I have an optimization problem where I want to minimize for the total cost of a system, so I write an objective function that is the sum of my different costs. The problem includes using one of three machines each one with different cost at a different threshold of usage. I define each machine (model.Machine#) as a binary variable and declare the parameters of each machine cost model.Cost#). I am trying to get the cost to be able to minimize it but when I write the constraint:
model.Cost1*model.Machine1 + model.Cost2*model.Machine2 + model.Cost3*model.Machine3 == model.MachineCost
Where I also write:
model.Machine1 + model.Machine2 + model.Machine3 == 1
Gurobi is telling me that it can't handle an quadratic function referring to the first constraint mentioned above. However it is parameters multiplied by binary variables there isn't anything quadratic.
I know the question is vague and part of a larger problem but I hope you can understand what I am referring to and help me!
Thank you so much for your assistance!
What is model.MachineCost? Is it an Expression component with some kind of quadratic expression stored inside of it?
If not, can you start commenting out things in your model until you get down to a minimal working example (that causes this error) and post that? Otherwise, we can't be sure that there are not other quadratic pieces of the model that you are not showing.
Related
This is a follow-up to a suggestion by #DCTLib in the post below.
Cudd_PrintMinterm, accessing the individual minterms in the sum of products
I've been pursuing part (b) of the suggestion and will share some pseudo-code in a separate post.
Meanwhile, in his part (b) suggestion, #DCTLib posted a link to https://github.com/VerifiableRobotics/slugs/blob/master/src/BFAbstractionLibrary/BFCudd.cpp. I've been trying to read this program. There is a recursive function in the classic Somenzi paper, Binary Decision Diagrams, which describes an algo to compute the number of satisfying assignments (below, Fig. 7). I've been trying to compare the two, slugs and Fig. 7. But having a hard time seeing any similarities. But then C is mostly inscrutable to me. Do you know if slugs BFCudd is based on Somenze fig 7, #DCTLib?
Thanks,
Gui
It's not exactly the same algorithm.
There are two main differences:
First, the "SatHowMany" function does not take a cube of variables to consider for counting. Rather, that function considers all variables. The fact that "recurse_getNofSatisfyingAssignments" supports cubes manifest in the function potentially returning NaN (not a number) if a variable is found in the BDD that does not appear in the cube. The rest of the differences seem to stem from this support.
Second, SatHowMany returns the number of satisfying assignments to all n variables for a node. This leads, for instance, to the division by 2 in line -4. "recurse_getNofSatisfyingAssignments" only returns the number of assignments for the remaining variables to be considered.
Both algorithms cache information - in "SatHowMany", it's called a table, in "recurse_getNofSatisfyingAssignments" it's called a buffer. Note that in line 24 of "recurse_getNofSatisfyingAssignments", there is a constant string thrown. This means that either the function does not work, or the code is never reached. Most likely it's the latter.
Function "SatHowMany" seems to assume that it gets a BDD node - it cannot be a pointer to a complemented BDD node. Function "recurse_getNofSatisfyingAssignments" works correctly with complemented nodes, as a DdNode* may store a pointer to a complemented node.
Due to the support for cubes, "recurse_getNofSatisfyingAssignments" supports flexible variable ordering (hence the lookup of "cuddI" which denotes for a variable where it is in the current BDD variable ordering). For function SatHowMany, the variable ordering does not make a difference.
Suppose we're given some sort of graph where the feasible region of our optimization problem is given. For example: here is an image
How would I go on about constructing these constraints in an integer optimization problem? Anyone got any tips? Thanks!
Mate, I agree with the others that you should be a little more specific than that paint-ish picture ;). In particular you are neither specifying any objective/objective direction nor are you giving any context, what about this graph should be integer-variable related, except for the existence of disjunctive feasible sets, which may be modeled by MIP-techniques. It seems like your problem is formalization of what you conceptualized. However, in case you are just being lazy and are just interested in modelling disjunctive regions, you should be looking into disjunctive programming techniques, such as "big-M" (Note: big-M reformulations can be problematic). You should be aiming at some convex-hull reformulation if you can attain one (fairly easily).
Back to your picture, it is quite clear that you have a problem in two real dimensions (let's say in R^2), where the constraints bounding the feasible set are linear (the lines making up the feasible polygons).
So you know that you have two dimensions and need two real continuous variables, say x[1] and x[2], to formulate each of your linear constraints (a[i,1]*x[1]+a[i,2]<=rhs[i] for some index i corresponding to the number of lines in your graph). Additionally your variables seem to be constrained to the first orthant so x[1]>=0 and x[2]>=0 should hold. Now, to add disjunctions you want some constraints that only hold when a certain condition is true. Therefore, you can add two binary decision variables, say y[1],y[2] and an additional constraint y[1]+y[2]=1, to tell that only one set of constraints can be active at the same time. You should be able to implement this with the help of big-M by reformulating the constraints as follows:
If you bound things from above with your line:
a[i,1]*x[1]+a[i,2]-rhs[i]<=M*(1-y[1]) if i corresponds to the one polygon,
a[i,1]*x[1]+a[i,2]-rhs[i]<=M*(1-y[2]) if i corresponds to the other polygon,
and if your line bounds things from below:
-M*(1-y[1])<=-a[i,1]*x[1]-a[i,2]+rhs[i] if i corresponds to the one polygon,
-M*(1-y[1])<=-a[i,1]*x[1]-a[i,2]+rhs[i] if i corresponds to the other polygon.
It is important that M is sufficiently large, but not too large to cause numerical issues.
That being said, I am by no means an expert on these disjunctive programming techniques, so feel free to chime in, add corrections or make things clearer.
Also, a more elaborate question typically yields more elaborate and satisfying answers ;) If you had gone to the effort of making up a true small example problem you likely would have gotten a full formulation of your problem or even an executable piece of code in no time.
I need to repeatedly compute derivatives at a given point of nonlinear pyomo constraints. According to this post, there are basically two options: one symbolic approach (which uses sympy) and one numeric approach via the NL file generated by Pyomo.
I have tried the symbolic approach which looks like:
def gradient(constr, variables):
grad_num = np.array([value(partial) for partial in
differentiate(constr.body, wrt_list=variables)])
if (not (is_leq_constr(constr))):
grad_num = -grad_num
return grad_num
In principle, this approach works. However, when dealing with larger problems, the function call takes very long and I would expect the numeric approach to be much faster.
Is this true? If so, can anyone help me, to "translate" the above function to the gjh_asl_json approach? I am grateful for any hint.
Is there a function in Julia that is similar to the solver function in Excel where I can provide and equation, and it will solve for the unknown variable? If not, does anybody know the math behind Excel's solver function?
I am not expecting anybody to solve the equation, but if it helps:
Price = (Earnings_1/(1+r)^1)+(Earnings_2/(1+r)^2)++(Earnings_3/(1+r)^3)+(Earnings_4/(1+r)^4)+(Earnings_5/(1+r)^5)+(((Earnings_5)(RiskFreeRate))/((1+r)^5)(1-RiskFreeRate))
The known variables are: Price, All Earnings, and RiskFreeRate. I am just trying to figure out how to solve for r.
Write this instead as an expression f(r) = 0 by subtracting Price over to the other side. Now it's a rootfinding problem. If you only have one variable you're solving for (looks to be the case), then Roots.jl is a good choice.
fzero(f, a::Real, b::Real)
will search for a solution between a and b for example, and the docs have more choices for algorithms when you don't know a range to start with and only give an initial condition for example.
In addition, KINSOL in Sundials.jl is good when you know you're starting close to a multidimensional root. For multidimensional and needing some robustness to the initial condition, I'd recommend using NLsolve.jl.
There's nothing out of the box no. Root finding is a science in itself.
Luckily for you, your function has an analytic first derivative with respect to r. That means that you can use Newton Raphson, which will be extremely stable for your function.
I'm sure you're aware your function behaves badly around r = -1.
I am having some strange problems with excel's solver. Basically what I am trying to do is curve fit my data. I have two different lines, one is my calibration line and the other is the derived line that I am attempting to match up to the calibration line. My line depends on 19 different variable parameters (Perhaps this is too many? I have tried fewer without result) and I am using solver to adjust these parameters to make the two lines as close as possible.
For Example:
The QP column contains the variables I would like changed, changing these will draw me closer or further from the calibration curve. Each subsequent value of QP must be greater than the first.
Col=B Col=C
Power .QP_'
1 ..... 57000
2 ..... 65000
3 ..... 70000
4 ..... 80000
5 ..... 80000
Therefore my excel solver parameters look like this: C1:C19>=0,C1:C19<=100000 and C2>=C1, C3>=C2,C4>=C3... I have also tried making another column of the differences between each value and then saying that these must be diff>=0.
To compare this with my calibration curve I have taken the calibration curve data and subtracted my data derived from QP and then squared that to create my sum of the squares error. For example:
(Calibration-DerivedQP)^2=SS(x) <- where x represents the row number
Sum(SS(x))=SSE
SSE is what I have set solver to minimize. And upon changing QP everything automatically updates. There are no if statements being used and no pivot tables are used.
If I remove the parameters similar to C2>=C1 everything works perfectly, except the derived values are not feasible. But when the solver is run with these parameters, nothing gets changed and no matter which guesses I used as starting values ( so that I can ensure I haven't guessed a local minimum), the solver cannot improve upon my solution. This has led me to believe that something in my parameters is being broken, since I can very easily improve on my solution by guess and check. The rest of solvers settings are at the defaults, and the evolutionary method is used since my curve isn't smooth (I don't think) I had this working in the past and now something seems to be broken. Any ideas are appreciated! Thank you so much! Sorry if I am missing any critical information. I am also familiar with matlab and R if there are better methods in those languages.
I found the solution to my problem. I don't know if this will be helpful to anyone else since my problem vague and pretty specific to me. That being said, my problem was in the constraints. I changed some data on my excel sheet to allow for fewer constraints. An example might look like this:
Guess..........Squared......Added..................Q
-12..............(-12)^2....... 0
-16..............(-16)^2.......=(-16)^2+0.............256
+7.................(7)^2..........=(7)^2+(-16)^2+0....305
Now I allow solver to guess any number subject to minimal constraints.
Essentially, what is happening now, is the excel sheet allows for any guess that solver makes to work. By squaring the numbers it give me positive values, and the added column ensures that each successive value is equal to or greater than the first. This means there are very few constraints. I also changed the solver option from evolutionary to GRG Nonlinear.
Tips for getting solver to work:
Try and use the spreadsheet to set constraints (other than bounds, bounds seem to be good) wherever possible, the more constraints that I set in solver, the less likely my solution was to work.
Hope that helps, sorry if I have provided any incorrect information.