Some feasible direction methods for the minimization of a linearly constrained convex function are studied. Special emphasis is placed on the analysis of the procedures which find the search direction, by developing active set methods which use orthogonal or Gauss-Jordan-like transformations.
Numerical experiments are performed on a class of quadratic problems depending on two parameters, related to the conditioning of the matrix associated with the quadratic form and the matrix of active constraints at the optimal point. Results are given for the rate of convergence and the average iteration time.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Zoutendijk, G.,Mathematical Programming Methods, North-Holland, Amsterdam, Holland, 1976.
Rosen, J. B.,The Gradient Projection Methods for Nonlinear Programming, Part 1, Linear Constraints, SIAM Journal on Applied Mathematics, Vol. 8, No. 1, 1960.
Topkis, D. M., andVeinott, A. F.,On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming, SIAM Journal on Control, Vol. 5, No. 2, 1967.
Zoutendijk, G.,Methods of Feasible Directions, Elsevier, New York, New York, 1960.
Arioli, M., Laratta, A., andMenchi, O.,On the Computation of the Least Distance from a Polyhedral Convex Set, Istituto di Elaborazione dell'Informazione, Pisa, Italy, Report No. B80–29, 1980.
Golub, G. H.,Numerical Methods for Solving Linear Least Squares Problems, Numerische Mathematik, Vol. 7, pp. 206–216, 1965.
Lawson, C. L., andHanson, R. J.,Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1974.
Mifflin, R.,A Stable Method for Solving Certain Constrained Least Squares Problems, Mathematical Programming, Vol. 16, No. 2, 1979.
Schittkowski, K., andStoer, J.,A Factorization Method for the Solution of Constrained Linear Least Squares Problems Allowing Subsequent Data Changes, Numerische Mathematik, Vol. 31, pp. 431–463, 1979.
Stoer, J.,On the Numerical Solution of Constrained Least Squares Problems, SIAM Journal on Numerical Analysis, Vol. 8, No. 2, 1971.
Wilkinson, J. M.,The Algebraic Eigenvalue Problem, Oxford University Press, London, England, 1965.
Boot, J. G.,Quadratic Programming, North-Holland, Amsterdam, Holland, 1964.
Rosen, J. B., andSuzuki, S.,Construction of Nonlinear Programming Test Problems, Communications of ACM, Vol. 8, No. 2, 1965.
Chaney, K. W.,On the Rate of Convergence of Some Feasible Direction Algorithms, Journal of Optimization Theory and Applications, Vol. 20, No. 1, 1973.
Pironneau, O., andPolak, E.,Rate of Convergence of Methods of Feasible Directions, SIAM Journal on Numerical Analysis, Vol. 10, No. 1, 1973.
Ortega, J. A., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Peters, G., andWilkinson, J. H.,On the Stability of Gauss-Jordan Elimination with Pivoting, Communications of ACM, Vol. 13, No. 3, 1975.
This research was partially supported by the Progetto Finalizzato Informatica, CNR, Rome, Italy.
Communicated by I. Galligani
About this article
Cite this article
Arioli, M., Laratta, A. & Menchi, O. Numerical study of some feasible direction methods in mathematical programming. J Optim Theory Appl 40, 1–23 (1983). https://doi.org/10.1007/BF00934629
- Nonlinear programming
- feasible directions
- linear least squares
- Householder orthogonal factorization
- Gauss-Jordan factorization