Abstract
We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is\(\left[ {\begin{array}{*{20}c} { - D^{ - 2} } & {A^T } \\ A & 0 \\ \end{array} } \right]\) instead of reducing to obtain the usualAD 2 A T system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the productAD 2 A T whenA has dense columns; and (2) it can easily be used to solve nonseparable quadratic programs since it requires only thatD be symmetric. We also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense.
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References
I. Adler, N.K. Karmarkar, M.G.C. Resende and G. Veiga, “An implementation of Karmarkar's algorithm for linear programming,”Mathematical Programming 44 (1989) 297–335.
E.R. Barnes, “A variation on Karmarkar's algorithm for solving linear programming problems,”Mathematical Programming 36 (1986) 174–182.
J.R. Bunch and B.N. Parlett, “Direct methods for solving symmetric indefinite systems of linear equations,”SIAM Journal on Numerical Analysis 8 (1971) 639–655.
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,” to appear in:ORSA Journal on Computing.
I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Soviet Mathematics Doklady 8 (1967) 674–675.
R. Fourer and S. Mehrotra, “Performance of an augmented system approach for solving least-squares problems in an interior point method for linear programming,”Mathematical Programming Society COAL Newsletter 19 (1991) 26–31.
D.M. Gay, “Electronic mail distribution of linear programming test problems,”Mathematical Programming Society COAL Newsletter 13 (1985) 10–12.
A. George and J. Liu,Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).
P.E. Gill, W. Murray, D.B. Ponceleón and M.A. Saunders, “Preconditioners for indefinite systems arising in optimization,” Technical Report SOL 90-8, Systems Optimization Laboratory, Stanford University (Stanford, CA, 1990).
P.E. Gill, W. Murray and M.H. Wright,Numerical Linear Algebra and Optimization, Vol. 1 (Addison-Wesley, Redwood City, CA, 1991).
N.K. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.
H. Konno and K. Suzuki, “A fast algorithm for solving large scale mean-variance models by compact factorization of covariance matrices,” Report IHSS 91-32, Institute of Human and Social Sciences, Tokyo Institute of Technology (Tokyo, 1991).
I.J. Lustig, “Feasibility issues in a primal—dual interior point method for linear programming,”Mathematical Programming 49 (1991) 145–162.
I.J. Lustig, R. 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.
I.J. Lustig, R.E. Marsten and D.F. Shanno, “On implementing Mehrotra's predictor—corrector interior point method for linear programming,”SIAM Journal on Optimization 2 (1992) 435–449.
H.M. Markowitz,Portfolio Selection: Efficient Diversification of Investments (Wiley, New York, 1959).
R.E. Marsten, M.J. Saltzman, D.F. Shanno, G.S. Pierce and J.F. Ballintijn, “Implementation of a dual affine interior point algorithm for linear programming,”ORSA Journal on Computing 1 (1989) 287–297.
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).
S. Mehrotra, “Handling free variables in interior point methods,” Technical Report 91-06, Department of Industrial Engineering and Management Sciences, Northwestern University (Evanston, IL, 1991).
R.D.C. Monteiro and I. Adler, “Interior path following primal—dual algorithms. Part I: Linear programming,”Mathematical Programming 44 (1989) 27–42.
R.D.C. Monteiro and I. Adler, “Interior path following primal—dual algorithms. Part II: Convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.
B.A. Murtagh and M.A. Saunders, “MINOS 5.1 user's guide,” Technical Report SOL 83-20R, Systems Optimization Laboratory, Stanford University (Stanford, CA, 1987).
D.B. Ponceléon, “Barrier methods for large-scale quadratic programming,” Ph.D. Thesis, Stanford University (Stanford, CA, 1990).
R.J. Vanderbei, “ALPO: Another linear program optimizer,” Technical Report, AT&T Bell Laboratories (Murray Hill, NJ, 1990).
R.J. Vanderbei, “A brief description of ALPO,” 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, M.S. Meketon and B.F. Freedman, “A modification of Karmarkar's linear programming algorithm,”Algorithmica 1 (1986) 395–407.
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Vanderbei, R.J., Carpenter, T.J. Symmetric indefinite systems for interior point methods. Mathematical Programming 58, 1–32 (1993). https://doi.org/10.1007/BF01581257
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DOI: https://doi.org/10.1007/BF01581257