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A majorized penalty approach to inverse linear second order cone programming problems
Quadratic optimization over one firstorder cone
1.  Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China 
2.  Industrial and Systems Engineering Department, North Carolina State University, Raleigh, NC 276957906, United States, United States 
References:
[1] 
F. Alizadeh and D. Goldfarb, Secondorder cone programming,, Mathematical Programming, 95 (2003), 3. doi: 10.1007/s1010700203395. 
[2] 
E. D. Andersen, C. Roos and T. Terlaky, Notes on duality in second order and $p$order cone optimization,, Optimization, 51 (2002), 627. doi: 10.1080/0233193021000030751. 
[3] 
P. Belotti, J. C. Góez, I. Pólik, T. K. Ralphs and T. Terlaky, On families of quadratic surfaces having fixed intersections with two hyperplanes,, Discrete Applied Mathematics, 161 (2013), 2778. doi: 10.1016/j.dam.2013.05.017. 
[4] 
A. BenTal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications,, MPS/SIAM Series on Optimization, (2001). doi: 10.1137/1.9780898718829. 
[5] 
E. Bishop and R. R. Phelps, The support functionals of a convex set,, in Proceedings of Symposia in Pure Mathematics, (1963), 27. 
[6] 
S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004). 
[7] 
S. Burer, On the copositive representation of binary and continous nonconvex quadratic programs,, Mathematical Programming, 120 (2009), 479. doi: 10.1007/s101070080223z. 
[8] 
S. Burer and K. M. Anstreicher, Secondordercone constraints for extended trustregion subproblems,, SIAM Journal on Optimization, 23 (2013), 432. doi: 10.1137/110826862. 
[9] 
Z. Deng, S.C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoidbased approximation scheme,, European Journal of Operational Research, 229 (2013), 21. doi: 10.1016/j.ejor.2013.02.031. 
[10] 
G. Eichfelder and J. Povh, On reformulations of nonconvex quadratic programs over convex cones by setsemidefinite constraints,, preprint, (2010). 
[11] 
Q. Jin, Quadratically Constrained Quadratic Programming Problems and Extensions,, Ph.D thesis, (2011). 
[12] 
Q. Jin, Y. Tian, Z. Deng, S.C. Fang and W. Xing, Exact computable representation of some secondorder cone constrained quadratic programming problems,, Journal of Operations Research Society of China, 1 (2013), 107. doi: 10.1007/s4030501300098. 
[13] 
C. Lu, Q. Jin, S.C. Fang, Z. Wang and W. Xing, An LMI based adaptive approximation scheme to cones of nonnegative quadratic functions,, working paper, (2011). 
[14] 
M. W. Margaret, Advances in conebased preference modeling for decision making with multiple criteria,, Decision Making in Manufacturing and Services, 1 (2007), 153. 
[15] 
Y. Nesterov and A. Nemirovsky, Interiorpoint Polynomial Methods in Convex Programming,, SIAM, (1994). doi: 10.1137/1.9781611970791. 
[16] 
R. T. Rockafellar, Convex Analysis,, 2nd edition, (1972). 
[17] 
J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods,, Optimization Methods and Software, 11/12 (1999), 625. doi: 10.1080/10556789908805766. 
[18] 
J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions,, Mathematics of Operations Research, 28 (2003), 246. doi: 10.1287/moor.28.2.246.14485. 
[19] 
Y. Tian, S.C. Fang, Z. Deng and W. Xing, Computable representation of the cone of nonnegative quadratic forms over a general secondorder cone and its application to completely positive programming,, Journal of Industrial and Management Optimization, 9 (2013), 703. doi: 10.3934/jimo.2013.9.703. 
[20] 
S. A. Vavasis, Nonlinear Optimization: Complexity Issues,, International Series of Monographs on Computer Science, (1991). 
[21] 
Y. Ye and S. Zhang, New results on quadratic minimization,, SIAM Journal on Optimization, 14 (2003), 245. doi: 10.1137/S105262340139001X. 
show all references
References:
[1] 
F. Alizadeh and D. Goldfarb, Secondorder cone programming,, Mathematical Programming, 95 (2003), 3. doi: 10.1007/s1010700203395. 
[2] 
E. D. Andersen, C. Roos and T. Terlaky, Notes on duality in second order and $p$order cone optimization,, Optimization, 51 (2002), 627. doi: 10.1080/0233193021000030751. 
[3] 
P. Belotti, J. C. Góez, I. Pólik, T. K. Ralphs and T. Terlaky, On families of quadratic surfaces having fixed intersections with two hyperplanes,, Discrete Applied Mathematics, 161 (2013), 2778. doi: 10.1016/j.dam.2013.05.017. 
[4] 
A. BenTal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications,, MPS/SIAM Series on Optimization, (2001). doi: 10.1137/1.9780898718829. 
[5] 
E. Bishop and R. R. Phelps, The support functionals of a convex set,, in Proceedings of Symposia in Pure Mathematics, (1963), 27. 
[6] 
S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004). 
[7] 
S. Burer, On the copositive representation of binary and continous nonconvex quadratic programs,, Mathematical Programming, 120 (2009), 479. doi: 10.1007/s101070080223z. 
[8] 
S. Burer and K. M. Anstreicher, Secondordercone constraints for extended trustregion subproblems,, SIAM Journal on Optimization, 23 (2013), 432. doi: 10.1137/110826862. 
[9] 
Z. Deng, S.C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoidbased approximation scheme,, European Journal of Operational Research, 229 (2013), 21. doi: 10.1016/j.ejor.2013.02.031. 
[10] 
G. Eichfelder and J. Povh, On reformulations of nonconvex quadratic programs over convex cones by setsemidefinite constraints,, preprint, (2010). 
[11] 
Q. Jin, Quadratically Constrained Quadratic Programming Problems and Extensions,, Ph.D thesis, (2011). 
[12] 
Q. Jin, Y. Tian, Z. Deng, S.C. Fang and W. Xing, Exact computable representation of some secondorder cone constrained quadratic programming problems,, Journal of Operations Research Society of China, 1 (2013), 107. doi: 10.1007/s4030501300098. 
[13] 
C. Lu, Q. Jin, S.C. Fang, Z. Wang and W. Xing, An LMI based adaptive approximation scheme to cones of nonnegative quadratic functions,, working paper, (2011). 
[14] 
M. W. Margaret, Advances in conebased preference modeling for decision making with multiple criteria,, Decision Making in Manufacturing and Services, 1 (2007), 153. 
[15] 
Y. Nesterov and A. Nemirovsky, Interiorpoint Polynomial Methods in Convex Programming,, SIAM, (1994). doi: 10.1137/1.9781611970791. 
[16] 
R. T. Rockafellar, Convex Analysis,, 2nd edition, (1972). 
[17] 
J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods,, Optimization Methods and Software, 11/12 (1999), 625. doi: 10.1080/10556789908805766. 
[18] 
J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions,, Mathematics of Operations Research, 28 (2003), 246. doi: 10.1287/moor.28.2.246.14485. 
[19] 
Y. Tian, S.C. Fang, Z. Deng and W. Xing, Computable representation of the cone of nonnegative quadratic forms over a general secondorder cone and its application to completely positive programming,, Journal of Industrial and Management Optimization, 9 (2013), 703. doi: 10.3934/jimo.2013.9.703. 
[20] 
S. A. Vavasis, Nonlinear Optimization: Complexity Issues,, International Series of Monographs on Computer Science, (1991). 
[21] 
Y. Ye and S. Zhang, New results on quadratic minimization,, SIAM Journal on Optimization, 14 (2003), 245. doi: 10.1137/S105262340139001X. 
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