How can I solve or re-write an optimization problem with equality constraints, some integer variables and a non-linear objective function?

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Dredging Production
Dredging Production 2018-12-19
编辑： Walter Roberson 2019-1-13
Hi,
I'm trying to optimize the following (very simplified) problem:
Non_linear_objective = 500 * (x(1)+x(2)+x(3)+x(4)) + 200 * (x(5)+x(6)+x(7)+x(8)) + 95 * ( x(9)-1)^2 + (x10-x9)^2 + (x11-x10)^2 + (x12-x11)^2 + (1-x12)^2 );
5 equality constraints:
x(1)+x(5)+x(9) = 1
x(2)+x(6)+x(10) = 1
x(3)+x(7)+x(11) = 1
x(4)+x(8)+x(12) = 1
40 *( x(1)+x(2)+x(3)+x(4)) = 80
4 integer variables: x(9), x(10),x(11),x(12) (which actually means that x(1)+x(5) (etc) is binary)
Why I'm currently not able to solve this:
  1. fmincon does not support integer variables
  2. I can rewrite the problem to a linear objective, with non-linear constraints, and 4 integer variables. But intlinprog does not support non-linear constraints
  3. I can use the Genetic Algorithm, but this finds a local mimum only. I have to solve this simple problem >300 times, to find the correct answer. (I'm using lb and ub of 0 and 1 for all variables)
Is my reasoning correct?
How could I solve these kind of problems?
Thank you!
  1 个评论
Torsten
Torsten 2018-12-19
编辑：Torsten 2018-12-19
Are there positivity constraints on the x(i), i.e. x >= 0 e.g. ?
If this is the case, x(9)-x(12) can only have values 0 or 1. There are 2^4 = 16 combinations for x(9)-x(12) with 0 or 1 for the variables. So just solve your problem by running it 16 times using "quadprog" (or "fmincon").
Best wishes
Torsten.

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Dredging Production
Dredging Production 2018-12-19
编辑：Dredging Production 2018-12-19
Thank you. Note sure whether I can use that software, but I will have a look.
I'm thinking about another solution: Creating the input (selection of integer variables) in such a way that the optimum solution can be found without calculating all the combinations. So, somehow the 'input creator' should learn from the output (machine learning). I have some ideas on how to do this, but I have never set-up something like this before.
Do you perhaps have somes ideas about this, or an example in which this is used ?
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Torsten
Torsten 2019-1-7
If this task is important for you, I'd consider licencing "CPLEX". It's the best software package you can get in the field of optimization, I guess.
Dredging Production
Dredging Production 2019-1-13
编辑：Dredging Production 2019-1-13
Great, had a quick look seems robust, fast and organized. Will have a proper look later.
Btw an update: I slightly rewrote the problem and it solves the problem much quicker now. I'm using the MOSEK solver. BMIBNB works as well, but a lot slower.
Will try to increase the size of my problem now, and see whether it works.
Thank you for all your help!

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更多回答（1 个）

Dredging Production
Dredging Production 2018-12-19
编辑：Dredging Production 2018-12-19
Thank you for the quick reply!
All x are >= 0 and <=1.
I understand that solving 16 problems, is a solution for this small problem (thanks). But actually this problem is quite big in reality. For instance, there are actually 48 instead of 4 integer variables (and some more other constraints). This would mean I need to solve the problem way too many times.
Is there a way to avoid solving multiple problems ( = preferable!), or at least to decrease the amount of problems to be solved?

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