## A Basic Branch and Bound Solver in Python using Cvxpy

Branch and bound is a useful problem solving technique. The idea is, if you have a minimization problem you want to solve, maybe there is a way to relax the constraints to an easier problem. If so, the solution of the easier problem is a lower bound on the possible solution of the hard problem. If the solution of the easier problem just so happens to also obey the more constrained hard problem, then it must also be the solution to the hard problem. You can also use the lower bound coming from a relaxed problem to prune your search tree for the hard problem. If even the relaxed problem doesn’t beat the current best found, don’t bother going down that branch.

A standard place this paradigm occurs is in mixed integer programming. The relaxation of a binary constraint (either 0 or 1) can be all the values in between (any number between 0 and 1). If this relaxed problem can be expressed in a form amenable to a solver like a linear programming solver, you can use that to power the branch and bound search, also using returned solutions for possible heuristics.

I built a basic version of this that uses cvxpy as the relaxed problem solver. Cvxpy already has much much faster mixed integer solvers baked in (which is useful to make sure mine is returning correct results), but it was an interesting exercise. The real reason I’m toying around is I kind of want the ability to add custom branching heuristics or inspect and maintain the branch and bound search tree, which you’d need to get into the more complicated guts of the solvers bound to cvxpy to get at. Julia might be a better choice.

A somewhat similar (and better) project is https://github.com/oxfordcontrol/miosqp which doesn’t use cvxpy explicitly, but does have the branch and bound control in the python layer of the solver. There are also other projects that can use fairly arbitrary solvers like Bonmin

As a toy problem I’m using a knapsack problem where we have objects of different sizes and different values. We want to maximize the value while keeping the total size under the capacity of the bag. This can be phrased linearly like so: $\max v \cdot x$ s.t. $\sum_i s_i x_i<= capacity$, $x \in {0,1}$. The basic heuristic I’m using is to branch on variables that are either 0 or 1 in even the relaxed solution. The alternative branch hopefully gets pruned fast.

This is at least solving the problem fairly quickly. It needs better heuristics and to be sped up, which is possible in lots of ways. I was not trying to avoid all performance optimizations. It takes maybe 5 seconds, whereas the cvxpy solver is almost instantaneous.

## Mixed Integer Programming & Quantization Error

I though of another fun use case of mixed integer programming the other day. The quantization part of a digital to analog converter is difficult to analyze by the techniques taught in a standard signals course (linear analysis, spectral techniques, convolution that sort of thing). The way it is usually done is via assuming the quantization error is a kind of randomized additive noise.

Mixed Integer programming does have the ability to directly encode some questions about this quantization though. We can directly encode the integer rounding relations by putting the constraint that the quantized signal is exactly +-1/2 a quantization interval away from the original signal. Then we can run further analysis on the signals and compare them. For example, I wrote down a quick cosine transform. Then I ask for the worst case signal that leads to the most error on the quantized transform versus the transform of the unquantized signal. My measure of worst case performance was the sum of the difference of the two transforms. I chose this because it is tractable as a mixed integer linear program. Not all reasonable metrics one might want will be easily encodable in a mixed integer framework it seems. Some of them are maximizing over a convex function, which is naughty. (for example trying to maximize the L2 error $\sum|x-y|^2$ )

In a variant of this, it is also possible to directly encode the digital signal process in terms of logic gate construction and compare that to the intended analog transform, although this will be a great deal more computational expensive.

This is interesting as a relatively straightforward technique for the analysis of quantization errors.

This also might be an interesting place to use the techniques of Vanderbei https://vanderbei.princeton.edu/tex/ffOpt/ffOptMPCrev4.pdf . He does a neato trick where he partially embeds the FFT algorithm into an optimization problem by adding auxiliary variables. Despite the expense of adding these variables, it greatly increases the sparsity of the constraint matrices, which will probably be a win. I wonder if one might do something similar with a Fast Multipole Method , Barnes Hut, or Wavelet transform? Seems likely. Would be neat, although I’m not sure what for. I was thinking of simulating the coulomb gas. That seems like a natural choice. Oooh. I should do that.

## Solving the XY Model using Mixed Integer Optimization in Python

There are many problems in physics that take the form of minimizing the energy. Often this energy is taken to be quadratic in the field. The canonical example is electrostatics. The derivative of the potential $\phi$ gives the electric field E. The energy is given as $\int (|\nabla \phi|^2 + \phi \rho) d^3 x$. We can encode a finite difference version of this (with boundary conditions!) directly into a convex optimization modelling language like so.

It is noted rarely in physics, but often in the convex optimization world that the barrier between easy and hard problems is not linear vs. nonlinear, it is actually more like convex vs. nonconvex. Convex problems are those that are bowl shaped, on round domains. When your problem is convex, you can’t get caught in valleys or on corners, hence greedy local methods like gradient descent and smarter methods work to find the global minimum. When you differentiate the energy above, it results in the linear Laplace equations $\nabla^2 \phi = \rho$. However, from the perspective of solvability, there is not much difference if we replace the quadratic energy with a convex alternative.

Materials do actually have non-linear permittivity and permeability, this may be useful in modelling that. It is also possible to consider the convex relaxation of truly hard nonlinear problems and hope you get the echoes of the phenomenology that occurs there.

Another approach is to go mixed integer. Mixed Integer programming allows you to force that some variables take on integer values. There is then a natural relaxation problem where you forget the integer variables have to be integers. Mixed integer programming combines a discrete flavor with the continuous flavor of convex programming. I’ve previously shown how you can use mixed integer programming to find the lowest energy states of the Ising model but today let’s see how to use it for a problem of a more continuous flavor.

As I’ve described previously, in the context of robotics, the non-convex constraint that variables lie on the surface of a circle can be approximated using mixed integer programming. We can mix this fairly trivially with the above to make a global solver for the minimum energy state of the XY model. The XY model is a 2d field theory where the value of the field is constrained to lie on a circle. It is a model of a number of physical systems, such as superconductivity, and is the playground for a number of interesting phenomenon, like the Kosterlitz-Thouless phase transition. Our encoding is very similar to the above except we make two copies of the field $phi$ and we then force them to lie on a circle. I’m trying to factor out the circle thing into my library cvxpy-helpers, which is definitely a work in progress.

Now, this isn’t really an unmitigated success as is. I switched to an absolute value potential because GLPK_MI needs it to be linear. ECOS_BB works with a quadratic potential, but it was not doing a great job. The commercial solvers (Gurobi, CPlex, Mosek) are supposed to be a great deal better. Perhaps switching to Julia, with it’s richer ecosystem might be a good idea too. I don’t really like how the solution of the absolute value potential looks. Also, even at such a small grid size it still takes around a minute to solve. When you think about it, it is exploring a ridiculously massive space and still doing ok. There are hundreds of binary variables in this example. But there is a lot of room for tweaking and I think the approach is intriguing.

Musings:

• Can one do steepest descent style analysis for low energy statistical mechanics or quantum field theory?
• Is the trace of the mixed integer program search tree useful for perturbative analysis? It seems intuitively reasonable that it visits low lying states
• The Coulomb gas is a very obvious candidate for mixed integer programming. Let the charge variables on each grid point = integers. Then use the coulomb potential as a quadratic energy. The coulomb gas is dual to the XY model. Does this exhibit itself in the mixed integer formalism?
• Nothing special about the circle. It is not unreasonable to make piecewise linear approximations or other convex approximations of the sphere or of Lie groups (circle is U(1) ). This is already discussed in particular about SO(3) which is useful in robotic kinematics and other engineering topics.

By absolute value potential, I mean using |del phi| as compared to a more ordinary quadratic |del phi|2.

This is where I’m getting confused. As you say later, you are actually using two fields, phi_x and phi_y. So I’m guessing your potential is the “L1 norm”

|del phi| = |del phi_x| + |del phi_y|

? This is the only linear thing I can think of.

I don’t feel that the exact specifics of the XY model actually matter all the much.

You should be careful here though. A key point in the XY model is the O(2) symmetry of the potential: you can multiply the vector (phi_x,phi_y) by a 2D rotation matrix and the Hamiltonian is unchanged. You have explicitly broken this symmetry down to Z_4 if your potential is as I have written above. In this case, the results of the famous JKKN paper and this followup by Kadanoff suggest that you’ll actually get a phase transition of the so-called “Ashkin-Teller” universality class. These are actually closely related to the Kosterlitz-Thouless transitions of the XY model; the full set of Ashkin-Teller phase transitions actually continuously link the XY transition with that of two decoupled Ising models.

You should still get an interesting phase transition in any case! Just wanted to give some background, as the physics here is extremely rich. The critical exponents you see will be different from the XY model, and you will actually get an ordered Z_4 phase at low temperatures rather than the quasi-long range order seen in the low temperature phase of the XY model. (You should be in the positive h_4 region of the bottom phase diagram of Figure 1 of the linked JKKN paper.)”

These are some interesting points and references.

## 2D Robot Arm Inverse Kinematics using Mixed Integer Programming in Cvxpy

Mixed Integer programming is crazy powerful. You can with ingenuity encode many problems into it. The following is a simplification of the ideas appearing in http://groups.csail.mit.edu/robotics-center/public_papers/Dai19.pdf . They do 3d robot arms, I do 2d. I also stick to completely linear approximations.

The surface of a circle is not a convex shape. If you include the interior of a circle it is. You can build a good approximation to the circle as polygons. A polygon is the union of it’s sides, each of which is a line segment. Line sgements are convex set. Unions of convex sets are encodable using mixed integer programming. What I do is sample N regular positions on the surface of a circle. These are the vertices of my polygon. Then I build boolean indicator variables for which segment we are on. Only one of them is be nonzero $\sum s_i == 1$. If we are on a segment, we are allowed to make positions $x$ that interpolate between the endpoints $x_i$ of that segment $x = \lambda_1 x_1 + \lambda_2 x_2$, where $\lambda_i >= 0$ and $\sum \lambda=1$. These $\lambda$ are only allowed to be nonzero if we are on the segment, so we suppress them with the indicator variables $\lambda_i <= s_i + s_{i+1}$. That’s the gist of it.

Given a point on the circle (basically sines and cosines of an angle) we can build a 2d rotation matrix $R$ from it. Then we can write down the equations connecting subsequent links on the arm. $p_{i+1}=p_{i} +Rl$. By using global rotations with respect to the world frame, these equations stay linear. That is a subtle point. $p$ and $R$ are variables, whereas $l$ is a constant describing the geometry of the robot arm. If we instead used rotation matrices connecting frame i to i+1 these R matrices would compound nonlinearly.

All in all, pretty cool!

## Casadi – Pretty Damn Slick

Casadi is something I’ve been aware of and not really explored much. It is a C++ / python / matlab library for modelling optimization problems for optimal control with bindings to IPOpt and other solvers. It can produce C code and has differentiation stuff. See below for some examples after I ramble.

I’ve enjoyed cvxpy, but cvxpy is designed specifically for convex problems, of which many control problems are not.

Casadi gives you a nonlinear modelling language and easy access to IPOpt, an interior point solver that works pretty good (along with some other solvers, many of which are proprietary however).

While the documentation visually looks very slick I actually found it rather confusing in contents at first. I’m not sure why. Something is off.

It also has a bunch of helper classes for DAE building and other things. They honestly really put me off. The documentation is confusing enough that I am not convinced they give you much.

The integrator classes give you access to external smart ode solvers from the Sundials suite. They give you good methods for difficult odes and dae (differential algebraic equations, which are ODEs with weird constraints like x^1 + y^1 == 1) Not clear to me if you can plug those in to an optimization, other than by a shooting method.

Casadi can also output C which is pretty cool.

I kind of wondered about Casadi vs Sympy. Sympy has lot’s of general purpose symbolic abilities. Symbolic solving, polynomial smarts, even some differential equation understanding. There might be big dividends to using it. But it is a little harder to get going. I feel like there is an empty space for a mathemtical modelling language that uses sympy as it’s underlying representation. I guess monkey patching sympy expressions into casadi expression might not be so hard. Sympy can also output fast C code. Sympy doesn’t really have any support for sparseness that I know of.

As a side note, It can be useful to put these other languages into numpy if you need extended reshaping abilities. The other languages often stop at matrices, which is odd to me.

Hmm. Casadi actually does have access to mixed integer programs via bonmin (and commercial solvers). That’s interesting. Check out lotka volterra minlp example

The optim interface makes some of this look better. optim.minimize and subject_to. Yeah, this is more similar to the interfaces I’m used to. It avoids the manual unpacking of the solution and the funky feel of making everything into implicit == 0 expressions.

Here is a simple harmonic oscillator example using the more raw casadi interface. x is positive, v is velocity, u is a control force. I’m using a very basic leap frog integration. You tend to have to stack things into a single vector with vertcat when building the final problem.

Let’s use the opti interface, which is pretty slick. Here is a basic cartpole https://web.casadi.org/blog/opti/

Very fast. Very impressive. Relatively readable code. I busted this out in like 15 minutes. IPopt solves the thing in the blink of an eye (about 0.05s self reported). Might be even faster if I warm start it with a good solution, as I would in online control (which may be feasible at this speed). Can add the initial condition as a parameter to the problem

I should try this on an openai gym.

## Cvxpy and NetworkX Flow Problems

Networkx outputs scipy sparse incidence matrices

https://networkx.github.io/documentation/networkx-1.9/reference/generated/networkx.linalg.graphmatrix.incidence_matrix.html#networkx.linalg.graphmatrix.incidence_matrix

https://docs.scipy.org/doc/scipy/reference/sparse.html

Networkx also has it’s own flow solvers, but cvxpy gives you some interesting flexibility, like turning the problem mixed integer, quadratic terms, and other goodies. Plus it is very easy to get going as you’ll see.

So here’s a basic example of putting these two together. Very straightforward and cool.

Here was a cool visual from a multi commodity flow problem (nx.draw_networkx_edges)

Nice, huh.

## A Simple Interior Point Linear Programming Solver in Python

This solver is probably not useful for anything. For almost all purposes, let me point you to cvxpy.

If you want an open source solver CBC/CLP and GLPK and OSQP are good.

If you want proprietary, you can get a variable number constrained trial license to Gurobi for free.

Having said that, here we go.

The simplex method gets more press, and certainly has it’s advantages, but the interior point method makes much more sense to me. What follows is the basic implementation described in Stephen Boyd’s course and book http://web.stanford.edu/~boyd/cvxbook/

In the basic interior point method, you can achieve your inequality constraints $\phi(x) \ge 0$ by using a logarithmic potential to punish getting close to them $-\gamma \ln (\phi(x))$ where $\gamma$ is a parameter we’ll talk about in a bit.  From my perspective, the logarithm is a somewhat arbitrary choice. I believe some properties of the logarithmic potential is necessary for some convergence guarantees.

The basic unconstrained newton step takes a locally quadratic approximation to the function you’re trying to optimize and finds the minimum of that. This basically comes down to taking a step that is the inverse hessian applied to the gradient.

$\min_{dx} f(x_0+dx) \approx f(x_0) + \nabla f(x_0)dx + \frac{1}{2} dx^T H dx$

$(H)_{ij} = \partial_{ij}f(x_0)$

$\nabla f(x_0) +H dx = 0 \rightarrow dx =- H^{-1}\nabla f$

We can maintain a linear constraint on the variable x during this newton step. Instead of setting the gradient to zero, we set it so that it is perpendicular to the constraint plane using the Lagrange multiplier procedure.

$\nabla f(x_0) +H dx = -A^T \lambda \rightarrow Hdx + A^T \lambda = - \nabla f$

$A(x_0 + dx) = b$

This is a block linear system

$\begin{bmatrix} H & A^T \\ A & 0 \\ \end{bmatrix} \begin{bmatrix} dx \\ \lambda \end{bmatrix} = \begin{bmatrix} -\nabla f \\ b - Ax_0 \end{bmatrix}$

Despite the logarithm potential, there is no guarantee that the newton step would not take us outside the allowed region. This is why we need a line search on top of the newton step. We scale the newton dx to $\alpha dx$. Because the function we’re optimizing is convex and the region we’re in is convex, there is some step length in that newton direction that will work. So if we keep decreasing the overall step size, we’ll eventually find something acceptable.

As part of the interior point method, once it has converged we decrease the parameter $\gamma$ applied to the logarithm potential. This will allow the inequality constraints to satisfied tighter and tighter with smaller gamma.

The standard form of an LP is

$\min c^T x$

$A x = b$

$x \ge 0$

This doesn’t feel like the form you’d want. One way you can construct this is by adding slack variables and splitting regular variables into a positive and negative piece

$x = x_+ - x_-$

$Ax \ge b \rightarrow Ax +s = b, s \ge 0$

The interior point formulation of this is

$\min c^T x- \gamma \sum_i \ln(x_i)$

$Ax = b$

The Hessian and gradient are quite simple here

$\nabla f = -\frac{\gamma}{x_i}$

$(H)_{ij} = \delta_{ij} \frac{\gamma}{x_i^2}$

The optimum conditions for this are

$\nabla (c^T x - \gamma \ln(x))= c - \gamma \frac{1}{x} = 0$

$Ax=b$

Now in the above, I’m not sure I got all the signs right, but I did implement it in python. The result seems to be correct and does work. I haven’t tested extensively, YMMV. It’s a useful starting point.

Musings:

I wanted to build this because I’ve been getting really into mixed integer programming and have been wondering how much getting deep in the guts of the solver might help. Given my domain knowledge of the problems at hand, I have probably quite good heuristics. In addition, I’ve been curious about a paper that has pointed out an interesting relatively unexploited territory, combining machine learning with mixed integer programming https://arxiv.org/pdf/1811.06128

For these purposes, I want a really simple optimization solver.

But this is silly. I should use CLP or OSQP as a black box if I really want to worry about the mixed integer aspect.

MIOSQP is interesting.

It is interesting how the different domains of discrete optimization and search seem to have relatively similar sets of methods. Maybe I’m crazy. Maybe at the loose level I’m gonna talk almost anything is like almost anything else.

Clause learning and Cutting plane addition feel rather similar.

Relaxation to LP and unit propagation are somewhat similar. Or is unit propagation like elimination?

Mixed integer programs build their own heuristics.

Fourier Motzkin and resolution are similar methods. In Fourier motzkin, you eliminate variables in linear inequalities by using algebra to bring that variable by itself on one side of the inequality and then matching up all the <= to all the uses of >= of that variable. There are packages that compute these things. See CDD or Polyhedra.jl

Resolution takes boolean formula. You can eliminate a variable q from a CNF formula by taking all the negated instances $\not q$ and combining them with all positive instances.

## Trajectory Optimization of a Pendulum with Mixed Integer Linear Programming

There is a reasonable piecewise linear approximation for the pendulum replacing the the sinusoidal potential with two quadratic potentials (one around the top and one around the bottom). This translates to a triangle wave torque.

Cvxpy curiously has support for Mixed Integer Programming.

Cbc is probably better than GLPK MI. However, GLPK is way easier to get installed. Just brew install glpk and pip install cvxopt.

Getting cbc working was a bit of a journey. I had to actually BUILD Cylp (god forbid) and fix some problems.

Special Ordered Set constraints are useful for piecewise linear approximations. The SOS2 constraints take a set of variables and make it so that only two consecutive ones can be nonzero at a time. Solvers often have built in support for them, which can be much faster than just blasting them with general constraints. I did it by adding a binary variable for every consecutive pair. Then these binary variables suppress the continuous ones. Setting the sum of the binary variables to 1 makes only one able to be nonzero.

One downside is that every evaluation of these non linear functions requires a new set of integer and binary variables, which is possibly quite expensive.

For some values of total time steps and step length, the solver can go off the rails and never return.

At the moment, the solve is not fast enough for real time control with CBC (~ 2s). I wonder how much some kind of warm start might or more fiddling with heuristics, or if I had access to the built in SOS2 constraints rather than hacking it in. Also commercial solvers are usually faster. Still it is interesting.

Blue is angle, orange is the applied torque. The torque is running up against the limits I placed on it.

## Gettin’ that Robot some Tasty Apples: Solving a simple geometrical puzzle in Z3 python

At work there is a monthly puzzler.

“Design a robot that can pick up all 9 apples arranged on a 3 by 3 rectangular grid, and spaced 1m apart. The robot parts they have are faulty. The robot can only turn three times”

I think the intent of the puzzle is that the robot can drive in forward and reverse, but only actually turn 3 times. It’s not very hard to do by hand. I decided to take a crack at this one using Z3 for funzies. Z3 is an SMT solver. It is capable of solving a interesting wide variety of problems.

I interpret this as “find 4 lines that touch all points in the grid, such that each subsequent line intersects.”

It is fairly easy to directly translate this into a Z3 model.

If we ask z3 to use only 3 lines, it returns unsat. Try to prove that by hand.

However, If the robot is on the projective plane, it is possible with 3 lines. It only needs to drive to infinity and turn twice. All lines intersect exactly once on the projective plane. How convenient.

The problem only seems somewhat difficult to computerize because of the seemingly infinite nature of geometry. If we only consider the lines that touch at least two points, all possible robot paths becomes extremely enumerable. Is there a proof that we only need these lines?

Another interesting approach might be to note that the points are described by the set of equations $x*(x-1)*(x-2)=0$ and $y*(y-1)*(y-2)=0$. I think we could then possibly use methods of nonlinear algebra (Groebner bases) to find the lines. Roughly an ideal containment question? Don’t have this one fully thought out yet. I think z3 might be doing something like this behind the scenes.

## More Reinforcement Learning with cvxpy

So I spent thanksgiving doing this and playing Zelda. Even though that sounds like a hell of a day, seems a little sad for thanksgiving :(. I should probably make more of an effort to go home next year.

I tried implementing a more traditional q-learning pipeline using cvxpy (rather than the inequality trick of the last time). Couldn’t get it to work as well. And it’s still kind of slow despite a lot of rearrangement to vectorize operations (through batching basically).

I guess I’m still entranced with the idea of avoiding neural networks. In a sense, that is the old boring way of doing things. The Deep RL is the new stuff. Using ordinary function approximators is way older I think. But I feel like it takes a problem out of the equation (dealing with training neural nets). Also I like modeling languages/libraries.

I kept finding show stopping bugs throughout the day (incorrectly written maxaction functions, typos, cross episode data points, etc.), so I wouldn’t be surprised if there is one still in here. It’s very surprising how one can convince oneself that it is kind of working when it is actually impossible it’s working. All these environments are so simple, that I suspect I could randomly sample controllers out of a sack for the time I’ve been fiddling with this stuff and find a good one.

I also did the easy cartpole environment using the inequality trick.  Seems to work pretty well.

I also have some Work in Progress on getting full swingup cartpole. Currently is not really working. Seems to kind of be pumping about right? The continuous force control easy cartpole does work though.

Now I feel that a thing that matters quite a bit is what is your choice of action for the next time step. Hypothetically you want a ton of samples here. I now think that using an action that is just slightly perturbed from the actual action works well because the actual action is tending to become roughly the optimal one. Subsequent time steps have roughly the same data in them.

One advantage of discrete action space is that you can really search it all.

Does that mean I should seriously investigate the sum of squares form? A semidefinite variable per data point sounds bad. I feel like I’d have to seriously limit the amount of data I’m using. Maybe I’ll be pleasantly surprised.

I haven’t even gotten to playing with different polynomials yet. The current implementation is exponentially sized in the number of variables. But in kind of a silly way. I think it would be better to use all terms of a bounded total degree.