I have a question about obtaining the sign of dual variables in the dual model of the set partitioning problem. As I know, when there is an equality constraint the dual variable corresponding to this constraint should be no restricted. Although, I found in an article a different interpretation, as shown by the following:
Primal model
Dual model
I'm wondering why dual variables are restricted to be nonnegative?
Thanks a much for any explanations.
Best regards
Related
Currently I'm working through Applied Linear Models $5^{th}$ ed - by Kutner, et al. A question I'm working on is asking me to perform an F-Test for Lack of Fit on my linear model. The linear model is a simple linear model of one parameter nothing too troublesome.
To perform the test one has to assess the difference between the full model and the reduced model. At this current junction the authors have stated to take the full model as $\hat{\mu_{j}} = \bar{Y_{j}}$. Specifically the screenshot below says the following:
The reduced model would be the simple linear model:
I have no problem being able to do this manually within R, by computing the necessary values where need be as I've done for other questions. But I'm trying to improve my R skill set and this is where my problem lies.
I have done some reading to other answers related to this and model comparison can be done directly in the anova() function. But I'm having issues stating my full model correctly to be able to leverage the anova() function. I thought about computing a "vector of means" for the subgroups of data (which I display here just for completeness)
But I'm going to run into the problem of the anova() function most likely not being able to compute the degrees of freedom correctly. My data set is very small and this seems like the sort of situation that would show up all the time. With huge data sets I wouldn't see it being feasible to compute things manually so surely there has to be a way for me to phrase my Full Model properly to allow for the computation of means from the subgroups of replicates. But how do I do so? is the question of the day.
For completeness and posterity an answer was given on a sister site I asked this question on:
https://stats.stackexchange.com/questions/539958/how-to-specify-a-vector-full-of-means-with-degrees-of-freedom-for-a-lack-of-fit
the mods can delete the question if they deem fit and doesn't contribute to the community.
Looking at Gurobi's expamples for programs, there is one for QCPs, and one for bilinear programs, and I was wondering how to add a constraint that is linear and bilinear (sorry if there's specific jargon for such a problem) in R (or any other language, if easier, but I am using R). Specifically, how would I add a matrix of constraints of the form (for example) that
xz + y - yz < c
where c is some constant. I think I could use mccormick relaxation to re-write this as a linear program (right?), but I was wondering if Gurobi has easy syntax for such constraints?
My current understanding of the syntax for QCPs and bilinear programs is that you use a sparse matrix construction of the form
And so you cannot refer to x,y,z on their own..
Figured it out. In case anyone else comes across a similar issue, you create a quadcon list and add it to the model, as described here. For an illustration of using quadcon, it is quite similar to quadratic constraints in this example, though this example is not explicitly of the type of constraint I asked about.
The following linear programming problem is not of canonical form. I am really stuck when trying to put this in regular form and feed it into the normal lp() function.
Does someone has experience with such weird form?
B and A are the blocker and antiblocker, respectively, which are simply two sets of inequalities.
I don't know what the "normal lp() function" is. Let's assume this is the lp function from the LpSolve package.
This function does not expect a canonical form. (Canonical usually means each constraint has the same fixed sign, e.g. Ax<=b; lp() allows different signs for each constraint).
lp() just wants one big constraint matrix: each column is an individual variable and each row is an individual constraint. This is conceptual simple, but often tedious in practice. Best thing to do is to get a large piece of paper and draw the layout of the LP matrix: which variables and constraints go where.
For some classes of models there are easier-to-use tools to express an LP model, such as OMPR, CVXR.
I am working on a problem which has two linear constraints including one equality constraint. I am using SPEA2 algorithm. The constraints are given below.
I have tried penalty function approach but had difficulty in selecting parameters. Secondly, I have used constrained dominance relation approach but again could not get feasible solutions. Please advise...
It is mentioned here that z3 uses Dual Simplex not Revised Simplex, which would mean that as of now, all rows and columns of the original (fixed) tableaux get accessed during every pivoting operation. Am I then correct to infer that implementing Revised Simplex in z3 would substantially improve performance for problems where the tableaux happens to be large?
Z3's implementation of Simplex uses Dual simplex and was tuned towards backtracking search in conjunction within a DPLL(T) framework. Other approaches include the specialized solvers
for difference logic, unit-two variable per inequality logic,
trying revised simplex (within DPLL(T) as you suggest or some other approach) or
procedures of the kind: "Dejan Jovanovic, Leonardo Mendonça de Moura: Cutting to the Chase -
Solving Linear Integer Arithmetic. J. Autom. Reasoning 51(1): 79-108 (2013)", gradient methods, interior point methods, etc.
There are many factors that play into what works more efficiently given an application area
depending on the branching ratio vs. number of variables and inequalities in a problem.