An exact penalty function method for nonlinear mixed discrete programming problems
In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additi...
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| Published in: | Optimization letters Vol. 7; no. 1; pp. 23 - 38 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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01.01.2013
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| ISSN: | 1862-4472, 1862-4480 |
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| Abstract | In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method. |
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| AbstractList | In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method. |
| Author | Yu, Changjun Bai, Yanqin Teo, Kok Lay |
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| Keywords | Nonlinear mixed integer programming Exact penalty function |
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| References | Murray, Ng (CR12) 2010; 47 Li, Chou (CR7) 1993; 22 CR16 CR15 CR14 Linderoth, Savelsbergh (CR8) 1999; 11 Loh, Papalambros (CR10) 1991; 113 Lucidi, Rinaldi (CR11) 2010; 145 Ng, Zhang (CR13) 2005; 31 Ringertz (CR18) 1988; 13 Gao (CR5) 2000 Gao, Ruan (CR6) 2010; 47 Sun, Li, Luo (CR21) 2010; 59 Sandgren (CR19) 1988; 112 Fang, Gao, Sheu, Wu (CR3) 2008; 4 Fu, Fenton, Cleghorn (CR4) 1991; 17 Shin, Gürdal, Griffin, O.H. (CR20) 1990; 16 Loh, Papalambros (CR9) 1991; 113 Tseng, Arora (CR22) 1988; 26 Zheng, Sun, Li (CR25) 2010; 146 Pardalos, Yatsenko (CR17) 2009 Wang, Teo, Lee (CR23) 1998; 30 Cai, Thierauf (CR2) 1993; 21 Bremicker, Papalambros, Loh (CR1) 1990; 37 Yu, Teo, Zhang, Bai (CR24) 2010; 6 J.T. Linderoth (391_CR8) 1999; 11 E. Sandgren (391_CR19) 1988; 112 H.T. Loh (391_CR9) 1991; 113 C. Tseng (391_CR22) 1988; 26 391_CR15 391_CR14 D.Y. Gao (391_CR5) 2000 X.J. Zheng (391_CR25) 2010; 146 H.T. Loh (391_CR10) 1991; 113 U.T. Ringertz (391_CR18) 1988; 13 391_CR16 D.K. Shin (391_CR20) 1990; 16 H.L. Li (391_CR7) 1993; 22 J.B. Cai (391_CR2) 1993; 21 S.C. Fang (391_CR3) 2008; 4 M. Bremicker (391_CR1) 1990; 37 X.L. Sun (391_CR21) 2010; 59 C.J. Yu (391_CR24) 2010; 6 C. Ng (391_CR13) 2005; 31 S. Lucidi (391_CR11) 2010; 145 W. Murray (391_CR12) 2010; 47 P.M. Pardalos (391_CR17) 2009 S. Wang (391_CR23) 1998; 30 J.F. Fu (391_CR4) 1991; 17 D.Y. Gao (391_CR6) 2010; 47 |
| References_xml | – volume: 113 start-page: 325 year: 1991 end-page: 334 ident: CR9 article-title: A sequential linearization approach for solving mixed-discrete nonlinear design optimization problems publication-title: J. Mech. Des. doi: 10.1115/1.2912786 – volume: 6 start-page: 895 issue: 4 year: 2010 end-page: 910 ident: CR24 article-title: A new exact penalty function method for continuous inequality constrained optimization problems publication-title: J. Ind. Manage. Optim. doi: 10.3934/jimo.2010.6.895 – volume: 21 start-page: 293 issue: 4 year: 1993 end-page: 306 ident: CR2 article-title: Discrete optimization of structures using an improved penalty function method publication-title: Eng. Optim. doi: 10.1080/03052159308940981 – volume: 112 start-page: 223 issue: 2 year: 1988 end-page: 229 ident: CR19 article-title: Nonlinear integer and discrete programming in mechanical design optimization publication-title: J. Mech. Des. doi: 10.1115/1.2912596 – ident: CR14 – volume: 22 start-page: 109 issue: 2 year: 1993 end-page: 122 ident: CR7 article-title: A global approach for nonlinear mixed discrete programming in design optimization publication-title: Eng. Optim. doi: 10.1080/03052159308941328 – ident: CR16 – volume: 16 start-page: 29 issue: 1 year: 1990 end-page: 42 ident: CR20 article-title: A penalty approach for nonlinear optimization with discrete design variables publication-title: Eng. Optim. doi: 10.1080/03052159008941163 – volume: 146 start-page: 463 issue: 2 year: 2010 end-page: 489 ident: CR25 article-title: Separable relaxation for nonconvex quadratic integer programming: integer diagonalization approach publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-010-9653-x – volume: 47 start-page: 257 issue: 2 year: 2010 end-page: 288 ident: CR12 article-title: An algorithm for nonlinear optimization problems with binary variables publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-008-9218-1 – volume: 26 start-page: 1365 issue: 6 year: 1988 end-page: 1402 ident: CR22 article-title: On implementation of computational algorithms for optimization design. Part I and Part publication-title: Int. J. Numer. Methods Eng. doi: 10.1002/nme.1620260610 – volume: 4 start-page: 125 issue: 1 year: 2008 end-page: 142 ident: CR3 article-title: Canonical dual approach to solving 0–1 quadratic programming problems publication-title: J. Ind. Manage. Optim. doi: 10.3934/jimo.2008.4.125 – year: 2009 ident: CR17 publication-title: Optimization and control of bilinear systems – volume: 30 start-page: 249 issue: 3 year: 1998 end-page: 262 ident: CR23 article-title: A new approach to nonlinear mixed discrete programming problems publication-title: Eng. Optim. doi: 10.1080/03052159808941246 – volume: 17 start-page: 263 issue: 4 year: 1991 end-page: 280 ident: CR4 article-title: A mixed integer-discretecontinuous programming method and its application to engineering design optimization publication-title: Eng. Optim. doi: 10.1080/03052159108941075 – volume: 37 start-page: 451 issue: 4 year: 1990 end-page: 461 ident: CR1 article-title: Solution of mixed-discrete structural optimization problems with a new sequential linearization algorithms publication-title: Comput. Struct. doi: 10.1016/0045-7949(90)90035-Z – volume: 31 start-page: 87 issue: 1 year: 2005 end-page: 115 ident: CR13 article-title: Discrete filled function method for discrete global optimization publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-005-0985-7 – volume: 13 start-page: 47 issue: 1 year: 1988 end-page: 64 ident: CR18 article-title: On methods for discrete structural optimization publication-title: Eng. Optim. doi: 10.1080/03052158808940946 – volume: 113 start-page: 335 issue: 3 year: 1991 end-page: 344 ident: CR10 article-title: Computational implementation and tests of a sequential linearization algorithm for mixed-discrete nonlinear design optimization publication-title: J. Mech. Des. doi: 10.1115/1.2912787 – ident: CR15 – year: 2000 ident: CR5 publication-title: Duality principles in nonconvex systems—theory, methods and applications – volume: 145 start-page: 479 issue: 3 year: 2010 end-page: 488 ident: CR11 article-title: Exact penalty functions for nonlinear integer programming problems publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-010-9700-7 – volume: 47 start-page: 463 issue: 3 year: 2010 end-page: 484 ident: CR6 article-title: Solutions to quadratic minimization problems with box and integer constraints publication-title: J. Global Optim. doi: 10.1007/s10898-009-9469-0 – volume: 11 start-page: 173 issue: 2 year: 1999 end-page: 188 ident: CR8 article-title: A computational study of search strategies for mixed integer programming publication-title: INFORMS J. Comput. Spring – volume: 59 start-page: 627 issue: 5 year: 2010 end-page: 641 ident: CR21 article-title: Convex relaxation and Lagrangian decomposition for indefinite integer quadratic programming publication-title: Optimization doi: 10.1080/02331930801987607 – volume: 47 start-page: 257 issue: 2 year: 2010 ident: 391_CR12 publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-008-9218-1 – volume: 30 start-page: 249 issue: 3 year: 1998 ident: 391_CR23 publication-title: Eng. Optim. doi: 10.1080/03052159808941246 – volume: 26 start-page: 1365 issue: 6 year: 1988 ident: 391_CR22 publication-title: Int. J. Numer. Methods Eng. doi: 10.1002/nme.1620260610 – volume: 47 start-page: 463 issue: 3 year: 2010 ident: 391_CR6 publication-title: J. Global Optim. doi: 10.1007/s10898-009-9469-0 – volume-title: Optimization and control of bilinear systems year: 2009 ident: 391_CR17 – volume: 11 start-page: 173 issue: 2 year: 1999 ident: 391_CR8 publication-title: INFORMS J. Comput. Spring doi: 10.1287/ijoc.11.2.173 – volume: 31 start-page: 87 issue: 1 year: 2005 ident: 391_CR13 publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-005-0985-7 – volume: 145 start-page: 479 issue: 3 year: 2010 ident: 391_CR11 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-010-9700-7 – ident: 391_CR14 doi: 10.1007/BFb0000035 – volume: 113 start-page: 325 year: 1991 ident: 391_CR9 publication-title: J. Mech. Des. doi: 10.1115/1.2912786 – volume: 22 start-page: 109 issue: 2 year: 1993 ident: 391_CR7 publication-title: Eng. Optim. doi: 10.1080/03052159308941328 – volume: 17 start-page: 263 issue: 4 year: 1991 ident: 391_CR4 publication-title: Eng. Optim. doi: 10.1080/03052159108941075 – volume: 16 start-page: 29 issue: 1 year: 1990 ident: 391_CR20 publication-title: Eng. Optim. doi: 10.1080/03052159008941163 – volume: 21 start-page: 293 issue: 4 year: 1993 ident: 391_CR2 publication-title: Eng. Optim. doi: 10.1080/03052159308940981 – volume-title: Duality principles in nonconvex systems—theory, methods and applications year: 2000 ident: 391_CR5 doi: 10.1007/978-1-4757-3176-7 – volume: 6 start-page: 895 issue: 4 year: 2010 ident: 391_CR24 publication-title: J. Ind. Manage. Optim. doi: 10.3934/jimo.2010.6.895 – ident: 391_CR16 doi: 10.1007/0-387-32942-0_2 – volume: 112 start-page: 223 issue: 2 year: 1988 ident: 391_CR19 publication-title: J. Mech. Des. doi: 10.1115/1.2912596 – volume: 113 start-page: 335 issue: 3 year: 1991 ident: 391_CR10 publication-title: J. Mech. Des. doi: 10.1115/1.2912787 – volume: 13 start-page: 47 issue: 1 year: 1988 ident: 391_CR18 publication-title: Eng. Optim. doi: 10.1080/03052158808940946 – volume: 59 start-page: 627 issue: 5 year: 2010 ident: 391_CR21 publication-title: Optimization doi: 10.1080/02331930801987607 – volume: 37 start-page: 451 issue: 4 year: 1990 ident: 391_CR1 publication-title: Comput. Struct. doi: 10.1016/0045-7949(90)90035-Z – ident: 391_CR15 doi: 10.1007/978-1-4899-0289-4_22 – volume: 146 start-page: 463 issue: 2 year: 2010 ident: 391_CR25 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-010-9653-x – volume: 4 start-page: 125 issue: 1 year: 2008 ident: 391_CR3 publication-title: J. Ind. Manage. Optim. doi: 10.3934/jimo.2008.4.125 |
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| Title | An exact penalty function method for nonlinear mixed discrete programming problems |
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