Abstract
We propose an interior point method for large-scale convex quadratic programming where no assumptions are made about the sparsity structure of the quadratic coefficient matrixQ. The interior point method we describe is a doubly iterative algorithm that invokes aconjugate projected gradient procedure to obtain the search direction. The effect is thatQ appears in a conjugate direction routine rather than in a matrix factorization. By doing this, the matrices to be factored have the same nonzero structure as those in linear programming. Further, one variant of this method istheoretically convergent with onlyone matrix factorization throughout the procedure.
Similar content being viewed by others
References
E.R. Barnes, “A variation on Karmarkar's algorithm for solving linear programming problems,”Math. Programming,36 (1986) 174–182.
A. Brooke, D. Kendrick, and A. Meeraus,GAMS: A User's Guide, The Scientific Press, Redwood City, CA, 1988.
T.J. Carpenter, I.J. Lustig, J.M. Mulvey, and D.F. Shanno,Separable quadratic programming via a primal-dual interior point method and its use in a sequential procedure, Technical Report SOR-90-2, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1990.
R.S. Dembo and T. Steihaug, “Truncated-Newton algorithms for large-scale unconstrained optimization,”Math. Programming 26 (1983), 190–212.
I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Soviet Math. Doklady 8 (1967) 674–675.
R. Fletcher,Practical Methods of Optimization, John Wiley and Sons, Chichester, England, 1987.
R. Fourer and S. Mehrotra, “Performance of an augmented system approach for solving least-squares problems in an interior point method for linear programming,”COAL Newsletter,19 (1991) 26–30.
P.E. Gill, W. Murray, D.B. Ponceleón, and M.A. Saunders,Solving reduced KKT systems in barrier methods for linear for quadratic programming, Technical Report SOL 91-7, Department of Operations Research, Stanford University, Stanford, CA, 1991.
P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, “On projected Newton methods for linear programming and equivalence to Karmarkar's projective method,”Math. Programming 36 (1986) 183–209.
P.E. Gill, W. Murray, and M.H. Wright,Practical Optimization, Academic Press, New York, NY, 1981.
D. Goldfarb and S. Liu, “AnO(n 3 L) primal interior point algorithm for convex quadratic programming,”Math. Programming 49 (1991) 325–340.
G.H. Golub and C.F. Van Loan,Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1983.
C.C. Gonzaga, “Large step path-following methods for linear programming, Part I: Barrier function method,”SIAM J. on Optimization,1 (1991) 268–279.
M.A. Jenkins, “DOMINO-An APL primitive function for matrix inversion-Its implementation and applications,”APL Quote-Quad 4 (1972) 4–15.
N.K. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica,4 (1984) 373–395.
D.G. Luenberger,Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, MA, 1984.
I.J. Lustig, R.E. Marsten, and D.F. Shanno, “On implementing Mehrotra's predictor-corrector interior point method,”SIAM J. on Optimization,2 (1992) 435–449.
I.J. Lustig, R.E. Marsten, and D.F. Shanno,Starting and restarting the primal-dual interior point method, Technical Report SOR-90-14, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1990.
I.J. Lustig, R.E. Marsten, and D.F. Shanno, “Computational experience with a primal-dual interior point method for linear programming,”Linear Algebra and its Applications,152 (1991) 191–222.
H.M. Markowitz,Portfolio Selection: Efficient Diversification of Investments, John Wiley, New York, NY, 1959.
K.A. McShane, C.L. Monma, and D.F. Shanno, “An implementation of a primal-dual interior point method for linear programming,”ORSA J. on Computing,1 (1989) 70–83.
S. Mehrotra,On the implementation of a (primal-dual) interior point method, Technical Report 90-03, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1990.
R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part I: Linear programming,”Math. Programming,44 (1989) 27–42.
R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part II: Convex quadratic programming,”Math. Programming,44 (1989) 43–66.
B.A. Murtaugh and M.A. Saunders,Minos 5.1 User's Guide, Technical Report SOL-83-20R, Systems Optimization Laboratory, Stanford University, Stanford, CA, 1983.
J.L. Nazareth, “Pricing criteria in linear programming,” in N. Megiddo, editor,Progress in Mathematical Programming: Interior Point and Related Methods, Springer-Verlag, New York, NY, 1989, 105–129.
D.B. Ponceleón,Barrier methods for large-scale quadratic programming, PhD thesis, Department of Computer Science, Stanford University, Standord, CA, 1990.
R. Tapia, Y. Zhang, M. Saltzman, and A. Weiser,The predictor-corrector interior-point method as a composite Newton method, Technical Report 90-6, Department of Mathematical Sciences, Rice University, Houston, TX, 1990.
R.J. Vanderbei,ALPO: Another Linear Program Optimizer, Technical Report, AT&T Bell Laboratories, Murray Hill, NJ, 1990.
R.J. Vanderbei,Symmetric quasi-definite matrices, Technical Report SOR-91-10, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1991.
R.J. Vanderbei and T.J. Carpenter,Symmetric indefinite systems for interior point methods, Technical Report SOR-91-7, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1991.
R.J. Vanderbei, M.S. Meketon, and B.F. Freedman, “A modification of Karmarkar's linear programming algorithm,”Algorithmica,1 (1986) 395–407.
Y. Ye and E. Tse, “An extension of Karmarkar's projective algorithm for convex quadratic programming,”Math. Programming,44 (1989) 157–179.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carpenter, T.J., Shanno, D.F. An interior point method for quadratic programs based on conjugate projected gradients. Comput Optim Applic 2, 5–28 (1993). https://doi.org/10.1007/BF01299140
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01299140