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2017 | OriginalPaper | Chapter

10. Quadratic Programming

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Abstract

One of the most important nonlinear optimization problems is quadratic programming, in which a quadratic objective function is minimized with respect to linear equality and inequality constraints. These kinds of problems are present in many methods as sub-problems and in real applications from different areas of activity as mathematical models of these applications. At the very beginning, we consider the equality-constrained quadratic programming, after which the inequality-constrained programming will be presented.

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Literature
go back to reference Andersen, E. D., & Andersen, K. D. (2000). The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm. In T. T. H. Frenk, K. Roos, & S. Zhang (Eds.), High performance optimization (pp. 197–232). New York, NY, USA: Kluwer Academic Publishers.CrossRef Andersen, E. D., & Andersen, K. D. (2000). The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm. In T. T. H. Frenk, K. Roos, & S. Zhang (Eds.), High performance optimization (pp. 197–232). New York, NY, USA: Kluwer Academic Publishers.CrossRef
go back to reference Andrei, N. (1999). Programarea matematică avansată. Teorie, Metode computaţionale, Aplicaţii. [Advanced Mathematical Programming. Theory, Computational Methods, Applications] Editura Tehnică, Bucureşti. Andrei, N. (1999). Programarea matematică avansată. Teorie, Metode computaţionale, Aplicaţii. [Advanced Mathematical Programming. Theory, Computational Methods, Applications] Editura Tehnică, Bucureşti.
go back to reference Andrei, N. (2003). Modele, Probleme de Test şi Aplicaţii de Programare Matematică. [Models, Test Problems and Applications for Mathematical Programming] Editura Tehnică, Bucureşti. Andrei, N. (2003). Modele, Probleme de Test şi Aplicaţii de Programare Matematică. [Models, Test Problems and Applications for Mathematical Programming] Editura Tehnică, Bucureşti.
go back to reference Andrei, N. (2011a) Critica raţiunii algoritmilor de programare liniară. [Criticism of the Linear Programming Algorithms Reasoning]. Editura Academiei Române, Bucureşti. Andrei, N. (2011a) Critica raţiunii algoritmilor de programare liniară. [Criticism of the Linear Programming Algorithms Reasoning]. Editura Academiei Române, Bucureşti.
go back to reference Bartholomew-Biggs, M. (2008). Nonlinear optimization with engineering applications. New York, NY, USA: Springer Science +Business Media.CrossRefMATH Bartholomew-Biggs, M. (2008). Nonlinear optimization with engineering applications. New York, NY, USA: Springer Science +Business Media.CrossRefMATH
go back to reference Bunch J. W., & Kaufman, L. (1977). Indefinite quadratic programming. (Computing Science Technical Report 61, Bell Labs., Murray Hill, NJ). Bunch J. W., & Kaufman, L. (1977). Indefinite quadratic programming. (Computing Science Technical Report 61, Bell Labs., Murray Hill, NJ).
go back to reference Daniel, J. W., Graggs, W. B., Kaufman, L., & Stewart, G. W. (1976). Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorizations. Mathematics of Computation, 30, 772–795.MathSciNetMATH Daniel, J. W., Graggs, W. B., Kaufman, L., & Stewart, G. W. (1976). Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorizations. Mathematics of Computation, 30, 772–795.MathSciNetMATH
go back to reference Fletcher, R. (1971). A general quadratic programming algorithm. Journal of the Institute of Mathematics and its Applications, 7, 76–91.MathSciNetCrossRefMATH Fletcher, R. (1971). A general quadratic programming algorithm. Journal of the Institute of Mathematics and its Applications, 7, 76–91.MathSciNetCrossRefMATH
go back to reference Gill, P. E., Golub, G. H., Murray, W., & Saunders, M. A. (1974). Methods for modifying matrix factorizations. Mathematics of Computation, 28, 505–535.MathSciNetCrossRefMATH Gill, P. E., Golub, G. H., Murray, W., & Saunders, M. A. (1974). Methods for modifying matrix factorizations. Mathematics of Computation, 28, 505–535.MathSciNetCrossRefMATH
go back to reference Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1984). User’s guide for SOL/QPSOL. (Technical Report SOL84–6, Department of Operations Research, Stanford University, Stanford, California). Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1984). User’s guide for SOL/QPSOL. (Technical Report SOL84–6, Department of Operations Research, Stanford University, Stanford, California).
go back to reference Goldfarb, D. (1975). Matrix factorizations in optimization of nonlinear functions subject to linear constraints. Mathematical Programming, 10, 1–31.MathSciNetCrossRefMATH Goldfarb, D. (1975). Matrix factorizations in optimization of nonlinear functions subject to linear constraints. Mathematical Programming, 10, 1–31.MathSciNetCrossRefMATH
go back to reference Goldfarb, D., & Idnani, A. (1983). A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming, 27, 1–33.MathSciNetCrossRefMATH Goldfarb, D., & Idnani, A. (1983). A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming, 27, 1–33.MathSciNetCrossRefMATH
go back to reference Gonçalves, A. S. (1972). A primal-dual method for quadratic programming with bounded variables. In F. A. Lootsma (Ed.), Numerical Methods for Nonlinear Optimization (pp. 255–263). London, UK: Academic. Gonçalves, A. S. (1972). A primal-dual method for quadratic programming with bounded variables. In F. A. Lootsma (Ed.), Numerical Methods for Nonlinear Optimization (pp. 255–263). London, UK: Academic.
go back to reference Gould, N. I. M., & Toint, P. L. (2002a). Numerical methods for large-scale non-convex quadratic programming. In A. H. Siddiqi & M. Kocvara (Eds.), Trends in industrial and applied mathematics (pp. 149–179). Dordrecht, Europe: Kluwer Academic Publishers.CrossRef Gould, N. I. M., & Toint, P. L. (2002a). Numerical methods for large-scale non-convex quadratic programming. In A. H. Siddiqi & M. Kocvara (Eds.), Trends in industrial and applied mathematics (pp. 149–179). Dordrecht, Europe: Kluwer Academic Publishers.CrossRef
go back to reference Gould, N. I. M., & Toint, P. L. (2002b). An iterative working-set method for large-scale non-convex quadratic programming. Applied Numerical Mathematics, 43, 109–128.MathSciNetCrossRefMATH Gould, N. I. M., & Toint, P. L. (2002b). An iterative working-set method for large-scale non-convex quadratic programming. Applied Numerical Mathematics, 43, 109–128.MathSciNetCrossRefMATH
go back to reference Gould, N. I. M., Toint, Ph. L. (2012). A quadratic programming bibliography, (RAL Numerical Analysis Group Internal Report 2000–1, March 28, 2012). Gould, N. I. M., Toint, Ph. L. (2012). A quadratic programming bibliography, (RAL Numerical Analysis Group Internal Report 2000–1, March 28, 2012).
go back to reference Gould, N. I. M., Orban, D., & Toint, P. L. (2005a). Numerical methods for large-scale nonlinear optimization. Acta Numerica, 14, 299–361.MathSciNetCrossRefMATH Gould, N. I. M., Orban, D., & Toint, P. L. (2005a). Numerical methods for large-scale nonlinear optimization. Acta Numerica, 14, 299–361.MathSciNetCrossRefMATH
go back to reference Guéret, C., Prins, C., & Sevaux, M. (2002). Applications of optimization with Xpress-MP. Dash Optimization. Guéret, C., Prins, C., & Sevaux, M. (2002). Applications of optimization with Xpress-MP. Dash Optimization.
go back to reference ILOG CPLEX 8.0. (2002). User’s manual. France, Europe: ILOG SA Gentilly. ILOG CPLEX 8.0. (2002). User’s manual. France, Europe: ILOG SA Gentilly.
go back to reference Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 8, 77–91. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 8, 77–91.
go back to reference Murty, K. G., & Kabadi, S. N. (1987). Some NP-complete problems in quadratic and nonlinear programming. Mathematical Programming, 19, 200–212.MathSciNetMATH Murty, K. G., & Kabadi, S. N. (1987). Some NP-complete problems in quadratic and nonlinear programming. Mathematical Programming, 19, 200–212.MathSciNetMATH
go back to reference Nocedal, J., & Wright, S. J. (2006). Numerical optimization, Springer series in operations research (2nd ed.). New York, NY, USA: Springer Science+Business Media.MATH Nocedal, J., & Wright, S. J. (2006). Numerical optimization, Springer series in operations research (2nd ed.). New York, NY, USA: Springer Science+Business Media.MATH
go back to reference Pant, M., Thangaraj, R., & Singh, V. P. (2009). Particle swarm optimization with crossover operator and its engineering applications. IAENG International Journal of Computer Science, 36, 112–121. Pant, M., Thangaraj, R., & Singh, V. P. (2009). Particle swarm optimization with crossover operator and its engineering applications. IAENG International Journal of Computer Science, 36, 112–121.
go back to reference Powell, M. J. D. (1983). ZQPCVX: A Fortran subroutine for convex quadratic programming. (Technical Report, Department of Applied Mathematics and Theoretical Physics, Cambridge University). Powell, M. J. D. (1983). ZQPCVX: A Fortran subroutine for convex quadratic programming. (Technical Report, Department of Applied Mathematics and Theoretical Physics, Cambridge University).
go back to reference Schittkowski, K. (2005). QL: A Fortran code for convex quadratic programming. User’s guide, Version 2.11. (Technical report, Department of Mathematics, University of Bayreuth, July 2005). Schittkowski, K. (2005). QL: A Fortran code for convex quadratic programming. User’s guide, Version 2.11. (Technical report, Department of Mathematics, University of Bayreuth, July 2005).
go back to reference Sun, W., & Yuan, Y. X. (2006). Optimization theory and methods. Nonlinear programming. New York, NY, USA: Springer Science + Business Media.MATH Sun, W., & Yuan, Y. X. (2006). Optimization theory and methods. Nonlinear programming. New York, NY, USA: Springer Science + Business Media.MATH
go back to reference Vanderbei, R. J. (2001). Linear programming: Foundations and extensions (2nd ed.). New York, NY, USA: Springer.CrossRefMATH Vanderbei, R. J. (2001). Linear programming: Foundations and extensions (2nd ed.). New York, NY, USA: Springer.CrossRefMATH
Metadata
Title
Quadratic Programming
Author
Neculai Andrei
Copyright Year
2017
DOI
https://doi.org/10.1007/978-3-319-58356-3_10