Hi,
I have a little knowledge about optimization and I have a simple question. If my convex optimization problem has an inequality constraint that ensures that the optimization variable can't be less than zero and greater than a maximum value and I have an iterative algorithm to solve this problem. My question is, can I handle this constraint with programming by forcing the value of the variable to not be less than zero and larger than the maximum value in each iteration (with the min() and max() functions for example)? or I must handle this constraint by two Lagrangian multipliers?
Dear Ali Zaher Assaygh,
I suggest you to see links and attached files on topic.
https://www.sciencedirect.com/topics/mathematics/inequality-constraint
https://www.epfl.ch/labs/la/wp-content/uploads/2018/08/ic-32_lectures-1-8.pdf
Article Linear Programming with Inequality Constraints Via Entropic ...
Article A Novel Method to Handle Inequality Constraints for Convex P...
Article A new equivalent transformation for interval inequality cons...
Best regards
Dear Ali
You don't specify if you are dealing with lineal or non-lineal programming.
If it is lineal programming, and as I understand your question, you have a constraint that must be larger than zero and less that a certain value.
If this is the case, yes, you can do that, at least in SIMUS, by simply creating two identical criteria, one affected by > for expressing that it has to be larger than 0, and the other con < to indicate that it has to be less than a certain value. I do this all the time, and usually the criterion takes an intermediate value between the 0 and the established value.
If your problem has no more that two alternatives, you can even do this graphically, albeit with only one line.
Look up "gradient projection method" - which is very simple to implement if you have a feasible set bounded by linear constraints. Moreover, when doing line searches you have an very easy way to control that your iteration points are feasible.
Based on "has an inequality constraint that ensures that the optimization variable can't be less than zero and greater than a maximum value," as I understand it, you are talking about simple lower and upper bounds. I don't understand how min() and max() functions will come into play. Yes, you have to force your variables to be between them. Using Lagrange multipliers will not guarantee that those constraints will be satisfied.
Practically every LP or NLP solver can handle simple bounds - have a look on the file interfaces LP or MPS, where you will find a BOUNDS section.
There you can specify a lower (lb) and/or upper bound (ub) for some or all variables, forcing the solver to maintain (at least up to numerical accuracy)
lb
Dear Ralf
You are right, however yo say that you need to modify the algorithm, something that I don 't discuss
What I proposed Ali, does not involve any modification of the Simplex.
It only involves replication a criterion twice, one asking for minimization for the lower bound, and the other for maximization for the upper bound. As you know, this creates a dimensional space where the two aspects may coexist
I am afraid that I don't concur with your last sentence. Certainly you have 0 and infinite default values , and what you do is to establish new extreme frontiers, using the greater or equal than, and the lower or equal than symbols
Dear Cenk
I refer to your fist sentence.
To express that an inequation must be larger than zero and greater than a certain value, you only need one expression, just indicate the second by the > symbol. Remember that in LP the default is that results can't be negative or less than zero, therefore, you don't need to establish it, unless you want
However, in so doing you are considering that the upper bound is infinite. If as probably, you need to establish a feasible upper bound, you need another inequation.
My guess is that Ali's question is not to how to enter lower and/or upper bounds on variables in a ready-to-use solver, but how to include them into a program created by him.
"I have an iterative algorithm to solve this problem. My question is, can I handle this constraint with programming by forcing the value of the variable to not be less than zero and larger than the maximum value in each iteration (with the min() and max() functions for example)?"
The word "programming" probaly stands here for some programming language, not linear/nonlinear programming and min()/max() for built-in functions of this language.
Dear Przenyslaw
If, as you guess, our friend Ali has his own program or method, and depending on its nature, it is possible, if he is using inequations, to use twice the same criterion: One maximizing () that the minimum value mots be higher than the lower bound
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