If I'm not wrong, there can be non linear programming problems solved by iteratively calling the simplex algorithm on a modified sub-problem.
Are there optimization problems which require both the primal and dual simplex?
Excellent question Marco!
I don't know about non-linear problems, but remember that the dual in linear problems gives the shadow prices that are normally valid within an acceptable variations of the criteria,
If the problem is slightly non-linear, the Simplex does not give you the shadow prices but the Lagrange multiplier which is valid for only one point of contact between the polygon and the objective convex curve.
Your question really interest me because the SIMUS method uses simultaneously the direct and the dual results for sensitivity analysis in this way:
1- The primal problem identifies the best alternative from a Pareto efficient matrix, developed by successive uses of the Simplex
2-This matrix shows the objectives/criteria that are satisfied by the best alternative
3 - With this data the dual problem identifies which are the criteria that form that solution and gives the shadow prices for each one, as well as the range of validity of each one
4- Considering that the shadow prices are also marginal utilities, they are used for sensitivity analysis in lieu of the artificial generated criteria 'weights'
5 - Varying these selected criteria, a part of SIMUS called IOSA (Input-output Sensitivity Analysis), draws the total utility curve for each objective
Most probable there are other applications for using the primal and the dual simultaneously, but I am not aware of that
If you are interested in SIMUS/IOSA, I can send you information if you give me your email. Mine is: [email protected]
Please check this:
http://mat.uab.cat/~alseda/MasterOpt/310trialtext.pdf
Dear Ahmed
Yes, remember that every LP problem gives the Primal and the Dual.
The latter is the inverse of the former
If you have solved one of the primal or dual LP problems, you can derive the solution to the other without having solved another problem - IF the problems are not degenerate - and that seldom appears in practice. But when they do occur, you have a simple solution to that bug also.
Dear Noha A. Mostafa , thank you for the link. It is an excellent book, indeed. Anyway, can you be more precise on whith chapter, at least, it is shown a situation where both the primal solver and the dual solver are used in combination? Thank you again
Dear Marco
I concur that the book suggested by Noah is excellent and very thorough. I have perused it and find extremelly useful, and one of the most complete works in Linear Programming
However, I don't remember seeing an example of primal and dual used jointly, but probably I am mistaken.
I can show you that symbiosis in a book of mine published in 2019, which is the base for the analysis of a solution obtained by the primal, and also the foundation for performing sensitivity analysis based on marginal values instead of arbitrary weights
Ok prof. Nolberto Munier , please find me some material here: [email protected].
Thank you so much!
MC
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