I am trying to use GLPK for solving an LP problem. My problem is the routing problem in a computer network. Given network topology and each link capacity and the traffic demand matrix for each source-destination pair in the network, I want to minimize maximum link utilization in the network. This is an LP problem and I know how to use GLPK to get the optimum solution.
My problem is that I want to get the sub-optimal solutions also. Is there any way that I can get say top 10 suboptimal solutions by GLPK?
Best
For a pure LP (with only continuous variables), the concept of finding "next best" solutions is very difficult (just move an epsilon away, and you have another solution). We can define this differently: find "next best" corner points (a.k.a. bases). This is not so easy to do, but there is a somewhat complex way by encoding bases using binary variables (link).
If the problem is actually a MIP (with binary variables) it is easier to find "next best" solutions. Some advanced solvers have built-in facilities for this (called: solution pool). Note: glpk does not have this option. Alternatively, we can also do this by adding a cut that forbids the best-found solution and then resolve (link). In this case we exploited some structure. A general cut for 0-1 variables is derived here. This can also be done for general integer variables, but then things get a bit messy.
Say I have to solve for a large system of equations where
A_i = f(B_i)
B_i = g(A_i)
for many different i. Now, this is a system of equations which are only pair-wise dependent. The lm algorythm has proven most stable to solve this.
Now, I could solve these either independently (i.e. loop over i many scipy.optimize.root, or stack them all together and solve at the same time). I'm unsure which will be the fastest, and it's difficult to know generally. I'm having the following arguments for and against:
The algorythm initially numerically approximates the Jacobian at the provided guess, increasing dimensionality exponentially increases the time it takes to find the Jacobian (speaks against stacking)
Once the Jacobian is found, most of the updating is linear matrix algebra, and therefore should be faster if stacked.
Does that make sense? My conclusion would in that case be "if solving it takes a long time (bad guess or irregular function), stack them, if it's quick, do not stack".
I am not sure I understand correctly; when you say that they are pairwise dependent do you mean that the full system can be decomposed in a collection of small 2x2 systems? If so, you should definitely opt for solving the smaller systems. If not, can you provide some equations?
I'm using scipy.optimize.minimize to find the minimum of a 4D function that is rather sensitive to the initial guess used. If I vary it a little bit, the solution will change considerably.
There are many questions similar to this one already in SO (e.g.: 1, 2, 3), but no real answer.
In an old question of mine, one of the developers of the zunzun.com site (apparently no longer online) explained how they managed this:
Zunzun.com uses the Differential Evolution genetic algorithm (DE) to find initial parameter estimates which are then passed to the Levenberg-Marquardt solver in scipy. DE is not actually used as a global optimizer per se, but rather as an "initial parameter guesser".
The closest I've found to this algorithm is this answer where a for block is used to call the minimizing function many times with random initial guesses. This generates multiple minimized solutions, and finally the best (smallest value) one is picked.
Is there something like what the zunzun dev described already implemented in Python?
There is no general answer for such question, as a problem of minimizing arbitrary function is impossible to solve. You can do better or worse on particular classes of functions, thus it is rather a domain for mathematician, to analyze how your function probably looks like.
Obviously you can also work with dozens of so called "meta optimizers", which are just bunch of heuristics, which might (or not) work for you particular application. Those include random sampling starting point in a loop, using genetic algorithms, or - which is as far as I know most mathematically justified approach - using Bayesian optimization. In general the idea is to model your function in the same time when you try to minimize it, this way you can make informed guess where to start next time (which is level of abstraction higher than random guessing or using genetic algorithms/differential evolution). Thus, I would order these methods in following way
grid search / random sampling - uses no information from previous runs, thus - worst results
genetic approach, evolutionary, basin-hooping, annealing - use information from previous runs as a (x, f(x)) pairs, for limited period of time (generations) - thus average results
Bayesian optimization (and similar methods) - use information from all previous experiences through modeling of the underlying function and performing sampling selection based on expected improvement - best results (at the cost of most complex methods)
I am building a script that generates input data [parameters] for another program to calculate. I would like to optimize the resulting data. Previously I have been using the numpy powell optimization. The psuedo code looks something like this.
def value(param):
run_program(param)
#Parse output
return value
scipy.optimize.fmin_powell(value,param)
This works great; however, it is incredibly slow as each iteration of the program can take days to run. What I would like to do is coarse grain parallelize this. So instead of running a single iteration at a time it would run (number of parameters)*2 at a time. For example:
Initial guess: param=[1,2,3,4,5]
#Modify guess by plus minus another matrix that is changeable at each iteration
jump=[1,1,1,1,1]
#Modify each variable plus/minus jump.
for num,a in enumerate(param):
new_param1=param[:]
new_param1[num]=new_param1[num]+jump[num]
run_program(new_param1)
new_param2=param[:]
new_param2[num]=new_param2[num]-jump[num]
run_program(new_param2)
#Wait until all programs are complete -> Parse Output
Output=[[value,param],...]
#Create new guess
#Repeat
Number of variable can range from 3-12 so something such as this could potentially speed up the code from taking a year down to a week. All variables are dependent on each other and I am only looking for local minima from the initial guess. I have started an implementation using hessian matrices; however, that is quite involved. Is there anything out there that either does this, is there a simpler way, or any suggestions to get started?
So the primary question is the following:
Is there an algorithm that takes a starting guess, generates multiple guesses, then uses those multiple guesses to create a new guess, and repeats until a threshold is found. Only analytic derivatives are available. What is a good way of going about this, is there something built already that does this, is there other options?
Thank you for your time.
As a small update I do have this working by calculating simple parabolas through the three points of each dimension and then using the minima as the next guess. This seems to work decently, but is not optimal. I am still looking for additional options.
Current best implementation is parallelizing the inner loop of powell's method.
Thank you everyone for your comments. Unfortunately it looks like there is simply not a concise answer to this particular problem. If I get around to implementing something that does this I will paste it here; however, as the project is not particularly important or the need of results pressing I will likely be content letting it take up a node for awhile.
I had the same problem while I was in the university, we had a fortran algorithm to calculate the efficiency of an engine based on a group of variables. At the time we use modeFRONTIER and if I recall correctly, none of the algorithms were able to generate multiple guesses.
The normal approach would be to have a DOE and there where some algorithms to generate the DOE to best fit your problem. After that we would run the single DOE entries parallely and an algorithm would "watch" the development of the optimizations showing the current best design.
Side note: If you don't have a cluster and needs more computing power HTCondor may help you.
Are derivatives of your goal function available? If yes, you can use gradient descent (old, slow but reliable) or conjugate gradient. If not, you can approximate the derivatives using finite differences and still use these methods. I think in general, if using finite difference approximations to the derivatives, you are much better off using conjugate gradients rather than Newton's method.
A more modern method is SPSA which is a stochastic method and doesn't require derivatives. SPSA requires much fewer evaluations of the goal function for the same rate of convergence than the finite difference approximation to conjugate gradients, for somewhat well-behaved problems.
There are two ways of estimating gradients, one easily parallelizable, one not:
around a single point, e.g. (f( x + h directioni ) - f(x)) / h;
this is easily parallelizable up to Ndim
"walking" gradient: walk from x0 in direction e0 to x1,
then from x1 in direction e1 to x2 ...;
this is sequential.
Minimizers that use gradients are highly developed, powerful, converge quadratically (on smooth enough functions).
The user-supplied gradient function
can of course be a parallel-gradient-estimator.
A few minimizers use "walking" gradients, among them Powell's method,
see Numerical Recipes p. 509.
So I'm confused: how do you parallelize its inner loop ?
I'd suggest scipy fmin_tnc
with a parallel-gradient-estimator, maybe using central, not one-sided, differences.
(Fwiw,
this
compares some of the scipy no-derivative optimizers on two 10-d functions; ymmv.)
I think what you want to do is use the threading capabilities built-in python.
Provided you your working function has more or less the same run-time whatever the params, it would be efficient.
Create 8 threads in a pool, run 8 instances of your function, get 8 result, run your optimisation algo to change the params with 8 results, repeat.... profit ?
If I haven't gotten wrong what you are asking, you are trying to minimize your function one parameter at the time.
you can obtain it by creating a set of function of a single argument, where for each function you freeze all the arguments except one.
Then you go on a loop optimizing each variable and updating the partial solution.
This method can speed up by a great deal function of many parameters where the energy landscape is not too complex (the dependency between the parameters is not too strong).
given a function
energy(*args) -> value
you create the guess and the function:
guess = [1,1,1,1]
funcs = [ lambda x,i=i: energy( guess[:i]+[x]+guess[i+1:] ) for i in range(len(guess)) ]
than you put them in a while cycle for the optimization
while convergence_condition:
for func in funcs:
optimize fot func
update the guess
check for convergence
This is a very simple yet effective method of simplify your minimization task. I can't really recall how this method is called, but A close look to the wikipedia entry on minimization should do the trick.
You could do parallel at two parts: 1) parallel the calculation of single iteration or 2) parallel start N initial guessing.
On 2) you need a job controller to control the N initial guess discovery threads.
Please add an extra output on your program: "lower bound" that indicates the output values of current input parameter's decents wont lower than this lower bound.
The initial N guessing thread can compete with each other; if any one thread's lower bound is higher than existing thread's current value, then this thread can be dropped by your job controller.
Parallelizing local optimizers is intrinsically limited: they start from a single initial point and try to work downhill, so later points depend on the values of previous evaluations. Nevertheless there are some avenues where a modest amount of parallelization can be added.
As another answer points out, if you need to evaluate your derivative using a finite-difference method, preferably with an adaptive step size, this may require many function evaluations, but the derivative with respect to each variable may be independent; you could maybe get a speedup by a factor of twice the number of dimensions of your problem. If you've got more processors than you know what to do with, you can use higher-order-accurate gradient formulae that require more (parallel) evaluations.
Some algorithms, at certain stages, use finite differences to estimate the Hessian matrix; this requires about half the square of the number of dimensions of your matrix, and all can be done in parallel.
Some algorithms may also be able to use more parallelism at a modest algorithmic cost. For example, quasi-Newton methods try to build an approximation of the Hessian matrix, often updating this by evaluating a gradient. They then take a step towards the minimum and evaluate a new gradient to update the Hessian. If you've got enough processors so that evaluating a Hessian is as fast as evaluating the function once, you could probably improve these by evaluating the Hessian at every step.
As far as implementations go, I'm afraid you're somewhat out of luck. There are a number of clever and/or well-tested implementations out there, but they're all, as far as I know, single-threaded. Your best bet is to use an algorithm that requires a gradient and compute your own in parallel. It's not that hard to write an adaptive one that runs in parallel and chooses sensible step sizes for its numerical derivatives.
I have to implement the solution to a 0/1 Knapsack problem with constraints.
My problem will have in most cases few variables (~ 10-20, at most 50).
I recall from university that there are a number of algorithms that in many cases perform better than brute force (I'm thinking, for example, to a branch and bound algorithm).
Since my problem is relative small, I'm wondering if there is an appreciable advantange in terms of efficiency when using a sophisticate solution as opposed to brute force.
If it helps, I'm programming in Python.
You can either use pseudopolynomial algorithm, which uses dynamic programming, if the sum of weights is small enough. You just calculate, whether you can get weight X with first Y items for each X and Y.
This runs in time O(NS), where N is number of items and S is sum of weights.
Another possibility is to use meet-in-the middle approach.
Partition items into two halves and:
For the first half take every possible combination of items (there are 2^(N/2) possible combinations in each half) and store its weight in some set.
For the second half take every possible combination of items and check whether there is a combination in first half with suitable weight.
This should run in O(2^(N/2)) time.
Brute force stuff would work fine for 10 variables, but for, say, 40 you'd get some 1000'000'000'000 possible solutions, which would probably take too long to enumerate. I'd consider approximate algorithms, e.g. the polynomial time algorithm (see, e.g. http://math.mit.edu/~goemans/18434S06/knapsack-katherine.pdf) or use a search algorithm such as branch-and-bound, maybe with an additional heuristic.
Brute force algorithms will always return the best solutions. The problem with them is that in exponential order problems they quickly become not feasible.
If you are guaranteed to have up to 20 variables, you will test no more than 1 million solutions (2^20= 1M). Hence, brute force is feasible and no other algorithm will return a better solution.
Heuristics are great, but they should be used only when we have no exact solution to the problem. There is a great book that might help you: How to Solve it, by Michalewicz.