Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization
- 140 Downloads
Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant.
For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: “what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?”. The differences of the results between the linear and convex quadratic optimization problems are highlighted too.
KeywordsParametric optimization Sensitivity analysis Linear optimization Optimal partitions
Unable to display preview. Download preview PDF.
- 1.Goldman, A.J., Tucker, A.W.: Theory of linear programming. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear Inequalities and Related Systems. Annals of Mathematical Studies, vol. 38, pp. 63–97. Princeton University Press, Princeton (1956) Google Scholar
- 5.Jarvis, J.J., Sherali, H.D., Bazaraa, M.S. (eds.): Linear Programming and Network Flows. Wiley, New York (1997) Google Scholar
- 10.Jansen, B., Roos, C., Terlaky, T.: An interior point approach to postoptimal and parametric analysis in linear programming. Report No. 92-21, Faculty of Technical Mathematics and Information/Computer science, Delft University of Technology (1992) Google Scholar
- 13.Ghaffari Hadigheh, A.R., Romanko, O., Terlaky, T.: Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors. Algorithmic Oper. Res. (2007, to appear) Google Scholar
- 17.Ghaffari Hadigheh, A.R., Mirnia, K., Terlaky, T.: Sensitivity analysis in linear and convex quadratic optimization: invariant active constraint set and invariant set intervals. Inf. Syst. Oper. Res. (INFOR) 44(2), 129–155 (2006) Google Scholar
- 19.Berkelaar, A.B., Jansen, B., Roos, C., Terlaky, T.: An interior point approach to parametric convex quadratic programming. Working paper, Erasmus University Rotterdam, Rotterdam, Netherlands (1997) Google Scholar
- 20.Nesterov, Y., Nemirovskii, A. (eds.): Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (1994) Google Scholar