# Computational schemes for large-scale problems in extended linear-quadratic programming

- 104 Downloads
- 44 Citations

## Abstract

Numerical approaches are developed for solving large-scale problems of extended linear-quadratic programming that exhibit Lagrangian separability in both primal and dual variables simultaneously. Such problems are kin to large-scale linear complementarity models as derived from applications of variational inequalities, and they arise from general models in multistage stochastic programming and discrete-time optimal control. Because their objective functions are merely piecewise linear-quadratic, due to the presence of penalty terms, they do not fit a conventional quadratic programming framework. They have potentially advantageous features, however, which so far have not been exploited in solution procedures. These features are laid out and analyzed for their computational potential. In particular, a new class of algorithms, called finite-envelope methods, is described that does take advantage of the structure. Such methods reduce the solution of a high-dimensional extended linear-quadratic program to that of a sequence of low-dimensional ordinary quadratic programs.

## Key words

Extended linear-quadratic programming large-scale numerical optimization finite-envelope methods stochastic programming nonsmooth optimization linear complementarity variational inequalities## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R.W. Cottle, F. Giannessi and J.-L. Lions, eds.,
*Variational Inequalities and Complementarity Problems*(Wiley, Chichester, 1980).Google Scholar - [2]D. Goldfarb and S. Liu, “An O(
*n*^{3}*L*) primal interior point algorithm for convex quadratic programming,” preprint.Google Scholar - [3]A. King, “An implementation of the Lagrangian finite generation method,” in: Y. Ermoliev and R.J.-B. Wets, eds.,
*Numerical Techniques for Stochastic Programming Problems*(Springer, Berlin, 1988).Google Scholar - [4]K.C. Kiwiel, Methods of Descent for Nondifferentiable Optimization,
*Lecture Notes in Mathematics, Vol. 1133*(Springer, Berlin, 1985).Google Scholar - [5]Y.Y. Lin and J.-S. Pang, “Iterative methods for large convex quadratic programs: a survey,”
*SIAM Journal on Control and Optimization*25 (1987) 383–411.Google Scholar - [6]R.C. Monteiro and I. Adler, “An O(
*n*^{3}*L*) interior point algorithm for convex quadratic programming,” preprint.Google Scholar - [7]W. Oettli, “Decomposition schemes for finding saddle points of quasi-convex-concave functions,” preprint.Google Scholar
- [8]J.-S. Pang, “Methods for quadratic programming: a survey,”
*Computers and Chemical Engineering*7 (1983) 583–594.Google Scholar - [9]R.T. Rockafellar, “Linear-quadratic programming and optimal control,”
*SIAM Journal on Control and Optimization*25 (1987) 781–814.Google Scholar - [10]R.T. Rockafellar and R.J.-B. Wets, “A Lagrangian finite-generation technique for solving linear-quadratic problems in stochastic programming,”
*Mathematical Programming Studies*28 (1986) 63–93.Google Scholar - [11]R.T. Rockafellar and R.J.-B. Wets, “Linear-quadratic problems with stochatic penalties: the finite generation algorithm,” in: V.I. Arkin, A. Shiraev and R.J.-B. Wets, eds.,
*Stochastic Optimization, Lecture Notes in Control and Information Sciences, 81*(Springer, Berlin, 1987) pp. 545–560.Google Scholar - [12]R.T. Rockafellar and R.J.-B. Wets, “Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time,”
*SIAM Journal on Control and Optimization*, to appear.Google Scholar - [13]R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”
*SIAM Journal on Control and Optimization*14 (1976) 877–898.Google Scholar - [14]R.T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming,”
*Mathematics of Operations Research*1 (1976) 97–116.Google Scholar - [15]R.T. Rockafellar, “Lagrange multipliers and variational inequalities,” in: [1] pp. 303–322.Google Scholar
- [16]R.T. Rockafellar,
*Convex Analysis*(Princeton University Press, Princeton, NJ, 1970).Google Scholar - [17]J.M. Wagner, “Stochastic programming with recourse applied to groundwater quality management,” Doctoral Dissertation, MIT (Cambridge, MA, 1988).Google Scholar
- [18]Y. Ye and E. Tse, “A polynomial-time algorithm for convex quadratic programming,” preprint.Google Scholar
- [19]C. Zhu and R.T. Rockafellar, “Finite-envelope gradient projection methods for extended linear-quadratic programming,” preprint.Google Scholar