I have an optimization problem that I could set up as either a rather complex univariate problem with a non-linear optimization function, or as a multivariate problem with linear optimization function.
Is it possible to somewhat generalize whether from a programming perspective, one is more efficient than the other? I.e. even without going into the details of the functions and constraints, are solvers more efficient with functionally complex univariate problems over simple multivariate problems or vice versa?
I tried to google the problem but the results I read were all looking at examples that didn't even require numerical solving. (e.g. this)
Once I figured out the best approach I'll implement it in python.
I am trying to solve the following problem using fipy but I am a bit overwhelmed. Hoping someone can point me in the right direction.
C and q are the dependent variables and I am solving in 2 dimensions, time (t) and length (z).
Is there any example I can follow? Where should I be looking at in the documentation? Or is there a some other library better suited for this problem?
Thanks in advance.
examples.diffusion.mesh1D is the starting point for anybody using FiPy. It runs through the basic concepts of how to set up a problem and it covers the transient and diffusive behavior of your first equation.
examples.convection.robin illustrates a static convection-diffusion-source problem with a Robin boundary condition. This covers all the terms on the right-hand side of your first equation and your first boundary condition.
The two equations can either be "swept" in succession until they converge, or they can be coupled. examples.diffusion.coupled illustrates both approaches for a pair of diffusion equations.
If these aren't enough to get you started, please come back with specific questions about where you're having trouble.
How can I code a bi-level optimization problem using scipy.optimize.minimize which consists of the following two levels, (1) being the upper-level problem, which is subject to (2) being the lower-level:
minimize linear programming problem, subject to a
minimize quadratic programming problem
I know how to write a single objective function for quadratic programming as a typical scipy.optimize.minimize routine (step 2 by itself), but how can I add an upper level minimization problem as well that has its own objective function? Can scipy handle a minimization problem subject to a lower minimization problem? A rough outline of the code using minimize and linprog would help
I don't think scipy solvers are very suited for this.
A standard approach for bilevel problems is to form the KKT conditions of the inner problem and add these as constraints to the outer problem. The complementarity conditions make the problem hard. You need a global NLP solver or a MIP solver to solve this thing to proven optimality. Here are some formulations for linear bilevel problems. If the inner problem is a (convex) QP problem, things become a little bit more difficult, but the same idea can be applied (form KKT conditions of the inner problem).
For more information see Bard, Practical Bilevel Optimization: Algorithms And Applications, Springer, 1998.
I'm trying to call upon the famous multilateration algorithm in order to pinpoint a radiation emission source given a set of arrival times for various detectors. I have the necessary data, but I'm still having trouble implementing this calculation; I am relatively new with Python.
I know that, if I were to do this by hand, I would use matrices and carry out elementary row operations in order to find my 3 unknowns (x,y,z), but I'm not sure how to code this. Is there a way to have Python implement ERO, or is there a better way to carry out my computation?
Depending on your needs, you could try:
NumPy if your interested in numerical solutions. As far as I remember, it could solve linear equations. Don't know how it deals with non-linear resolution.
SymPy for symbolic math. It solves symbolically linear equations ... according to their main page.
The two above are "generic" math packages. I doubt you will find (easily) any dedicated (and maintained) library for your specific need. Their was already a question on that topic here: Multilateration of GPS Coordinates
I'm looking for a good library that will integrate stiff ODEs in Python. The issue is, scipy's odeint gives me good solutions sometimes, but the slightest change in the initial conditions causes it to fall down and give up. The same problem is solved quite happily by MATLAB's stiff solvers (ode15s and ode23s), but I can't use it (even from Python, because none of the Python bindings for the MATLAB C API implement callbacks, and I need to pass a function to the ODE solver). I'm trying PyGSL, but it's horrendously complex. Any suggestions would be greatly appreciated.
EDIT: The specific problem I'm having with PyGSL is choosing the right step function. There are several of them, but no direct analogues to ode15s or ode23s (bdf formula and modified Rosenbrock if that makes sense). So what is a good step function to choose for a stiff system? I have to solve this system for a really long time to ensure that it reaches steady-state, and the GSL solvers either choose a miniscule time-step or one that's too large.
If you can solve your problem with Matlab's ode15s, you should be able to solve it with the vode solver of scipy. To simulate ode15s, I use the following settings:
ode15s = scipy.integrate.ode(f)
ode15s.set_integrator('vode', method='bdf', order=15, nsteps=3000)
ode15s.set_initial_value(u0, t0)
and then you can happily solve your problem with ode15s.integrate(t_final). It should work pretty well on a stiff problem.
(See also Link)
Python can call C. The industry standard is LSODE in ODEPACK. It is public-domain. You can download the C version. These solvers are extremely tricky, so it's best to use some well-tested code.
Added: Be sure you really have a stiff system, i.e. if the rates (eigenvalues) differ by more than 2 or 3 orders of magnitude. Also, if the system is stiff, but you are only looking for a steady-state solution, these solvers give you the option of solving some of the equations algebraically. Otherwise, a good Runge-Kutta solver like DVERK will be a good, and much simpler, solution.
Added here because it would not fit in a comment: This is from the DLSODE header doc:
C T :INOUT Value of the independent variable. On return it
C will be the current value of t (normally TOUT).
C
C TOUT :IN Next point where output is desired (.NE. T).
Also, yes Michaelis-Menten kinetics is nonlinear. The Aitken acceleration works with it, though. (If you want a short explanation, first consider the simple case of Y being a scalar. You run the system to get 3 Y(T) points. Fit an exponential curve through them (simple algebra). Then set Y to the asymptote and repeat. Now just generalize to Y being a vector. Assume the 3 points are in a plane - it's OK if they're not.) Besides, unless you have a forcing function (like a constant IV drip), the MM elimination will decay away and the system will approach linearity. Hope that helps.
PyDSTool wraps the Radau solver, which is an excellent implicit stiff integrator. This has more setup than odeint, but a lot less than PyGSL. The greatest benefit is that your RHS function is specified as a string (typically, although you can build a system using symbolic manipulations) and is converted into C, so there are no slow python callbacks and the whole thing will be very fast.
I am currently studying a bit of ODE and its solvers, so your question is very interesting to me...
From what I have heard and read, for stiff problems the right way to go is to choose an implicit method as a step function (correct me if I am wrong, I am still learning the misteries of ODE solvers). I cannot cite you where I read this, because I don't remember, but here is a thread from gsl-help where a similar question was asked.
So, in short, seems like the bsimp method is worth taking a shot, although it requires a jacobian function. If you cannot calculate the Jacobian, I will try with rk2imp, rk4imp, or any of the gear methods.