Skip to main content
    • Aa
    • Aa

Interior-point methods for optimization

  • Arkadi S. Nemirovski (a1) and Michael J. Todd (a2)

This article describes the current state of the art of interior-point methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

F. Alizadeh (1995), ‘Interior point methods in semidefinite programming with applications to combinatorial optimization’, SIAM J. Optim. 5, 1351.

F. Alizadeh , J.-P. A. Haeberly and M. L. Overton (1998), ‘Primal–dual interiorpoint methods for semidefinite programming: Convergence rates, stability and numerical results’, SIAM J. Optim. 8, 746768.

E. D. Andersen and Y. Ye (1999), ‘On a homogeneous algorithm for monotone complementarity system’, Math. Program. 84, 375399.

H. H. Bauschke , O. Güler , A. S. Lewis and H. S. Sendov (2001), ‘Hyperbolic polynomials and convex analysis’, Canad. J. Math. 53, 470488.

A. Ben-Tal and A. S. Nemirovski (2001), Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia.

R. E. Bixby (2002), ‘Solving real-world linear programs: A decade and more of progress’, Oper. Res. 50, 315.

I. M. Bomze (1988), ‘On standard quadratic optimization problems’, J. Global Optim. 13, 369387.

I. M. Bomze and E. de Klerk (2002), ‘Solving standard quadratic optimization problems via semidefinite and copositive programming’, J. Global Optim. 24, 163185.

I. M. Bomze , M. Dür , E. de Klerk , C. Roos , A. J. Quist and T. Terlaky (2000), ‘On copositive programming and standard quadratic optimization problems’, J. Global Optim. 18, 301320.

S. Boyd and L. Vandenberghe (2004), Convex Optimization, Cambridge University Press.

S. Boyd , L. El Ghaoui , E. Feron and V. Balakrishnan (1994), Linear Matrix Inequalities in System and Control Theory, Vol. 15 of Studies in Applied Mathematics, SIAM.

R. H. Byrd , J. Nocedal and R. A. Waltz (2006), KNITRO: An integrated package for nonlinear optimization. In Large-Scale Nonlinear Optimization ( G. di Pillo and M. Roma , eds), Springer, New York, pp. 3559.

C. W. Carroll (1961), ‘The created response surface technique for optimizing nonlinear, restrained systems’, Oper. Res. 9, 169185.

J. S. Chai and K. C. Toh (2007), ‘Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming’, Comput. Optim. Appl. 36, 221247.

L. Chen and D. Goldfarb (2006), ‘Interior-point l2-penalty methods for nonlinear programming with strong convergence properties’, Math. Program. 108, 136.

R. Courant (1943), ‘Variational methods for the solution of problems of equilibrium and vibrations’, Bull. Amer. Math. Soc. 49, 123.

L. Faybusovich (1997), ‘Linear systems in Jordan algebras and primal–dual interiorpoint algorithms’, J. Comput. Appl. Math. 86, 149175.

A. Forsgren , P. E. Gill and M. H. Wright (2002), ‘Interior methods for nonlinear optimization’, SIAM Review 44, 525597.

R. W. Freund , F. Jarre and S. Schaible (1996), ‘On self-concordant barrier functions for conic hulls and fractional programming’, Math. Program. 74, 237246.

D. M. Gay , M. L. Overton and M. H. Wright (1998), A primal–dual interior method for nonconvex nonlinear programming. In Advances in Nonlinear Programming ( Y. Yuan , ed.), Kluwer, pp. 3156.

P. E. Gill , W. Murray , M. A. Saunders , J. A. Tomlin and M. H. Wright (1986), ‘On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method’, Math. Program. 36, 183209.

M. X. Goemans (1997), ‘Semidefinite programming in combinatorial optimization’, Math. Program. 79, 143161.

C. C. Gonzaga (1989), An algorithm for solving linear programming problems in O(n3L) operations. In Progress in Mathematical Programming: Interior Point and Related Methods ( N. Megiddo , ed.), Springer, New York, pp. 128.

N. I. M. Gould , D. Orban and P. L. Toint (2005), Numerical methods for large-scale nonlinear optimization. In Acta Numerica, Vol. 14, Cambridge University Press, pp. 299361.

O. Güler (1996), ‘Barrier functions in interior-point methods’, Math. Oper. Res. 21, 860885.

O. Güler (1997), ‘Hyperbolic polynomials and interior-point methods for convex programming’, Math. Oper. Res. 22, 350377.

C. Helmberg , F. Rendl , R. Vanderbei and H. Wolkowicz (1996), ‘An interior-point method for semidefinite programming’, SIAM J. Optim. 6, 342361.

N. Karmarkar (1984), ‘A new polynomial-time algorithm for linear programming’, Combinatorica 4, 373395.

E. de Klerk , C. Roos and T. Terlaky (1997), ‘Initialization in semidefinite programming via a self-dual skew-symmetric embedding’, Oper. Res. Letters 20, 213221.

M. Kojima , S. Mizuno and A. Yoshise (1989), ‘A polynomial–time algorithm for a class of linear complementarity problems’, Math. Program. 44, 126.

M. Kojima , S. Shindoh and S. Hara (1997), ‘Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices’, SIAM J. Optim. 7, 86125.

J. B. Lasserre (2001), ‘Global optimization with polynomials and the problem of moments’, SIAM J. Optim. 11, 796817.

A. S. Lewis and M. L. Overton (1996), Eigenvalue optimization. In Acta Numerica, Vol. 5, Cambridge University Press, pp. 149160.

M. S. Lobo , L. Vandenberghe , S. Boyd and H. Lebret (1998), ‘Applications of second-order cone programming’, Linear Algebra Appl. 284, 193228.

Z.-Q. Luo , J. F. Sturm and S. Zhang (2000), ‘Conic convex programming and self-dual embedding’, Optim. Methods Software 14, 169218.

N. Megiddo (1989), Pathways to the optimal set in linear programming. In Progress in Mathematical Programming: Interior Point and Related Methods ( N. Megiddo , ed.), Springer, New York, pp. 131158.

S. Mizuno (1994), ‘Polynomiality of infeasible-interior-point algorithms for linear programming’, Math. Program. 67, 109119.

R. D. C. Monteiro (1997), ‘Primal–dual path-following algorithms for semidefinite programming’, SIAM J. Optim. 7, 663678.

R. D. C. Monteiro and I. Adler (1989), ‘Interior path following primal–dual algorithms I: Linear programming’, Math. Program. 44, 2741.

S. G. Nash (1998), ‘SUMT (revisited)’, Oper. Res. 46, 763775.

A. Nemirovski and L. Tunçel (2005), ‘“Cone-free” primal–dual path-following and potential-reduction polynomial time interior-point methods’, Math. Program. 102, 261295.

Y. Nesterov (1997), ‘Long-step strategies in interior-point primal–dual methods’, Math. Program. 76, 4794.

Y. Nesterov and A. Nemirovski (1994), Interior Point Polynomial Time Methods in Convex Programming, SIAM, Philadelphia.

Y. E. Nesterov and M. J. Todd (1997), ‘Self-scaled barriers and interior-point methods for convex programming’, Math. Oper. Res. 22, 142.

Y. E. Nesterov and M. J. Todd (1998), ‘Primal–dual interior-point methods for self-scaled cones’, SIAM J. Optim. 8, 324364.

and (), , , . J. Nocedal S. J. Wright 2006 Numerical Optimization SpringerNew York

P. A. Parrilo (2003), ‘Semidefinite programming relaxations for semialgebraic problems’, Math. Program. 96, 293320.

F. A. Potra and R. Sheng (1998), ‘On homogeneous interior-point algorithms for semidefinite programming’, Optim. Methods Software 9, 161184.

J. Renegar (2001), A Mathematical View of Interior-Point Methods in Convex Optimization, SIAM, Philadelphia.

J. Renegar (2006), ‘Hyperbolic programs, and their derivative relaxations’, Found. Comput. Math. 6, 5979.

S. H. Schmieta and F. Alizadeh (2001), ‘Associative and Jordan algebras, and polynomial-time interior-point algorithms for symmetric cones’, Math. Oper. Res. 26, 543564.

M. J. Todd (2001), Semidefinite optimization. In Acta Numerica, Vol. 10, Cambridge University Press, pp. 515560.

M. J. Todd and Y. Ye (1990), ‘A centered projective algorithm for linear programming’, Math. Oper. Res. 15, 508529.

M. J. Todd , K.-C. Toh and R. H. Tütüncü (1998), ‘On the Nesterov–Todd direction in semidefinite programming’, SIAM J. Optim. 8, 769796.

K. C. Toh (2007), ‘An inexact primal–dual path-following algorithm for convex quadratic SDP’, Math. Program. 112, 221254.

L. Tunçcel (1998), ‘Primal–dual symmetry and scale-invariance of interior-point algorithms for convex programming’, Math. Oper. Res. 23, 708718.

R. J. Vanderbei (2007), Linear Programming: Foundations and Extensions, Springer, New York.

R. J. Vanderbei and D. F. Shanno (1999), ‘An interior-point algorithm for nonconvex nonlinear programming’, Comput. Optim. Appl. 13, 231252.

R. A. Waltz , J. L. Morales , J. Nocedal and D. Orban (2006), ‘An interior algorithm for nonlinear optimization that combines line search and trust region steps’, Math. Program. 107, 391408.

A. Wächter and L. T. Biegler (2006), ‘On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming’, Math. Program. 106, 2557.

H. Wolkowicz , R. Saigal and L. Vanderberghe , eds (2000), Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer, Boston.

M. H. Wright (1992), Interior methods for constrained optimization. In Acta Numerica, Vol. 1, Cambridge University Press, pp. 341407.

S. J. Wright (1997), Primal–Dual Interior-Point Methods, SIAM, Philadelphia.

X. Xu , P. F. Hung and Y. Ye (1996), ‘A simplified homogeneous self-dual linear programming algorithm and its implementation’, Ann. Oper. Res. 62, 151171.

Y. Ye (1991), ‘An O(n3L) potential reduction algorithm for linear programming’, Math. Program. 50, 239258.

Y. Ye (1997), Interior Point Algorithms: Theory and Analysis, Wiley.

Y. Ye , M. J. Todd and S. Mizuno (1994), ‘An O()-iteration homogeneous and self-dual linear programming algorithm’, Math. Oper. Res. 19, 5367.

Y. Zhang (1994), ‘On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem’, SIAM J. Optim. 4, 208227.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 30 *
Loading metrics...

Abstract views

Total abstract views: 138 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th June 2017. This data will be updated every 24 hours.