Hi,
I would like to know that whether it is possible to solve the optimization problems including continuous integrals using typical optimization software or not?
See the following example, please. Can I solve it using typical optimization software such as GAMS, Lingo, Lindo, AIMMS, AMPL, etc.?
Any help would be appreciated.
Morteza
http://mortezash.blogfa.com/
The analysis of the goal function z shows that it has two variables – the matrix (w [i, t]) and the matrix ( y [i, t] ). If this is so, it is possible to calculate continuous integrals previously as their sub-integral functions don't depend on these variables. Besides, it is possible to transform previously these continuous integrals, using the formula: Integral (k ** (-g) dk) = k ** (1-g)/(1-g) + C. Further it is possible to perform optimization, disregarding integrals.
you can do this . Even if your optimization code works on continuous values, you may put a command like 'round' (Matlab) before the variables.
I suggest you use discretization (say, using orthognal collocation) to transform the problem into a nonlinear optimization problem. Then you can use either GAMS or Matlab to solve the problem.
Hi,
@Igor: Many thanks for your answer. You are right. I mean if one of the bounds of integral is a variable and the integral cannot be solved using typical mathematical methods, how can we solve such optimization problems with continuous integrals?
@Ms Maroufmashat: Many thanks for your answer. Is it really possible to solve optimization problems using Maple or Mathematica? I did not know that! How?
@Ujjwal: Many thanks for your answer. I did not understand what you mean about using 'round()' function in this optimization problem. Could you please elaborate your answer?
@Abebe: Thank you.
Regards,
Morteza
@Morteza: using 'round' will make your variables as the nearest integer.
@Ujjwal: As far as I know, in optimization software such as GAMS, using a variable as an argument for the function 'round()' makes the problem nonlinear or even it may not be permitted in some solvers.
I guess this can be viewed as a nonlinear program (NLP), but one that happens to have a rather nasty objective function. Many of the algorithms for NLP (such as penalty-barrier methods) only require the user to provide a function that computes the gradient and Hessian at a given point. So, if you can find a closed formula for the gradient and Hessian, or at least write a function that computes them via numerical integration, then you should be okay.
It's not clear to me whether the NLP is convex or not. If it is convex, then you should be able to find a global optimum. Otherwise, you may have to be content with just a local optimum.
Dear Professor Letchford,
I am of exactly the same opinion. To my mind and as mentioned above, an appropriate way to deal with such problems could be 'numerical approximation' of the integral (discretization) and using SUM() function instead of integral. In this way, the problem can be simply optimized by GAMS or other optimization tools.
Kindest regards,
Morteza Shabanzadeh
http://mortezash.blogfa.com/
@Ujjwal: I don't think he is referring to variables that have to take integer values. Rather, he seems to be referring to the presence of the integration symbol in the objective function.
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