An optimization problem must be expressed mathematically with the least number of variables or the mathematical formulation that allows to demonstrate its optimal solution, however it requires to increase the number of variables. How should it be expressed?
Rafael: It is not necessarily the case at all that the optimization model that you construct to represent your real problem should have, as you state, "the least number of variables". In fact, in combinatorial optimization - which is a rather large subset of the reality-based models that we want to solve - there are many successful approaches based on what is known as "column generation" methods. Such methods will simply - well, not that simply - enable fruitful variables in the original space of variables to be generated, and used to derive an optimal solution to the explicit optimization problem. Such methods have been involved in many successful applications during the last three or so decades. Just "google" the term and perhaps the problem type that you are interested in, and see if you get a hit! :-)
Dear Rafael Bernardo Carmona Benitez ,
Your question is very general.
Optimization includes many techniques varies from simple problems to difficult ones. This based on the nature of the problem.
General talking, if we have some objective function ( or multi-objective functions) that subject to several constraints. The constraints may be inequalities or equations.
To extremize the objective function, there are several techniques that provide the maximum(or the minimum) value, mainly calculus of variation are essential in this domain. Sometimes we add an artificial variable to simplify the solution as in the simplex methods. But this dependence on the nature of the problem.
If you have any specific question in optimization, show the model, this will help to give a more specific answer.
Best regards
The optimization problem should be formulated in such a way that you can solve this problem at least approximately. As for the detailed description of the model, it depends on your mathematical potential. Creating a model is not an end in itself, because the next step is the solution method. If you do not have this method, then the model must be reformulated so that the new version of the model could be a solvable method of solution that you know.
Rafael: It is not necessarily the case at all that the optimization model that you construct to represent your real problem should have, as you state, "the least number of variables". In fact, in combinatorial optimization - which is a rather large subset of the reality-based models that we want to solve - there are many successful approaches based on what is known as "column generation" methods. Such methods will simply - well, not that simply - enable fruitful variables in the original space of variables to be generated, and used to derive an optimal solution to the explicit optimization problem. Such methods have been involved in many successful applications during the last three or so decades. Just "google" the term and perhaps the problem type that you are interested in, and see if you get a hit! :-)
Thank you all for your reply. My problem is that I do not know if it is necessary to write two times the same model because I can solve the model, in fact I found the global optimum by adding one continous variable to the main model. What I did is, I calculate an integer variable with the division of two continuous variables, to do so, I add to the model an extra continous variable. Ii is possible because I can express the integer variables in terms of the relation of two continous variables, but I do not eliminate the integer variable, because I need to add a restriction that make sures the result of the division of the two continous variables is integer, then I do so with an extra contraint, so i do not eliminate the integer variable, but i took it out from the objective function, an from all contraints, except for one restriction, the restriction I just add to the mathematical formulation that makes sure the division of the two continous variable is an integer value.
What I am trying to know is if in the best practices, it is necessary to write the first model, or the model with integer variables in the objective function, and then write the model with the extra continuous variable (the model with an objective function without integer variables), is it necessary? or I just go direct and write the the model without the integer variables in the objective function and the extra constraints becuase it is the one I can solve or find the optimum.
I think from Gennady respond, I just have to write the model in the formulation I can solve.
Thanks all for your replay
First and foremost, your model represents the real problem correctly. Second, your decision variables to be optimized should be independent to each other. If possible, your model should not have redundant constraints.
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