Abstract
The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. Interior-point methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear programs. We review some core developments in the area.
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F. Alizadeh, J.-P.A. Haeberly and M.L. Overton (1998), Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability, and numerical results, SIAM Journal on Optimization, 8, pp. 746–768.
K.D. Andersen (1996), A modified Schur complement method for handling dense columns in interior-point methods for linear programming, ACM Transaction on Mathematical Software, 22 (3), pp. 348–356.
J.R. Birge and F. Louveaux (1997), Introduction to Stochastic Programming, Springer Series in Operations Research, Springer.
J.R. Birge and L. Qi (1988), Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34, pp. 1472–1479.
R.H. Byrd, M. Hribar and J. Nocedal (1997), An interior point algorithm for large scale nonlinear programming, OTC Technical Report 97/05, Optimization Technology Center.
J. Castro (1998), A specialized interior-point algorithms for multi-commodity network flows, Technical report, Statistics and Operations Research, Universitat Rovira i Virgili, Tarragona, Spain.
A.R. Conn, N.I.M. Gould, D. Orban and P. Toint (1999), A primal-dual trust-region algorithm for minimizing a non-convex function subject to general inequality and linear equality constraints, Technical Report RAL-TR-1999–054, Atlas Centre, Rutherford Appleton Laboratory.
L. Faybusovich (1997), Linear systems in Jordan algebras and primal-dual interior-point algorithms, Journal of Computational and Applied Mathematics, 86, pp. 149–175.
A.V. Fiacco and G.P. McCormick (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York. Reprinted by SIAM Publications, 1990.
A. Forsgren and P.E. Gill (1998), Primal-dual interior-point methods for nonconvex nonlinear programming, SIAM Journal on Optimization, 8 (4), pp. 1132–1152.
D.M. Gay, M.L. Overton and M.H. Wright (1997),A primal–dual interior method for nonconvex nonlinear programming, Technical Report 97–4–08,Computing Sciences Research, Bell Laboratories, Murray Hill, NJ.
M.X. Goemans and D.P. Williamson (1995), Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the Association for Computing Machinery, 42 (6), pp. 1115–1145.
J. Goffin and J. Vial (1999),Convex nondifferentiable optimization: A survey based on the analytic center cutting plane method, Technical Report 99.02,Logilab, HEC, Section of Management Studies, University of Geneva.
J. Gondzio (1996), Multiple centrality corrections in a primal-dual method for linear programming, Computational Optimization and Applications, 6, pp. 137–156.
J. Gondzio and R. Kouwenberg (1999) High performance computing for asset liability management, Technical Report MS-99–004,Department of Mathematics and Statistics, The University of Edinburgh.
J. Gondzio and R. Sarkissian (1996), Column generation with a primal-dual method, Technical Report 96. 9, Logilab, HEC, Section of Management Studies, University of Geneva. Revised October, 1997.
C. Gonzaga (1991), Large-step path-following methods for linear programming, SIAM Journal on Optimization, 1, pp. 268–279.
N.I.M. Gould and P.L. Toint (1999), SQP methods for large-scale nonlinear programming, Technical Report RAL-TR-1999–055, Atlas Centre, Rutherford Appleton Laboratory.
J. Haeberly, M.V. Nayakkankuppam and M.L. Overton (1999), Extending Mehrotra and Gondzio higher order methods to mixed semidefinite-quadratic-linear programming, to appear in Optimization Methods and Software.
N. Karmarkar (1984), A new polynomial-time algorithm for linear programming, Combinatorica, 4, pp. 373–395.
M.S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret (1998), Applications of second-order cone programming, Linear Algebra and Its Applications, 248, pp. 193–228.
S. Mehrotra (1992), Asymptotic convergence in a generalized predictor-corrector method, Technical Report, Dept. of Industrial Engineering and Management Science, Northwestern University, Evanston, Ill.
S. Mehrotra and J.-S. Wang (1995), Conjugate gradient based implementation of interior point methods for network flow problems, Technical Report 95–70, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Ill.
J.E. Mitchell (1997), Computational experience with an interior-point cutting plane algorithm, Technical report, Mathematical Sciences Department, Rensselaer Polytechnic Institute. Revised March 1999.
J.E. Mitchell (1999), Restarting after branching in the SDP approach to MAX-CUT and similar combinatorial optimization problems, Technical report, Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, NY.
J.E. Mitchell, P.M. Pardalos and M. Resende (1998), Interior-point methods for combinatorial optimization, in D. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 1, Kluwer Academic Publishers.
R.D.C. Monteiro and S.J. Wright (1994), Local convergence of interior-point algorithms for degenerate monotone LCP, Computational Optimization and Applications, 3, pp. 131–155.
Y.E. Nesterov and A.S. Nemirovskii (1994), Interior Point Polynomial Methods in Convex Programming, SIAM Publications, Philadelphia.
Y.E. Nesterov and M.J. Todd (1997), Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research, 22, pp. 1–42.
Y.E. Nesterov and M.J. Todd (1998), Primal-dual interior-point methods for self-scaled cones, SIAM Journal on Optimization, 8, pp. 324–362.
S. Portnoy and R. Koenker (1997), The Gaussian hare and the Laplacian tortoise: Computability of squared-error vs. absolute-error estimators, Statistical Science, 12, pp. 279–300.
D. Ralph and S.J. Wright (1996), Superlinear convergence of an interior-point method despite dependent constraints, Preprint ANL.MCS-P622–1196, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill.
C.V. Rao, S.J. Wright and J.B. Rawlings (1998), Application of interior-point methods to model predictive control, Journal of Optimization Theory and Applications, 99, pp. 723–757.
J. Renegar (1999), A mathematical view of interior-point methods in convex optimization, Unpublished notes.
C. Roos, J.-P. Vial and T. Terlaky (1997), Theory and Algorithms for Linear Optimization: An Interior Point Approach, WileyInterscience Series in Discrete Mathematics and Optimization, John Wiley and Sons.
H. Takehara (1993), An interior-point algorithm for large-scale portfolio optimization, Annals of Operations Research, 45, pp. 373–386.
M.J. Todd (1999),A study of search directions in primal-dual interior-point methods for semidefinite programming, Technical report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY.
L. Vandenberghe and S. Boyd (1996), Semidefinite programming, SIAM Review, 38, pp. 49–95.
H. Wolkowicz (2000),Semidefinite and Lagrangian relaxation algorithms for hard combinatorial problems, Proceedings of the 19th IFIP TC7 Conference on System Modeling and Optimization, July, 1999, Cambridge,Kluwer Academic Publishers (to appear).
M.H. Wright (1992), Interior methods for constrained optimization, in Acta Numerica 1, pp. 341–407.
S.J. Wright (1993). Interior point methods for optimal control of discrete-time systems, Journal of Optimization Theory and Applications, 77, pp. 161–187.
S.J. Wright (1997), Primal-Dual Interior-Point Methods, SIAM Publications, Philadelphia.
S.J. Wright (1999), Modified Cholesky factorizations in interior-point algorithms for linear programming, SIAM Journal on Optimization, 9, pp. 1159–1191.
Y. Ye (1997), Interior Point Algorithms: Theory and Analysis, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley and Sons.
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Wright, S.J. (2000). Recent Developments in Interior-Point Methods. In: Powell, M.J.D., Scholtes, S. (eds) System Modelling and Optimization. CSMO 1999. IFIP — The International Federation for Information Processing, vol 46. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35514-6_14
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DOI: https://doi.org/10.1007/978-0-387-35514-6_14
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