B., 2014. Reconciling Sales and Operations Management with Distant Suppliers in the Automotive Industry: A Simulation Approach. International Journal of Production Economics, 151, pp. 20-36. 60. Lin, M.H., Tsai, J.F. and Yu, C.S., 2012. A Review of Deterministic Optimization Methods in Engineering and Management. Mathematical Problems in Engineering, 2012, pp. 1-15. 61. Lu, C.N. and Unum, M.R., 1993. Network Constrained Security Control Using an Interior Point Algorithm. IEEE Transactions on Power System, 8(3), pp. 1068-1076. 62. Luc, D.T., 2011. On Duality in Multiple Objective Linear Programming. European Journal of Operational Research, 210(2), pp. 158-168. 63. Luenberger, D.G. and Ye, Y.Y., 2008. Linear and Nonlinear Programming, 3rd ed., USA: Stanford University. 64. Ma, G.F., Yang, J.H. and Zhang, P.C., 2012. Achieving Optimal Solution of Linear Programming Based on Mobile Agent Technology. Physics Procedia, 24, pp. 1364-1368. 65. Martins, S., Amorim, P., Figueira, G. and Almada-Lobo, B., 2017. An Optimization-simulation Approach to the Network Redesign Problem of Pharmaceutical Wholesalers. Computer & Industrial Engineering, 106, pp. 315-328. 66. Mehlhorn, K. and Saxena, S., 2016. A Still Simpler Way of Introducing Interior-point method for Linear Programming. Computer Science Review, 22, pp. 1-11. 67. Mitchell, J.E., Pardalos, P.M. and Resende, M.G., 1998. Handbook of Combinatorial Optimization, Springer, Boston, MA. 68. Morigi, S., Reichel, L. and Sgallari, F., 2010. An Interior-point Method for Large Constrained Discrete Ill-posed Problem. Journal of Computational and Applied Mathematics, 233(5), pp. 1288-1297. 69. Nasseri, S.H., Ebrahimnejad, A. and Mizuno, S., 2010. Duality in Fuzzy Linear Programming with Symmetric Trapezoidal Numbers. Applications and Applied Mathematics: An International Journal (AAM), 5(10), pp. 1469-1484. 70. Oliveira, A.R.L. and Sorensen, D.C., 2005. A New Class of Preconditioners for Large-Scale Linear Systems from Interior Point Methods for Linear Programming. Linear Algebra and its Applications, 394, pp. 1-24. 71. Papadimitrious, C.H. and Steiglitz, K., 1998. Combinatorial Optimization: Algorithms and Complexity, Mineola, New York: Dover. 72. Pires, M.C. and Frazzon, E.M., 2016. On the Research of Linear Programming Solving Methods for Non-hierarchical Spare Parts Supply Chain Planning. IFAC-Papers Online, 49(30), pp. 198-203. 73. Ponnambalam, K., Quintana, V.H. and Vannelli, A., 1991. A Fast Algorithm for Power System Optimization Problems Using an Interior Point Method. Power Industry Computer Application Conference 1991, pp. 393-400. 74. Ramik, J., 2006. Duality in Fuzzy Linear Programming with Possibility and Necessity Relations. Fuzzy Sets and Systems, 157, pp. 1283-1302. 75. Robinson-Mosher, A., Schroeder, C. and Fedkiw, R., 2011. A Symmetric Positive Definite Formulation for Monolithic Fluid Structure Interaction. Journal of Computational Physics, 230(4), pp. 1547-1566. 76. Robinson-Mosher, A., Shinar, T., Gretarsson, J., Su, J. and Fedkiw, R., 2008. Two-way Coupling of Fluids to Rigid and Deformable Solids and Shells. ACM Transactions on Graphics, 27 (3), pp. 46:1-46:9. 77. Romli, I., 2013. An assessment on the effect of different characteristics of symmetric matrices to primal and dual solutions in linear programming problems. [online]. Available at: https://www.google.com/search?q=romli+study+on+indefinite+matrix+&ie=utf-8&oe=utf-8&client=firefox-b-ab [Accessed on 10 January 2016]. 78. Schott, J.R., 2016. Matrix Analysis for Statistics, 3rd ed., USA: John Wiley & Sons. 79. Schrijver, A., 1998. Theory of Linear and Integer Programming, USA: John Wiley & Sons. 80. Shao, J.W. and Hou, X.R., 2010. Positive Definiteness of Hermitian Interval Matrices. Linear Algebra and Its Applications, 432, pp. 970-979. 81. Simon, C.P. and Blume, L., 1994. Mathematics for Economists, New York: W. W. Norton & Company. 82. Skiborowski, M., Rautenberg, M. and Marquardt, W., 2015. A Hybrid Evolutionary-deterministic Optimization Approach for Conceptual Design. Industrial & Engineering Chemistry, 54(41), pp. 10054-10072. 83. Son, J.H. and Scriber, B.H., 2005. Problem Solution of Linear Programming using Dual Simplex Method Neural Network. 2005 International Conference on Electrical Machines and Systems, 3, pp. 2045-2047. 84. Soylu, B., 2015. Heuristic Approaches for Bi-objective Mixed 0-1 Integer Linear Programming Problems. European Journal of Operational Research, 245(3), pp. 690-703. 85. Soyster, A.L. and Murphy, F.H., 2013. A Unifying Framework for Duality and Modelling in Robust Linear Programs. Omega, 41, pp. 984-997. 86. Su, T.S., 2017. A Fuzzy Multi-objective Linear Programming Model for Solving Remanufacturing Planning Problems with Multiple Products and Joint Components. Computers & Industrial Engineering, 110, pp. 242-254. 87. Taha, H.A., 2011. Operations Research: An Introduction, 9th ed., New Jersey, USA: Prentice Hall. 88. Todd, M.J., 2002. The Many Facets of Linear Programming. Mathematical Programming, 91, pp. 417 - 436. 89. Torres, D., Crichigno, J., Padilla, G. and Rivera, R., 2014. Scheduling Coupled Photovoltaic, Battery and Conventional Energy Sources to Maximize Profit using Linear Programming. Renewable Energy, 2, pp. 284-290. 90. Touil, I., Benterki, D. and Yassine, A., 2017. A Feasible Primal-dual Interior Point Method for Linear Semidefinite Programming. Journal of Computational and Applied Mathematics, 312, pp. 216-230. 91. Uctug, F.G. and Yukseltan, E., 2012. A Linear Programming Approach to Household Energy Conservation: Efficient Allocation of Budget. Energy and Buildings, 49, pp. 200-208. 92. Vanderbei, R.J., 2001. Linear Programming: Foundations and Extensions, 2nd ed., Dordrecht, London: Kluwer Academic. 93. Vasant, P., Weber, G.W. and Dieu, V.N., 2016. Handbook of Research on Modern Optimization Algorithms and Applications in Engineering and Economics, USA: Engineering Science Reference. 94. Vidal, T., Crainic, T.G., Gendreau, M., Lahrichi, N. and Rei, W., 2012. A Hybrid Genetic Algorithm for Multi-depot and Periodic Vehicle Routing Problems. Operations Research, 60(3), pp. 611-624. 95. Vinh, N.T., Kim, D.S., Tam, N.N. and Yen, N.D., 2016. Duality Gap Function in Infinite Dimensional Linear Programming. Journal of Mathematical Analysis and Applications, 437, pp. 1-15. 96. Wang, G.Q. and Bai, Y.Q., 2009. Primal-dual Interior-point Algorithm for Convex Quadratic Semidefinite Optimization. Nonlinear Analysis, 71, pp. 3389-3402. 97. Wang, G.Q., Wu, Z.C., Zheng, Z.T. and Cai, X.Z., 2015. Complexity Analysis of Primal-dual Interior-point Methods for Semidefinite Optimization Based on a Parametric Kernel Function with A Trigonometric Barrier Term. Numerical Algebra Control and Optimization, 5(2), pp. 101-113. 98. Wang, N., Jin, B., Zhang, Z.Z. and Lim, A., 2017. A Feasibility-based Heuristic for the Container Pre-marshalling Problem. European Journal of Operational Research, 256, pp. 90-101. 99. Weber, H.J. and Arfken, G.B., 2013. Essentials of Math Methods for Physicists, London, United Kingdom: American Press. 100. Wolsey, L.A., 1998. Integer Programming, USA: John Wiley & Sons. 101. Wu, Y., 2010. A Time Staged Linear Programming Model for Production Loading Problems with Import Quota Limit in a Global Supply Chain. Computers & Industrial Engineering, 59, pp. 520-529. 102. Zhou, T., 2017. A Hybrid Stochastic-deterministic Optimization Approach for Integrated Solvent and Process Design. Chemical Engineering Science, 159, pp. 207-216. 103. Zhu, F. and Ukkusuri, S.V., 2015. A Linear Programming Formulation for Autonomous Intersection Control within a Dynamic Traffic Assignment and Connected Vehicle Environment. Transportation Research Part C, 55, pp. 363-378. |