An Intro to Integer Programming for Engineers: Simplified Bus Scheduling

With Z3, unfortunately, the issue is that we are really bad at gauging what makes for a hard SMT problem. Often, knowing that a particular problem is hard is the same as solving it! That said, you can do some really interesting things with SMT solvers if you can formulate the problem the right way.

Gurobi, CPLEX, and friends are less magical if you have enough mathematical maturity. Typically, if you are comfortable with linear algebra (a subject woefully ignored in most undergraduate curricula), you can work with them. In my experience, working with optimization engines is a two step process. First, you need to figure out how to formulate the problem in a smart way (typically this means that it's convex with constraints in a certain format), which is normally the hard part. Then you scroll through the list of available algorithms that the engine has at your disposal and pick one with properties that you like (they often have suggestions if you have no strong preference). Normally, if you can formulate the problem so it's convex, you will at least have a good shot of doing well with your optimization.

Oh for sure. You could go down a real rabbit hole if you had to implement that stuff yourself.

Constraint programming can be viewed as looking for feasible solutions to optimization problems.

So, what is optimization? For positive integers m and n, the real numbers R, and functions

f: R^n --> R

g: R^n --> R^m

find x in R^n to (problem MP1)

maximize f(x)

subject to

g(x) where g(x) is regarded as an m x 1 matrix and the 0 is also m x 1 but with all its components 0.

The constraints are the

g(x) We say that we have m constraints.

The subject is essentially the same using

g(x) >= 0

or

g(x) = 0

instead or minimizing instead of maximizing.

Point x in R^n is feasible provided it satisfies the constraints, that is,

g(x) The feasible region is the set F of all feasible x.

A point x is optimal if it is feasible and makes f(x) as large as possible. That is, x is optimal provided for any feasible y we have f(x) >= f(y).

If one or more of the functions f or g is non-linear, then we have non-linear programming. Here we may use the Kuhn-Tucker conditions to seek a point that satisfies necessary conditions for optimality. For an algorithm, we may take gradients of the constraints to set up a linear program, solve that, do a line search, and try again.

If functions f and g are linear, easily written using matrix algebra, then the optimization problem is linear programming (here programming is used in the British sense of operational planning).

Function f is convex provided for any x, y in R^n and any t in [0, 1] we have

t f(x) + (1 - t) f(y) >= f( t x + (1 - t) y )

That is, in a graph of f, a straight line between (x, f(x) and (y, f(y) ) is on or over the graph and never under the graph. So, the straight line over approximates the function.

If f is convex, then it is continuous (proof is not very difficult but is in Fleming, Functions of Several Variables). IIRC convex implies differentiable almost everywhere (maybe proof in R. T. Rockafellar, Convex Analysis). The epigraph is the graph of f and everything above it, and it is a convex set so has supporting hyperplanes and is the intersection of all the closed half spaces from its supporting (tangent) hyperplanes (IIRC, proof in Fleming). Such a supporting hyperplane is called a subgradient and can be quite useful in optimization, e.g., Lagrangian relaxation. There is a closely related, really nice result in Hilbert space: Every non-empty, closed convex subset has a unique element of minimum norm. That can be very nice to know!

To be more clear, a convex function, while continuous, need not be differentiable. Why not? Intuitively, at a point, the function makes a sharp turn. Similarly, a cube is a convex set but has 10 sharp edges and 8 sharp points that have supporting (tangent) planes, but these planes are not unique. So, a subgradient at a point would be a gradient and a tangent plane if at that point it was the only tangent plane. So, intuitively, a subgradient is nearly a tangent plane.

Function f is concave provided that -f is convex. If f is convex and -f is convex, then f is linear -- so, intuitively, a function that is convex (or concave) is half of being linear.

We have some astoundingly powerful methods for solving linear programming problems.

A significant fraction of the Nobel prizes in economics have been from economic models based on linear programming.

Some special cases, e.g., least cost network flows, maximum matchings, are special cases with even better results for the algorithms.

If in MP1 function f is concave and funtion g is linear, then again we have some good methods.

If in MP1 function f is quadratic and concave and function g is linear, then we have the problem of Markowitz in investment portfolio management and his Nobel prize in economics.

If we are maximizing a convex function subject to linear constraints, then we have, IIRC, a problem in NP-complete.

If in linear programming we require in addition that some or all of the components of x are required to be integers, then we have some integer constraints and a problem in integer linear programming (ILP). ILP was one of the earliest problems shown to be in NP-complete.

So, that's a nutshell version of optimization.

Can also extend to accommodating randomness and doing much the same over time.

The over time part is control theory, e.g., how to fly an airplane from one point to another in least time -- there are books, e.g., by Athans and Falb. Mike Athans was long at MIT and Falb, at the Brown Division of Applied Mathematics. Control theory had a flurry of interest in the space program. There is an extension, differential game theory, which is like how best for a missile to chase an airplane where the airplane is best trying to avoid the missile. Typically in control theory and differential game theory we look for necessary conditions for optimality which usually turn out to be in terms of Lagrange multipliers. Optimization over time with randomness is essentially stochastic optimal control, and the main technique is a version of dynamic programming. An early proponent was Richard Bellman.

The Black-Scholes Nobel prize work in option pricing is a relatively simple case of stochastic optimal control.

So, in terms of MP1, constraint programming is, say, find x in R^n so that

g(x) that is, just find a point in the feasible region R.

An example might be, yesterday killed 5000 hogs and hung and chilled the results. Today cut the hogs and put the pieces in boxes in 18 wheel trucks, about 40,000 pounds of load per truck, for delivery to customers about 900 miles away.

Have only three bays in the loading dock. So, want to know the order in which to load the trucks to get them all loaded in time.

So, have a constraint satisfacation problem and not really an optimization problem.

Fundamentally, i.e., in the sense of computational complexity, finding a feasible point is as difficult as finding an optimal point. So, really, fundamentally, constraint programming is no easier than optimization.

IIRC at one time SAP and ILOG were interested in CPLEX as means of solving constraint satisfaction problems. So, to solve a constraint satisfaction problem, start with some optimization software.