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Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs
A conic approximation method for the 0-1 quadratic knapsack problem
1. | Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China, China, China, China |
References:
[1] |
A. Ben-Tal and A. Nemirovski, "Lectures On Modern Convex Optimization, Analysis, Algorithms and Engineering Applications," $1^{st}$ edition, MPS/SIAM Series on Optimization, Philadelphia, 2001.
doi: 10.1137/1.9780898718829. |
[2] |
A. Billionnet and F. Calmels, Linear programming for the 0-1 quadratic knapsack problem, European Journal of Operation Research, 92 (1996), 310-325.
doi: 10.1016/0377-2217(94)00229-0. |
[3] |
A. Billionnet and E. Soutif, An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem, European Journal of Operational Research, 157 (2004), 565-575.
doi: 10.1016/S0377-2217(03)00244-3. |
[4] |
A. Billionnet and E. Soutif, Using a mixed integer programming tool for solving the 0-1 quadratic knapsack problem, INFORMS Journal on Computing, 16 (2004), 188-197.
doi: 10.1287/ijoc.1030.0029. |
[5] |
A. Caprara, D. Pisinger and P. Toth, Exact solution of quadratic knapsack problem, INFORMS Journal on Computing, 11 (1999), 125-139.
doi: 10.1287/ijoc.11.2.125. |
[6] |
G. Dijkhuizen and U. Faigle, A cutting-plane approach to the edge-weighted maximal clique problem, European Journal Operational Research, 69 (1993), 121-130.
doi: 10.1016/0377-2217(93)90097-7. |
[7] |
G. Gallo, P. L. Hammer and B. Simeone, Quadratic knapsack problems, Mathematical Programming Study, 12 (1980), 132-149.
doi: 10.1007/BFb0120892. |
[8] |
D. J. Grainger and A. N. Letchford, "Improving a Formulation of the Quadratic Knapsack Problem," Available from: http://www.optimization-online.org/DB_HTML/2007/06/1678.html. |
[9] |
M. Grant and S. Boyed, CVX: Matlab software for disciplined convex programming, version 2.0(2012), Available from: http://cvxr.com/cvx. |
[10] |
C. Helmberg, F. Rendl and R. Weismantel, A semidefinite programming approach to the quadratic knapsack problem, Journal of Combinatorial Optimization, 4 (2000), 197-215.
doi: 10.1023/A:1009898604624. |
[11] |
K. Holmström, A. O. Göran and M. M. Edvall, "User's Guide for Tomlab 7," Available from: http://tomopt.com/tomlab/. |
[12] |
H. Kellerer and V. A. Strusevich, Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications, Algorithmica, 57 (2010), 769-795.
doi: 10.1007/s00453-008-9248-1. |
[13] |
L. Létocart, A. Nagih and G. Plateau, Reoptimization in Lagrangian methods for the 0-1 quadratic knapsack problem, Computers and Operations Research, 39 (2012), 12-18.
doi: 10.1016/j.cor.2010.10.027. |
[14] |
C. Lu, S.-C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, SIAM Journal on Optimization, 21 (2011), 1475-1490.
doi: 10.1137/100793955. |
[15] |
C. Lu, Q. Jin, S.-C. Fang, Z. Wang and W. Xing, Adaptive computable approximation to cones of nonnegative quadratic functions, Submitted to Optimization, (2011). |
[16] |
C. Lu, Z. Wang, W. Xing and S.-C. Fang, Extended canonical duality and conic programming for solving 0-1 quadratic programming problems, Journal of Industrial and Management Optimization, 6 (2010), 779-793.
doi: 10.3934/jimo.2010.6.779. |
[17] |
P. Michelon and L. Veilleux, Lagrangian methods for the 0-1 quadratic knapsack problem, European Journal of Operational Research, 92 (1996), 326-341.
doi: 10.1016/0377-2217(94)00286-X. |
[18] |
P. M. Pardalos and S. A. Vavasis, Quadratic programming with one negative eigenvalue is NP-Hard, Journal of Global Optimization, 1 (1991), 15-22.
doi: 10.1007/BF00120662. |
[19] |
K. Park, K. Lee and S. Park, An extended formulation approach to the edge-weighted maximal clique problem, European Journal of Operational Research, 95 (1996), 671-682.
doi: 10.1016/0377-2217(95)00299-5. |
[20] |
D. Pisinger, The quadratic knapsack problem-a survey, Discrete Applied Mathematics, 155 (2007), 623-648.
doi: 10.1016/j.dam.2006.08.007. |
[21] |
J. Rhys, A selection problem of shared fixed costs and network flows, Management Science, 17 (1970), 200-207.
doi: 10.1287/mnsc.17.3.200. |
[22] |
J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
[23] |
C. Witzgall, "Mathematical Methods of Site Selection for Electronic Message System(EMS)," Technical Report, NBS Internal Report, 1975. |
[24] |
X. J. Zheng, X. L. Sun and D. Li, On the reduction of duality gap in quadratic knapsack problems, Journal of Global Optimization, 54 (2012), 325-339.
doi: 10.1007/s10898-012-9872-9. |
show all references
References:
[1] |
A. Ben-Tal and A. Nemirovski, "Lectures On Modern Convex Optimization, Analysis, Algorithms and Engineering Applications," $1^{st}$ edition, MPS/SIAM Series on Optimization, Philadelphia, 2001.
doi: 10.1137/1.9780898718829. |
[2] |
A. Billionnet and F. Calmels, Linear programming for the 0-1 quadratic knapsack problem, European Journal of Operation Research, 92 (1996), 310-325.
doi: 10.1016/0377-2217(94)00229-0. |
[3] |
A. Billionnet and E. Soutif, An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem, European Journal of Operational Research, 157 (2004), 565-575.
doi: 10.1016/S0377-2217(03)00244-3. |
[4] |
A. Billionnet and E. Soutif, Using a mixed integer programming tool for solving the 0-1 quadratic knapsack problem, INFORMS Journal on Computing, 16 (2004), 188-197.
doi: 10.1287/ijoc.1030.0029. |
[5] |
A. Caprara, D. Pisinger and P. Toth, Exact solution of quadratic knapsack problem, INFORMS Journal on Computing, 11 (1999), 125-139.
doi: 10.1287/ijoc.11.2.125. |
[6] |
G. Dijkhuizen and U. Faigle, A cutting-plane approach to the edge-weighted maximal clique problem, European Journal Operational Research, 69 (1993), 121-130.
doi: 10.1016/0377-2217(93)90097-7. |
[7] |
G. Gallo, P. L. Hammer and B. Simeone, Quadratic knapsack problems, Mathematical Programming Study, 12 (1980), 132-149.
doi: 10.1007/BFb0120892. |
[8] |
D. J. Grainger and A. N. Letchford, "Improving a Formulation of the Quadratic Knapsack Problem," Available from: http://www.optimization-online.org/DB_HTML/2007/06/1678.html. |
[9] |
M. Grant and S. Boyed, CVX: Matlab software for disciplined convex programming, version 2.0(2012), Available from: http://cvxr.com/cvx. |
[10] |
C. Helmberg, F. Rendl and R. Weismantel, A semidefinite programming approach to the quadratic knapsack problem, Journal of Combinatorial Optimization, 4 (2000), 197-215.
doi: 10.1023/A:1009898604624. |
[11] |
K. Holmström, A. O. Göran and M. M. Edvall, "User's Guide for Tomlab 7," Available from: http://tomopt.com/tomlab/. |
[12] |
H. Kellerer and V. A. Strusevich, Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications, Algorithmica, 57 (2010), 769-795.
doi: 10.1007/s00453-008-9248-1. |
[13] |
L. Létocart, A. Nagih and G. Plateau, Reoptimization in Lagrangian methods for the 0-1 quadratic knapsack problem, Computers and Operations Research, 39 (2012), 12-18.
doi: 10.1016/j.cor.2010.10.027. |
[14] |
C. Lu, S.-C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, SIAM Journal on Optimization, 21 (2011), 1475-1490.
doi: 10.1137/100793955. |
[15] |
C. Lu, Q. Jin, S.-C. Fang, Z. Wang and W. Xing, Adaptive computable approximation to cones of nonnegative quadratic functions, Submitted to Optimization, (2011). |
[16] |
C. Lu, Z. Wang, W. Xing and S.-C. Fang, Extended canonical duality and conic programming for solving 0-1 quadratic programming problems, Journal of Industrial and Management Optimization, 6 (2010), 779-793.
doi: 10.3934/jimo.2010.6.779. |
[17] |
P. Michelon and L. Veilleux, Lagrangian methods for the 0-1 quadratic knapsack problem, European Journal of Operational Research, 92 (1996), 326-341.
doi: 10.1016/0377-2217(94)00286-X. |
[18] |
P. M. Pardalos and S. A. Vavasis, Quadratic programming with one negative eigenvalue is NP-Hard, Journal of Global Optimization, 1 (1991), 15-22.
doi: 10.1007/BF00120662. |
[19] |
K. Park, K. Lee and S. Park, An extended formulation approach to the edge-weighted maximal clique problem, European Journal of Operational Research, 95 (1996), 671-682.
doi: 10.1016/0377-2217(95)00299-5. |
[20] |
D. Pisinger, The quadratic knapsack problem-a survey, Discrete Applied Mathematics, 155 (2007), 623-648.
doi: 10.1016/j.dam.2006.08.007. |
[21] |
J. Rhys, A selection problem of shared fixed costs and network flows, Management Science, 17 (1970), 200-207.
doi: 10.1287/mnsc.17.3.200. |
[22] |
J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267.
doi: 10.1287/moor.28.2.246.14485. |
[23] |
C. Witzgall, "Mathematical Methods of Site Selection for Electronic Message System(EMS)," Technical Report, NBS Internal Report, 1975. |
[24] |
X. J. Zheng, X. L. Sun and D. Li, On the reduction of duality gap in quadratic knapsack problems, Journal of Global Optimization, 54 (2012), 325-339.
doi: 10.1007/s10898-012-9872-9. |
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