Abstract
We describe recent developments in interior-point algorithms for global optimization. We will focus on the algorithmic research for nonconvex quadratic programming, linear complementarity problem, and integer programming. We also outline directions in which future progress might be made.
Similar content being viewed by others
References
D. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming, I: Affine and projective scaling trajectories, Trans. Am. Math. Soc. 314 (1989) 499–581.
E.M.L. Beale, On quadratic programming, Naval Res. Logistics Quarterly 6 (1959) 227–243.
R.W. Cottle and G.B. Dantzig, Complementary pivot theory of mathematical programming, Linear Algebra Appl. 1 (1968) 103–125.
R.W. Cottle, J.S. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra Appl. 114/115 (1989) 231–249.
R.W. Cottle and A.F. Veinott, Jr., Polyhedral sets having a least element, Math. Programming 3 (1969) 238–249.
I.I. Dikin, Iterative solution of problems of linear and quadratic programming, Sov. Math. Doklady 8 (1967) 674–675.
R. Fletcher, A general quadratic programming algorithm, J. Inst. Math. Appl. 7 (1971) 76–91.
M.R. Garey and D.S. Johnson,Computers and Intractability, A Guide to the Theory of NP-Completeness (Freeman, New York, 1968).
D.M. Gay, Computing optimal locally constrained steps, SIAM J. Sci. Statis. Comput. 2 (1981) 186–197.
P.E. Gill and W. Murray, Newton type methods for unconstrained and linearly constrained optimization, Math. Programming 7 (1974) 311–350.
D. Goldfarb, Extensions of Newton's method and simplex methods for solving quadratic programs, in:Numerical Methods for Nonlinear Optimization, ed. F.A. Lootsma (Academic Press, London, 1972).
F.S. Hillier, Efficient heuristic procedures for integer linear programming with an interior, Oper. Res. 17 (1969) 600–637.
B. Kalantari, Canonical problems for quadratic programming and projective methods for their solution, manuscript, Department of Computer Science, Rutgers University, New Brunswick, NJ (1988).
B. Kalantari and J.B. Rosen, An algorithm for global minimization of linearly constrained concave quadratic functions, Math. Oper. Res. 12 (1987) 544–561.
A.P. Kamath, N. Karmarkar, K.G. Ramakrishnan and M.G.C. Resende, Computational experience with an interior point algorithm on the satisfiability problem, Technical Report, AT & T Bell Laboratories, Murray Hill, NJ (1989).
S. Kapoor and P. Vaidya, Fast algorithms for convex quadratic programming and multicommodity flows,Proc. 18th Annual ACM Symp. on the Theory of Computing (1989) pp. 147–159.
N. Karmarkar, An interior-point approach to NP-complete problems, extended abstract, AT&T Bell Laboratories, Murray Hill, NJ (1988).
N. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4 (1984) 373–395.
N. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, An interior point algorithm to solve computationally difficult set covering problems, Technical Report, AT&T Bell Laboratories. Murray Hill, NJ (1989).
N. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, An interior point approach to the maximum independent set problem in dense random graphs. Technical Report, AT&T Bell Laboratories, Murray Hill, NJ (1989).
L.G. Khachiyan, A polynomial algorithm for linear programming, Sov. Math. Doklady 20 (1979) 191–194.
M. Kojima, N. Megiddo and T. Noma, Homotopy continuation methods for complementarity problems, manuscript, IBM Almaden Research Center, San Jose, CA (1988).
M. Kojima, N. Mizuno and T. Noma, A new continuation method for complementarity problems with uniform P-functions, manuscript (1988), to appear in Math. Programming.
M. Kojima, N. Megiddo and Y. Ye, An interior point potential reduction algorithm for the linear complementarity problem. Research Report RJ 6486, IBM Almaden Research Center. San Jose, CA (1988), to appear in Math. Programming.
M. Kojima, S. Mizuno and A. Yoshise, An\(O(\sqrt n L)\) iteration potential reduction algorithm for linear complementarity problems, Research Report on Information Sciences B-217, Tokyo Institute of Technology, Tokyo, Japan (1988).
M. Kojima, S. Mizuno and A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems, Math. Programming 44 (1989) 1–26.
M.K. Kozlov, S.P. Tarasov and L.G. Khachiyan, Polynomial solvability of convex quadratic programming, Sov. Math. Doklady 20 (1979) 1108–1111.
C.E. Lemke, Bimatrix equilibrium points and mathematical programming, Management Sci. 11 (1965) 681–689.
O.L. Mangasarian, Linear complementarity problems solvable by a single linear program, Math. Programming 10 (1977) 263–270.
N. Megiddo, Pathways to the optimal set in linear programming, in:Progress in Mathematical Programming, ed. N. Megiddo (Springer, 1988) pp. 131–158.
R.C. Monteiro and I. Adler, AnO(n 3 L) primal-dual interior point algorithm for convex quadratic programming, Math. Programming 44 (1989) 27–42.
J.J. Moré and D.C. Sorensen, Computing a trust region step, SIAM J. Sci. Statist. Comput. 4 (1983) 553–572.
J.J. Moré and V. Vavasis, On the solution of concave knapsack problems, Preprint MCS-P40-1288, Argonne National Laboratory, Argonne, IL (1988).
K.G. Murty and S.N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Math. Programming 39 (1987) 117–129.
G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley, New York, 1988).
Yu.E. Nesterov and A.S. Nemirovsky,Self-Concordant Functions and Polynomial-Time Methods in Convex Programming (USSR Academy of Sciences, Central Economic and Mathematic Institute, Moscow, 1989).
P.M. Pardalos and J.B. Rosen,Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Sciences, vol. 268 (Springer, 1987).
P.M. Pardalos and G. Schnitger, Checking local optimality in constrained quadratic programming is NP-hard, Oper. Res. Lett. 7 (1988) 33–35.
P.M. Pardalos, Y. Ye and C.-G. Han, Algorithms for the solution of quadratic knapsack problems, Technical Report CS-89-10, Department of Computer Science, The Pennsylvania State University, University Park, PA (1989).
J. Renegar, A polynomial-time algorithm based on Newton's method for linear programming, Math. Programming 40 (1988) 59–93.
G. Sonnevend, An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,Proc. 12th IFIP Conf. on System Modeling and Optimization, Budapest, Hungary (1985).
R.E. Stone, Geometric aspects of the linear complementarity problem, Technical Report SOL 81-6, Department of OR, Stanford University, Stanford, CA (1981).
M.J. Todd and Y. Ye, A centered projective algorithm for linear programming, Technical Report 763, School of ORIE, Cornell University, Ithaca, NY (1987), to appear in Math. Oper. Res.
P. Wolfe, The simplex algorithm for quadratic programming, Econometrica 27 (1959) 382–398.
Y. Ye, Interior point algorithms for quadratic programming, Working Paper 89-29, College of Business Administration, The University of Iowa, Iowa City, IA (1989).
Y. Ye, AnO(n 3 L) potential reduction algorithm for linear programming, manuscript (1988), to appear in Math. Programming.
Y. Ye, On affine scaling algorithms for nonconvex quadratic programming, manuscript (1988), submitted to Math. Programming.
Y. Ye, A further result on the potential reduction algorithm for the P-matrix linear complementarity problem, Working Paper (1988).
Y. Ye and P. Pardalos, A class of linear complementarity problems solvable in polynomial time, manuscript (1989), submitted to Linear Algebra Appl.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
YE, Y. Interior-point algorithms for global optimization. Ann Oper Res 25, 59–73 (1990). https://doi.org/10.1007/BF02283687
Issue Date:
DOI: https://doi.org/10.1007/BF02283687