What algorithm is used by the matlab function linopt::minimize ? I have to write the source code for solving a Mixed Integer Linear Programming problem. Please tell me the algorithms I can work upon.
It is not specified in the documentation of Mathlab (c.f. http://www.mathworks.fr/fr/help/symbolic/mupad_ref/linopt-minimize.html). However, they do provide some references at the end of the page (you should have a look there).
My guesses are the following (based on my humble knowledge regarding state-of-the-art methods for solving LP/MILP) :
1-If the problem is not MILP, we usually use the simplex algorithm (http://en.wikipedia.org/wiki/Simplex_algorithm).
2-If the problem is MILP (i.e. there is some integer variables), one usually use a Branch and Bound algorithm in which we might use simplex for improving the search (we call it Branch and Cut)
P.S. This is not an exhaustive response since other methods can be used but the above are the most known ones.
Related
I am very new to MatLab. Thus I am sorry if this is very basic.
I use a function called fmincon to do find a solution for minimizing a function. Why do I get different solutions for running fmincon?
I would like to know a satisfying or convincing mathematical or programming explanation for having different solutions using fmincon.
Check these limitations in the MATLAB documentation.
fmincon is a gradient-based method that is designed to work on problems where the objective and constraint functions are both continuous and have continuous first derivatives.
The function is very delicate and it is best if you can avoid it. It only works neatly on problems that are neatly defined to begin with. Any deviation can lead to local instead of global minima, and these can depend (among other things) on your initial solution estimate or starting point.
As fmincon is sensitive to initial point, If you set different start point for the fmincon, you might get a different solution in each apply. You can find one of the algorithms of fmincon here.
The Matlab function interpn does n-grid interpolation. According to the documentation page:
In a future release, interpn will not accept mixed combinations of row and column vectors for the sample and query grids.
This page provides a bit more information but is still kind of cryptic.
My question is this: Why is this modification being implemented? In particular, are there any pitfalls to using interpn?
I am writing a program in fortran that is supposed to produce similar results to a Matlab program that uses interpn as a crucial component. I'm wondering if the Matlab program might have a problem that is related to this modification.
No, I don't think this indicates that there is any sort of problem with using interpn, or any of the other MATLAB interpolation functions.
Over the last few releases MathWorks has been introducing some new/better functionality for interpolation (for example the griddedInterpolant, scatteredInterpolant and delaunayTriangulation classes). This has been going on in small steps since R2009a, when they replaced the underlying QHULL libraries for computational geometry with CGAL.
It seems likely to me that interpn has for a long time supported an unusual form of input arguments (i.e. mixed row and column vectors to define the sample grid) that is probably a bit confusing for people, hardly ever used, and a bit of a pain for MathWorks to support. So as they move forward with the newer functionality, they're just taking the opportunity to simplify some of the syntaxes supported: it doesn't mean that there is any problem with interpn.
I'm quite surprised not to find one in standard library. Is there some reason it is missing or I just need to use a specific toolbox?
Implementing it myself would be very problematic because of the complexity of algorithm involved.
Are there any faster and more efficient solvers other than fmincon? I'm using fmincon for a specific problem and I run out of memory for modest sized vector variable. I don't have any supercomputers or cloud computing options at my disposal, either. I know that any alternate solution will still run out of memory but I'm just trying to see where the problem is.
P.S. I don't want a solution that would change the way I'm approaching the actual problem. I know convex optimization is the way to go and I have already done enough work to get up until here.
P.P.S I saw the other question regarding the open source alternatives. That's not what I'm looking for. I'm looking for more efficient ones, if someone faced the same problem adn shifted to a better solver.
Hmmm...
Without further information, I'd guess that fmincon runs out of memory because it needs the Hessian (which, given that your decision variable is 10^4, will be 10^4 x numel(f(x1,x2,x3,....)) large).
It also takes a lot of time to determine the values of the Hessian, because fmincon normally uses finite differences for that if you don't specify derivatives explicitly.
There's a couple of things you can do to speed things up here.
If you know beforehand that there will be a lot of zeros in your Hessian, you can pass sparsity patterns of the Hessian matrix via HessPattern. This saves a lot of memory and computation time.
If it is fairly easy to come up with explicit formulae for the Hessian of your objective function, create a function that computes the Hessian and pass it on to fmincon via the HessFcn option in optimset.
The same holds for the gradients. The GradConstr (for your non-linear constraint functions) and/or GradObj (for your objective function) apply here.
There's probably a few options I forgot here, that could also help you. Just go through all the options in the optimization toolbox' optimset and see if they could help you.
If all this doesn't help, you'll really have to switch optimizers. Given that fmincon is the pride and joy of MATLAB's optimization toolbox, there really isn't anything much better readily available, and you'll have to search elsewhere.
TOMLAB is a very good commercial solution for MATLAB. If you don't mind going to C or C++...There's SNOPT (which is what TOMLAB/SNOPT is based on). And there's a bunch of things you could try in the GSL (although I haven't seen anything quite as advanced as SNOPT in there...).
I don't know on what version of MATLAB you have, but I know for a fact that in R2009b (and possibly also later), fmincon has a few real weaknesses for certain types of problems. I know this very well, because I once lost a very prestigious competition (the GTOC) because of it. Our approach turned out to be exactly the same as that of the winners, except that they had access to SNOPT which made their few-million variable optimization problem converge in a couple of iterations, whereas fmincon could not be brought to converge at all, whatever we tried (and trust me, WE TRIED). To this day I still don't know exactly why this happens, but I verified it myself when I had access to SNOPT. Once, when I have an infinite amount of time, I'll find this out and report this to the MathWorks. But until then...I lost a bit of trust in fmincon :)
I have to solve a multiobjective problem but I don't know if I should use CPLEX or Matlab. Can you explain the advantage and disadvantage of both tools.
Thank you very much!
This is really a question about choosing the most suitable modeling approach in the presence of multiple objectives, rather than deciding between CPLEX or MATLAB.
Multi-criteria Decision making is a whole sub-field in itself. Take a look at: http://en.wikipedia.org/wiki/Multi-objective_optimization.
Once you have decided on the approach and formulated your problem (either by collapsing your multiple objectives into a weighted one, or as series of linear programs) either tool will do the job for you.
Since you are familiar with MATLAB, you can start by using it to solve a series of linear programs (a goal programming approach). This page by Mathworks has a few examples with step-by-step details: http://www.mathworks.com/discovery/multiobjective-optimization.html to get you started.
Probably this question is not a matter of your current concern. However my answer is rather universal, so let me post it here.
If solving a multiobjective problem means deriving a specific Pareto optimal solution, then you need to solve a single-objective problem obtained by scalarizing (aggregating) the objectives. The type of scalarization and values of its parameters (if any) depend on decision maker's preferences, e.g. how he/she/you want(s) to prioritize different objectives when they conflict with each other. Weighted sum, achievement scalarization (a.k.a. weighted Chebyshev), and lexicographic optimization are the most widespread types. They have different advantages and disadvantages, so there is no universal recommendation here.
CPLEX is preferred in the case, where (A) your scalarized problem belongs to the class solved by CPLEX (obviously), e.g. it is a [mixed integer] linear/quadratic problem, and (B) the problem is complex enough for computational time to be essential. CPLEX is specialized in the narrow class of problems, and should be much faster than Matlab in complex cases.
You do not have to limit the choice of multiobjective methods to the ones offered by Matlab/CPLEX or other solvers (which are usually narrow). It is easy to formulate a scalarized problem by yourself, and then run appropriate single-objective optimization (source: it is one of my main research fields, see e.g. implementation for the class of knapsack problems). The issue boils down to finding a suitable single-objective solver.
If you want to obtain some general information about the whole Pareto optimal set, I recommend to start with deriving the nadir and the ideal objective vectors.
If you want to derive a representation of the Pareto optimal set, besides the mentioned population based-heuristics such as GAs, there are exact methods developed for specific classes of problems. Examples: a library implemented in Julia, a recently published method.
All concepts mentioned here are described in the comprehensive book by Miettinen (1999).
Can cplex solve a pareto type multiobjective one? All i know is that it can solve a simple goal programming by defining the lexicographical objs, or it uses the weighted sum to change weights gradually with sensitivity information and "enumerate" the pareto front, which highly depends on the weights and looks very subjective.
You can refer here as how cplex solves the bi-objetive one, which seems not good.
For a true pareto way which includes the ranking, i only know some GA variants can do like NSGA-II.
A different approach would be to use a domain-specific modeling language for mathematical optimization like YALMIP (or JUMP.jl if you like to give Julia a try). There you can write your optimization problem with Matlab with some extra YALMIP functionalities and use CPLEX (or any other supported solver as a backend) without restricting to one solver.