In this paper, a modified version of the Classical
Lagrange Multiplier method is developed for convex quadratic optimization
problems. The method, which is evolved from the first order derivative test for
optimality of the Lagrangian function with respect to the primary variables of
the problem, decomposes the solution process into two independent ones, in
which the primary variables are solved for independently, and then the
secondary variables, which are the Lagrange multipliers, are solved for,
afterward. This is an innovation that leads to solving independently two
simpler systems of equations involving the primary variables only, on one hand,
and the secondary ones on the other. Solutions obtained for small sized
problems (as preliminary test of the method) demonstrate that the new method is
generally effective in producing the required solutions.
References
[1]
Luenburger, D.G. and Ye, Y. (2008) Dual and Cutting Plane Methods. Linear and Nonlinear Programming, Vol. 116. Springer, New York, 435-468. https://doi.org/10.1007/978-0-387-74503-9_14
[2]
Koopman, T.C. (2003) Activity Analysis of Production Allocation. Wiley & Chapman-Hall, New York, 339-347.
[3]
Bazaraa, M.S., Sherali, H.D. and Shetty, C.M. (2006) Nonlinear Programming, Theory & Algorithms. John Wiley & Sons Inc., New York. https://doi.org/10.1002/0471787779
[4]
Boyd, S. and Vandenberghe, L. (2004) Convex Optimization. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511804441
[5]
Nocedal, J. and Wright, S.J. (2006) Quadratic Programming. Numerical Optimization, Springer, New York, 448-492.
[6]
Shustrova, A. (2015) Primal-Dual Interior Point Methods for Quadratic Programming. Doctor’s Thesis, University of California, San Diego.
[7]
Hoffman, L.D. and Bradley, G.L. (1992) Calculus for Business, Economics, and the Social and Life Sciences. McGraw-Hill, New York.
[8]
Beavis, B. and Dobbs, I. (1990) “Static Optimization” Optimization and Stability Theory for Economic Analysis. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511559402
[9]
Hillier, F.S. and Price, C.C. (2012) International Series in Operations Research & Management Science. Springer, Berlin/Heidelberg
[10]
Jakovetic, D., Xavier, J., and Moure, J.M. (2014) Fast Distributed Gradient Methods. IEEE Transaction on Automatic Control, 59, 1131-1146. https://doi.org/10.1109/TAC.2014.2298712
[11]
Gao, X., Hao, Y.Z., Wang, Y., Zuo, X., and Chen, T. (2021) A Lagrange Relaxation-Based Decomposition Algorithm for Large-Scale Off Shore Oil Production Planning Optimization. Processes, 9, Article 1257. https://doi.org/10.3390/pr9071257
[12]
Curtis, F.E., Jiang, H. and Robinson, D.P. (2015) An Adaptive Augmented Lagrangian Method for Large-Scale Problems. Mathematical Programming Series A, 152, 201-245. https://doi.org/10.1007/s10107-014-0784-y
[13]
Liang, X., Hu, J., Zhong, W. and Qian, J. (2008) A Modified Augmented Lagrange Multiplier Method for Large Scale Optimization. Developments in Chemical Engineering and Mineral Processing, 9, 115-124. https://doi.org/10.1002/apj.5500090214
[14]
Burman, E., Hausbo, P. and Larson, M.G. (2023) The Augmented Lagrange Method as a Framework for Stabilized Methods in Computational Mechanics. Archives of Computational Methods in Engineering, 30, 2579-2604. https://doi.org/10.1007/s11831-022-09878-6
Fortin, M. and Glovinski, R. (1983) Augmented Lagrangian Methods: Numerical Solution of Boundary-Value Problems. Elsevier Science Problems, New York.
[17]
Komzsik, L. and Chaing, K. (1993) The Effects of a Lagrange Multiplier Approach in MSC/MASTIZAN on Large-Scale Parallel Application. Computing Systems in Engineering, 4, 399-403. https://doi.org/10.1016/0956-0521(93)90008-K
[18]
Schittkowski, K. (2012) More Test Examples for Nonlinear Programming Codes. Vol. 282, Springer Science & Business Media, Berlin.
