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2021 | OriginalPaper | Chapter

5. Basics in Linear and Convex Optimization

Author : Roman A. Polyak

Published in: Introduction to Continuous Optimization

Publisher: Springer International Publishing

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Abstract

In the first part of this chapter we cover basic LP facts, including LP duality, special LP structure, as well as two main tools for solving LP: simplex and interior point methods (IPMs).

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Literature
go back to reference Adler, I., Monteiro, R.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Program. 50, 29–51 (1991)MathSciNetMATHCrossRef Adler, I., Monteiro, R.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Program. 50, 29–51 (1991)MathSciNetMATHCrossRef
go back to reference Adler, I., Karmarkar, N., Resende, M., Veiga, G.: An implementation of Karmarkar’s algorithm for linear programming. Math. Program. 44, 297–335 (1989). Errata in Math. Program. 50, 415 (1991) Adler, I., Karmarkar, N., Resende, M., Veiga, G.: An implementation of Karmarkar’s algorithm for linear programming. Math. Program. 44, 297–335 (1989). Errata in Math. Program. 50, 415 (1991)
go back to reference Anstreicher, K.M.: Potential reduction algorithms, In: Terlaky, T. (ed.) Interior Point Methods of Mathematical Programming, pp. 125–158. Kluwer Academic Publishers, Dordrecht (1996)CrossRef Anstreicher, K.M.: Potential reduction algorithms, In: Terlaky, T. (ed.) Interior Point Methods of Mathematical Programming, pp. 125–158. Kluwer Academic Publishers, Dordrecht (1996)CrossRef
go back to reference Barnes, E.R.: Some results concerning convergence of the affine scaling algorithm. Contemp. Math. 114, 131–139 (1990)MathSciNetCrossRef Barnes, E.R.: Some results concerning convergence of the affine scaling algorithm. Contemp. Math. 114, 131–139 (1990)MathSciNetCrossRef
go back to reference Beck, A., Teboulle, M.: Fast gradient based algorithm for constrained total variation image denoising and deblurring problems. Trans. Imag. Proc. 18, 2419–2434 (2009)MathSciNetMATHCrossRef Beck, A., Teboulle, M.: Fast gradient based algorithm for constrained total variation image denoising and deblurring problems. Trans. Imag. Proc. 18, 2419–2434 (2009)MathSciNetMATHCrossRef
go back to reference Beck, A., Teboulle, M.: A fast iterative shrinkage—thresholding algorithm for linear inverse problem. SIAM J. Imag. Sci. 2(1), 183–202 (2009a)MathSciNetMATHCrossRef Beck, A., Teboulle, M.: A fast iterative shrinkage—thresholding algorithm for linear inverse problem. SIAM J. Imag. Sci. 2(1), 183–202 (2009a)MathSciNetMATHCrossRef
go back to reference Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)MATH Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)MATH
go back to reference Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999) Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)
go back to reference Bixby, R.E.: A brief history of linear and mixed—integer programming computation. In: Documenta Mathematica Optimization Stories, Berlin (2012) Bixby, R.E.: A brief history of linear and mixed—integer programming computation. In: Documenta Mathematica Optimization Stories, Berlin (2012)
go back to reference Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71, 51–69 (1995)MathSciNetMATHCrossRef Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71, 51–69 (1995)MathSciNetMATHCrossRef
go back to reference Cheney, W., Goldstein, A.: Note on paper by Zuhovickii concerning the Tchebycgeff problem for linear equations. J. Soc. Indust. Appl. Math. 6(3), 233–239 (1958)MathSciNetMATHCrossRef Cheney, W., Goldstein, A.: Note on paper by Zuhovickii concerning the Tchebycgeff problem for linear equations. J. Soc. Indust. Appl. Math. 6(3), 233–239 (1958)MathSciNetMATHCrossRef
go back to reference Dantzig, G., Wolfe, P.: The decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)MATHCrossRef Dantzig, G., Wolfe, P.: The decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)MATHCrossRef
go back to reference Dantzig, G., Fulkerson, D., Johnson, S.: Solution of a large—scale traveling—salesman problem. J. Oper. Res. Soc. Am. 2(4), 393–410 (1954)MathSciNetMATH Dantzig, G., Fulkerson, D., Johnson, S.: Solution of a large—scale traveling—salesman problem. J. Oper. Res. Soc. Am. 2(4), 393–410 (1954)MathSciNetMATH
go back to reference Dantzig, G., Ford, L., Fulkerson, D.: In: Kuhn, H., Tucker, A. (eds.) Algorithm for Simultaneous Solution the Primal and Dual Linear Programming Problem in Linear Inequalities and Related System (1956) Dantzig, G., Ford, L., Fulkerson, D.: In: Kuhn, H., Tucker, A. (eds.) Algorithm for Simultaneous Solution the Primal and Dual Linear Programming Problem in Linear Inequalities and Related System (1956)
go back to reference Demyanov, V., Rubinov, A.: Approximate Methods in Optimization Problems. American Elsevier Publishing, New York (1970) Demyanov, V., Rubinov, A.: Approximate Methods in Optimization Problems. American Elsevier Publishing, New York (1970)
go back to reference Den Hertog, D., Roos, C., Vial, J.: A complexity reduction for the long-step path-following algorithm for linear programming. SIAM J. Optim. 2, 71–87 (1992)MathSciNetMATHCrossRef Den Hertog, D., Roos, C., Vial, J.: A complexity reduction for the long-step path-following algorithm for linear programming. SIAM J. Optim. 2, 71–87 (1992)MathSciNetMATHCrossRef
go back to reference Dikin, I.: Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR 174, 747–748 (1967). Translated in Soviet Mathematics Doklady 8, 674–675 (1967) Dikin, I.: Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR 174, 747–748 (1967). Translated in Soviet Mathematics Doklady 8, 674–675 (1967)
go back to reference Dikin, I.: On the convergence of an iterative process. Upravlyaemye Sistemi 12, 54–60 (1974, in Russian) Dikin, I.: On the convergence of an iterative process. Upravlyaemye Sistemi 12, 54–60 (1974, in Russian)
go back to reference Farkas, J.: Teorie der einfachen Ungleivhungen. J. Reine und Angewandte Mathematik 124, 1–27 (1902) Farkas, J.: Teorie der einfachen Ungleivhungen. J. Reine und Angewandte Mathematik 124, 1–27 (1902)
go back to reference Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia (1990)MATHCrossRef Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia (1990)MATHCrossRef
go back to reference Freund, R.: Theoretical efficiency of a shifted barrier function algorithm for linear programming. Linear Algebra Appl. 152, 19–41 (1991)MathSciNetMATHCrossRef Freund, R.: Theoretical efficiency of a shifted barrier function algorithm for linear programming. Linear Algebra Appl. 152, 19–41 (1991)MathSciNetMATHCrossRef
go back to reference Garber, D., Hazan, E.: Faster Rates for the Frank–Wolfe Method over Strongly Convex Sets (2014, 2015). arxiv: 1406.1305v2[mathOC] Garber, D., Hazan, E.: Faster Rates for the Frank–Wolfe Method over Strongly Convex Sets (2014, 2015). arxiv: 1406.1305v2[mathOC]
go back to reference Gill, P., Murray, W., Saunders, J., Tomlin, J., Wright, M.: On projected Newton Barrier methods for linear programming and an equivalence to Karmarkar’s projective method. Math. Program. 36, 183–209 (1986)MathSciNetMATHCrossRef Gill, P., Murray, W., Saunders, J., Tomlin, J., Wright, M.: On projected Newton Barrier methods for linear programming and an equivalence to Karmarkar’s projective method. Math. Program. 36, 183–209 (1986)MathSciNetMATHCrossRef
go back to reference Goldshtein, E., Tretiakov, N.: Modified Lagrangian Functions. Nauka, Moscow (1989) Goldshtein, E., Tretiakov, N.: Modified Lagrangian Functions. Nauka, Moscow (1989)
go back to reference Goldstein, A.: Convex programming in gilbert space. Bull. Am. Math Soc. 70, 709–710 (1964)MATHCrossRef Goldstein, A.: Convex programming in gilbert space. Bull. Am. Math Soc. 70, 709–710 (1964)MATHCrossRef
go back to reference Gonzaga, C.: An algorithm for solving linear programming problems in O(n 3 L) operations. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 1–28. Springer, New York (1989) Gonzaga, C.: An algorithm for solving linear programming problems in O(n 3 L) operations. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 1–28. Springer, New York (1989)
go back to reference Griva, I., Shanno, D., Vanderbei, R., Benson, H.: Global convergence of a primal-dual interior-point method for nonlinear programming. Algorithmic Oper. Res. 3(1), 12–29 (2008)MathSciNetMATH Griva, I., Shanno, D., Vanderbei, R., Benson, H.: Global convergence of a primal-dual interior-point method for nonlinear programming. Algorithmic Oper. Res. 3(1), 12–29 (2008)MathSciNetMATH
go back to reference Grossman, K., Kaplan, A.A.: Nonlinear programming based on unconstrained minimization. Novosibirsk, Nauka (1981), in Russian Grossman, K., Kaplan, A.A.: Nonlinear programming based on unconstrained minimization. Novosibirsk, Nauka (1981), in Russian
go back to reference Guler, O., den Hertog, D., Roos, C., Terlaky, T., Tsuvhiya, T.: Degeneracy in interior point methods for linear programming: a survey. Ann. Oper. Res. 46, 107–138 (1993)MathSciNetMATHCrossRef Guler, O., den Hertog, D., Roos, C., Terlaky, T., Tsuvhiya, T.: Degeneracy in interior point methods for linear programming: a survey. Ann. Oper. Res. 46, 107–138 (1993)MathSciNetMATHCrossRef
go back to reference Hoffman, A.: On approximate solution of system of linear inequalities. J. Res. Nat. Bureau Stand. 49, 263–265 (1952)MathSciNetCrossRef Hoffman, A.: On approximate solution of system of linear inequalities. J. Res. Nat. Bureau Stand. 49, 263–265 (1952)MathSciNetCrossRef
go back to reference Ioffe, A., Tichomirov, V.: Duality of convex functions and extremum problems. Uspexi Mat. Nauk 23(6)(144), 51–116 (1968) Ioffe, A., Tichomirov, V.: Duality of convex functions and extremum problems. Uspexi Mat. Nauk 23(6)(144), 51–116 (1968)
go back to reference Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (2009)MATH Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (2009)MATH
go back to reference Jensen, B., Roos, C., Terlaky, T., Vial, J.: Primal-dual algorithms for linear programming based on th logarithmic barrier method. J. Optim. Theory Appl. 83, 1–26 (1994)MathSciNetMATHCrossRef Jensen, B., Roos, C., Terlaky, T., Vial, J.: Primal-dual algorithms for linear programming based on th logarithmic barrier method. J. Optim. Theory Appl. 83, 1–26 (1994)MathSciNetMATHCrossRef
go back to reference Jensen, B., Roos, C., Terlaky, T.: A polynomial dikin-type primal-dual algorithm for linear programming. Math. Oper. Res. 21, 341–353 (1996)MathSciNetMATHCrossRef Jensen, B., Roos, C., Terlaky, T.: A polynomial dikin-type primal-dual algorithm for linear programming. Math. Oper. Res. 21, 341–353 (1996)MathSciNetMATHCrossRef
go back to reference Kantorovich, L.: Mathematical methods of organizing and planning production. Manage. Sci. 6(4), 366–422 (1939). JSTOR 2627082 Kantorovich, L.: Mathematical methods of organizing and planning production. Manage. Sci. 6(4), 366–422 (1939). JSTOR 2627082
go back to reference Kantorovich, L.:The Best Use of Economic Resources. Pergamon Press, New York (1959) Kantorovich, L.:The Best Use of Economic Resources. Pergamon Press, New York (1959)
go back to reference Khachiyan, L.: A polynomial algorithm in linear programming. Doklady Akademiia Nauk SSSR 244, 1093–1096 (1979). (Translated into English in Soviet Mathematics Doklady 20, 191–194) Khachiyan, L.: A polynomial algorithm in linear programming. Doklady Akademiia Nauk SSSR 244, 1093–1096 (1979). (Translated into English in Soviet Mathematics Doklady 20, 191–194)
go back to reference Kojima, M., Mizuno, S., Yoshise, A.: A polynomial-time algorithm for a class of linear complementarity problems. Math. Program. 44, 1–26 (1989)MathSciNetMATHCrossRef Kojima, M., Mizuno, S., Yoshise, A.: A polynomial-time algorithm for a class of linear complementarity problems. Math. Program. 44, 1–26 (1989)MathSciNetMATHCrossRef
go back to reference Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming, In: Megiddo, N. (ed.) Progress in mathematical programming: interior point and related methods, pp 29–47. Springer, New York (1989a)CrossRef Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming, In: Megiddo, N. (ed.) Progress in mathematical programming: interior point and related methods, pp 29–47. Springer, New York (1989a)CrossRef
go back to reference Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach to interior point algorithms for linear complementarity problems. Lecture Notes in Computer Science, vol. 538. Springer, Berlin (1991) Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach to interior point algorithms for linear complementarity problems. Lecture Notes in Computer Science, vol. 538. Springer, Berlin (1991)
go back to reference Koopmans, T.C.: Optimum utilization of the transportation system. Econometrica 17, 136–146 (1949)CrossRef Koopmans, T.C.: Optimum utilization of the transportation system. Econometrica 17, 136–146 (1949)CrossRef
go back to reference Kuhn, H., Tucker, A.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp. 481–492 (1951) Kuhn, H., Tucker, A.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp. 481–492 (1951)
go back to reference Levitin, E., Polyak, B.: Minimization methods in the presence of constraints. Z. Vycisl. Mat. i Mat Fizz 6, 787–823 (1966)MathSciNetMATH Levitin, E., Polyak, B.: Minimization methods in the presence of constraints. Z. Vycisl. Mat. i Mat Fizz 6, 787–823 (1966)MathSciNetMATH
go back to reference Lustig, I., Marsten, R., Shanno, D.: Computational experience with a primal-dual interior point method for linear programming. Linear Algebra Appl. 152, 191–222 (1991)MathSciNetMATHCrossRef Lustig, I., Marsten, R., Shanno, D.: Computational experience with a primal-dual interior point method for linear programming. Linear Algebra Appl. 152, 191–222 (1991)MathSciNetMATHCrossRef
go back to reference Lustig, I., Marsten, R., Shanno, D.: On implementing Mehrotra’s predictor-corrector interior point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)MathSciNetMATHCrossRef Lustig, I., Marsten, R., Shanno, D.: On implementing Mehrotra’s predictor-corrector interior point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)MathSciNetMATHCrossRef
go back to reference Lustig, I., Marsten, R., Shanno, D.: Interior point methods for linear programming: computational state of the art. ORSA J. Comput. 6(1), 1–14 (1994)MathSciNetMATHCrossRef Lustig, I., Marsten, R., Shanno, D.: Interior point methods for linear programming: computational state of the art. ORSA J. Comput. 6(1), 1–14 (1994)MathSciNetMATHCrossRef
go back to reference Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming—Interior Point and Related Methods. Springer, Berlin, pp. 131–158 (1989) Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming—Interior Point and Related Methods. Springer, Berlin, pp. 131–158 (1989)
go back to reference Mehrotra, S.: Higher order methods and their performance. Technical Report 90-16R1, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, IL 60208, USA (1990). Revised July 1991 Mehrotra, S.: Higher order methods and their performance. Technical Report 90-16R1, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, IL 60208, USA (1990). Revised July 1991
go back to reference Mizuno, S., Todd, M.: An O(n 3 L) adaptive path following algorithm for a linear complementarity problem. Math. Program. 52, 587–595 (1991)MathSciNetMATHCrossRef Mizuno, S., Todd, M.: An O(n 3 L) adaptive path following algorithm for a linear complementarity problem. Math. Program. 52, 587–595 (1991)MathSciNetMATHCrossRef
go back to reference Monteiro, R., Adler, I., Resende, M.: A polynomial - time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res. 15, 191–214 (1990)MathSciNetMATHCrossRef Monteiro, R., Adler, I., Resende, M.: A polynomial - time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res. 15, 191–214 (1990)MathSciNetMATHCrossRef
go back to reference Nemirovski, A., Yudin, D.: Informational Complexity and Efficient Methods for Solution of Convex Extremal Problems. Wiley, New York (1983) Nemirovski, A., Yudin, D.: Informational Complexity and Efficient Methods for Solution of Convex Extremal Problems. Wiley, New York (1983)
go back to reference Nesterov, Y.: A method for solving convex programming problems with convergence rate O(k −2). Dokl. Akad. Navk. SSSR 269(3), 543–547 (1983, in Russian) Nesterov, Y.: A method for solving convex programming problems with convergence rate O(k −2). Dokl. Akad. Navk. SSSR 269(3), 543–547 (1983, in Russian)
go back to reference Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, Norwell, (2004)MATHCrossRef Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, Norwell, (2004)MATHCrossRef
go back to reference Pardalos, P., Resende, M.: Interior point methods for global optimization. In: Terlaky, T. (ed.) IPM of Mathematical Programming Applied Optimization, vol. 5. Kluwer, Dordrecht, pp. 467–500 (1996) Pardalos, P., Resende, M.: Interior point methods for global optimization. In: Terlaky, T. (ed.) IPM of Mathematical Programming Applied Optimization, vol. 5. Kluwer, Dordrecht, pp. 467–500 (1996)
go back to reference Pardalos, P., Wolkowicz, H.: Topics in Semidefinite and Interior - Point Methods. Fields Institute Communications Series 18, American Mathematical Society, New York (1998) Pardalos, P., Wolkowicz, H.: Topics in Semidefinite and Interior - Point Methods. Fields Institute Communications Series 18, American Mathematical Society, New York (1998)
go back to reference Pardalos, P., Ye, Y., Han, G.: Computational Aspects of an Interior Point Algorithm for Quadratic Problems with box Constraints. In: Large-Scale Numerical Optimization. SIAM, Philadelphia, pp. 92–112 (1990) Pardalos, P., Ye, Y., Han, G.: Computational Aspects of an Interior Point Algorithm for Quadratic Problems with box Constraints. In: Large-Scale Numerical Optimization. SIAM, Philadelphia, pp. 92–112 (1990)
go back to reference Polak E.: Computational Methods in Optimization: A Unified Approach. Academic Press, New York (1971) Polak E.: Computational Methods in Optimization: A Unified Approach. Academic Press, New York (1971)
go back to reference Polyak, B.: Introduction to Optimization. Optimization Software, New York (1987)MATH Polyak, B.: Introduction to Optimization. Optimization Software, New York (1987)MATH
go back to reference Polyak, R.: Algorithm for Simultaneous Solution Primal and Dual Convex Programming Problems, Transaction of the Scientific Seminar. Cibernetic Institute, Kiev (1966) Polyak, R.: Algorithm for Simultaneous Solution Primal and Dual Convex Programming Problems, Transaction of the Scientific Seminar. Cibernetic Institute, Kiev (1966)
go back to reference Polyak, R.: The Legendre Transformation in Modern Optimization in “Optimization and its Applications in Control and Data Sciences”. Springer, Berlin (2016) Polyak, R.: The Legendre Transformation in Modern Optimization in “Optimization and its Applications in Control and Data Sciences”. Springer, Berlin (2016)
go back to reference Potra, F.: A quadratically convergent predictor - corrector method for solving linear programs from infeasible starting points. Math. Program. 67(3), 383–406 (1994)MathSciNetMATHCrossRef Potra, F.: A quadratically convergent predictor - corrector method for solving linear programs from infeasible starting points. Math. Program. 67(3), 383–406 (1994)MathSciNetMATHCrossRef
go back to reference Pshenichnyj, B.: The Linearization Method for Constrained Optimization. Springer, Berlin (1994)MATHCrossRef Pshenichnyj, B.: The Linearization Method for Constrained Optimization. Springer, Berlin (1994)MATHCrossRef
go back to reference Ray, A., Majumder, S.: Derivation of some new distributions in statistical mechanics using maximum entropy approach. Yugoslav J. Oper. Res. 24(1), 145–155 (2014)MathSciNetMATHCrossRef Ray, A., Majumder, S.: Derivation of some new distributions in statistical mechanics using maximum entropy approach. Yugoslav J. Oper. Res. 24(1), 145–155 (2014)MathSciNetMATHCrossRef
go back to reference Renegar, J.: A mathematical view of interior - point methods in convex optimization. In: MPS-SIAM Series on Optimization. SIAM, New York (2001) Renegar, J.: A mathematical view of interior - point methods in convex optimization. In: MPS-SIAM Series on Optimization. SIAM, New York (2001)
go back to reference Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained minimization. Math. Program. 5, 354–373 (1973)MATHCrossRef Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained minimization. Math. Program. 5, 354–373 (1973)MATHCrossRef
go back to reference Rockafellar, R.T., Wets R.J.-B: Variational Analysis. Springer, Berlin (2009)MATH Rockafellar, R.T., Wets R.J.-B: Variational Analysis. Springer, Berlin (2009)MATH
go back to reference Roos, C., Terlaky, T.: Advances in linear optimization. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliography in Combinatorical Optimization, Chapter 7. Wiley, New York (1997) Roos, C., Terlaky, T.: Advances in linear optimization. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliography in Combinatorical Optimization, Chapter 7. Wiley, New York (1997)
go back to reference Roos, C., Vial, J.-Ph.: Analytic centers in linear programming. Technical Report 88–74, Faculty of Mathematics and Computer Science, TU Delft, NL-2628 BL Delft, The Netherlands (1988) Roos, C., Vial, J.-Ph.: Analytic centers in linear programming. Technical Report 88–74, Faculty of Mathematics and Computer Science, TU Delft, NL-2628 BL Delft, The Netherlands (1988)
go back to reference Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Wiley, Chichester (1997)MATH Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Wiley, Chichester (1997)MATH
go back to reference Rozen, J.: The gradient projection method for nonlinear programming part 1: linear constraints. SIAM Journal Appl. Math. 8, 181–218 (1960)CrossRef Rozen, J.: The gradient projection method for nonlinear programming part 1: linear constraints. SIAM Journal Appl. Math. 8, 181–218 (1960)CrossRef
go back to reference Rozen, J.: The gradient projection method for nonlinear programming part 2: nonlinear constraints. SIAM J. Appl. Math. 9, 514–553 (1961)CrossRef Rozen, J.: The gradient projection method for nonlinear programming part 2: nonlinear constraints. SIAM J. Appl. Math. 9, 514–553 (1961)CrossRef
go back to reference Shor, N.: Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic, Boston (1998)MATHCrossRef Shor, N.: Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic, Boston (1998)MATHCrossRef
go back to reference Stiefel, E.: Note on Jordan elimination, linear programming and Tchebycheff approximation. Numerische Matematik 2(1), 1–17 (1960)MathSciNetMATHCrossRef Stiefel, E.: Note on Jordan elimination, linear programming and Tchebycheff approximation. Numerische Matematik 2(1), 1–17 (1960)MathSciNetMATHCrossRef
go back to reference Todd, M.: Recent development and new directions in linear programming. In: Iri, M., Tanabe, K. (eds.) Mathematical programming: recent developments and applications, pp. 109–157. Kluwer Academic Press, Dordrecht (1989) Todd, M.: Recent development and new directions in linear programming. In: Iri, M., Tanabe, K. (eds.) Mathematical programming: recent developments and applications, pp. 109–157. Kluwer Academic Press, Dordrecht (1989)
go back to reference Todd, M.: A lower bound on the number of iterations of primal-dual interior-point methods for linear programming. In: Griffits, G.F. (ed.) Numerical Analysis (1993). Pitman Research, vol. 303 Todd, M.: A lower bound on the number of iterations of primal-dual interior-point methods for linear programming. In: Griffits, G.F. (ed.) Numerical Analysis (1993). Pitman Research, vol. 303
go back to reference Todd, M., Ye, Y.: A lower bound on the number of iterations of long-step and polynomial interior-point linear programming algorithms. Ann. Oper. Res. 62, 233–252 (1996)MathSciNetMATHCrossRef Todd, M., Ye, Y.: A lower bound on the number of iterations of long-step and polynomial interior-point linear programming algorithms. Ann. Oper. Res. 62, 233–252 (1996)MathSciNetMATHCrossRef
go back to reference Tsuchiya, T.: Global convergence of the affine scaling methods for degenerate linear programming problems. Math. Program. 52, 377–404 (1991)MathSciNetMATHCrossRef Tsuchiya, T.: Global convergence of the affine scaling methods for degenerate linear programming problems. Math. Program. 52, 377–404 (1991)MathSciNetMATHCrossRef
go back to reference Tsuchiya, T.: Affine scaling algorithm. In: Terlaky, T. (ed.) Interior point methods of mathematical programming, pp. 35–82. Kluwer Academic Publishers, Dordrecht (1996)CrossRef Tsuchiya, T.: Affine scaling algorithm. In: Terlaky, T. (ed.) Interior point methods of mathematical programming, pp. 35–82. Kluwer Academic Publishers, Dordrecht (1996)CrossRef
go back to reference Tsuchiya, T., Muramatsu, M.: Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems. SIAM J. Optim. 5(3), 525–551 (1995)MathSciNetMATHCrossRef Tsuchiya, T., Muramatsu, M.: Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems. SIAM J. Optim. 5(3), 525–551 (1995)MathSciNetMATHCrossRef
go back to reference Vaidya, P.: An algorithm for linear programming which requires O((m + n)n 2 + (m + n)1.5 nL) arithmetic operations. Math. Program. 47, 175–201 (1990) Vaidya, P.: An algorithm for linear programming which requires O((m + n)n 2 + (m + n)1.5 nL) arithmetic operations. Math. Program. 47, 175–201 (1990)
go back to reference Vanderbei, R.: Linear Programming: Foundations and Extensions. Kluwer Academic, Boston (1996)MATH Vanderbei, R.: Linear Programming: Foundations and Extensions. Kluwer Academic, Boston (1996)MATH
go back to reference Vanderbei, R., Meketon, M., Freedman, B.: A modification of Karmarkar’s linear programming algorithm. Algorithmica 1(4), 395–407 (1986)MathSciNetMATHCrossRef Vanderbei, R., Meketon, M., Freedman, B.: A modification of Karmarkar’s linear programming algorithm. Algorithmica 1(4), 395–407 (1986)MathSciNetMATHCrossRef
go back to reference Ye, Y., Pardalos, P.: A class of linear complementarity problems solvable in polynomial time. Linear Algebra Appl. 152, 3–17 (1991)MathSciNetMATHCrossRef Ye, Y., Pardalos, P.: A class of linear complementarity problems solvable in polynomial time. Linear Algebra Appl. 152, 3–17 (1991)MathSciNetMATHCrossRef
go back to reference Ye, Y., Guler, O., Tapia, R., Zhang, Y.: A quadratically convergent \(O(\sqrt {n}L)\)-iteration algorithm for linear programming. Math. Program. 59, 151–162 (1993) Ye, Y., Guler, O., Tapia, R., Zhang, Y.: A quadratically convergent \(O(\sqrt {n}L)\)-iteration algorithm for linear programming. Math. Program. 59, 151–162 (1993)
go back to reference Ye, Y., Todd, M., Mizuno, S.: An \(O(\sqrt {n}L)\) - iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994) Ye, Y., Todd, M., Mizuno, S.: An \(O(\sqrt {n}L)\) - iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994)
go back to reference Zhang, Y., Tapia, R.: Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. J. Optim. Theory Appl. 73(2), 229–242 (1992)MathSciNetMATHCrossRef Zhang, Y., Tapia, R.: Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. J. Optim. Theory Appl. 73(2), 229–242 (1992)MathSciNetMATHCrossRef
go back to reference Zoutendijk, G.: Methods of Feasible Directions. Elsevier, Amsterdam (1960)MATH Zoutendijk, G.: Methods of Feasible Directions. Elsevier, Amsterdam (1960)MATH
go back to reference Zuchovitsky, S.: Algorithm for finding chebushev approximation of overdetermined system of linear equations. Dokl. Acad. Nauk SSSR 79(4), 561–564 (1951) Zuchovitsky, S.: Algorithm for finding chebushev approximation of overdetermined system of linear equations. Dokl. Acad. Nauk SSSR 79(4), 561–564 (1951)
go back to reference Zukhovitsky, S., Polyak, R., Primak, M.: An algorithm for solving convex Chebyshev approximation problem. Dokl. Acad. Nauk SSSR 151(1), 27–30 (1963)MathSciNet Zukhovitsky, S., Polyak, R., Primak, M.: An algorithm for solving convex Chebyshev approximation problem. Dokl. Acad. Nauk SSSR 151(1), 27–30 (1963)MathSciNet
go back to reference Zukhovitsky, S., Polyak, R., Primak, M.: An algorithm for solving convex programming problem. Dokl. Acad. Nauk SSSR 153(3), 991–994 (1963a)MathSciNet Zukhovitsky, S., Polyak, R., Primak, M.: An algorithm for solving convex programming problem. Dokl. Acad. Nauk SSSR 153(3), 991–994 (1963a)MathSciNet
go back to reference Zukhovitsky, S., Polyak, R., Primak, M.: Numerical method for solving convex programming problem in Hilbert space. Dokl. Acad. Nauk SSSR 163(2) (1965) Zukhovitsky, S., Polyak, R., Primak, M.: Numerical method for solving convex programming problem in Hilbert space. Dokl. Acad. Nauk SSSR 163(2) (1965)
Metadata
Title
Basics in Linear and Convex Optimization
Author
Roman A. Polyak
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-68713-7_5

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