[Math] What are common Mathematical Programming Languages out there

integer programminglinear programmingoptimization

I've seen the term used Mathematical Programming to describe a superset of:

Linear programming
Quadratic programming
Nonlinear programming
Mixed-integer programming
Mixed-integer nonlinear programming
Second-order cone programming
Global optimization
Constraint programming

I'm looking for a list of programming languages that your can use for solving such problems.

NOTE:

Although it is true that the word "programming" as used in optimization doesn't mean "computer programming". And, languages like Matlab, python, C++, Julia, etc, can be used to solve optimization problems
What I'm looking for a language that allows the easy specification of Math Programming problems and may (Or may-not) use a Matlab, python, C++, Julia, etc. backend for the actual solution

Best Answer

Matematical "programming" doesn't mean the same thing as computer programming. Programming in this context simply means optimization.

A good choice that is pretty simple to learn, mainly because its syntax very closely resembles mathematical notation, is AMPL. It is built to model exactly the problems you mention. Note that AMPL is only a modeling language however - you will need additional software to actually solve the problems. These are compatible with AMPL and not hard to set up either. CPLEX is an example of a commercial solver, but there are also free solvers available, like Gurobi.

Related Solutions

MINLP problem with linear constraints and piecewise term

Let me start by saying that my solution is in no way the best solution. In fact, it's probably the second worst next to brute force, but it might be a solution nonetheless.

Have you tried a genetic algorithm based on bit-string affinity? The main idea is:

You generate a "population", or rather a bunch of randomly generated solutions in a (non-feasible) space. Each individual in the population is encoded into a bit-string.
Evaluate each of these solutions against a fitness function (the objective) and get rid of the worst 50%.
Create a new set of individuals based on some product rule to take the place of the ones you deleted.
Repeat from step 1 until ~90% of the bit-strings match.

Because you use a bit-string encoding instead of smooth numbers, you can choose the number of bits to encode a variable so they are truly integers. i.e. If you use $2^3$ bits, since $2^3 = 8$ you have 8 potential values the parameter can take on by choosing a lower and upper bound as $0$ and $7$ respectively. the parameters can only take on integer values by definition. It also doesn't assume differentiability or linearity so those points are satisfied. You can read more about this type of algorithm here, and I've also programmed an open-source GA based on bit-string affinity here.

Some notes:

You'll need to implement the constraints into the cost function yourself using some sort of large penalty multiplier. So $x^Tc \leq 3000$ will become:

if x[i]*c[i] > 3000:
    cost += 9001

or some other massive penalty.

I've personally used this to solve a medium-sized problem with integer parameters, nonlinear objective, and nonlinear constraints so I know for a fact it can solve your problem. However, I have the freedom to run it off-line and can let it chug away overnight. If time is a constraint and you want a solution <5 minutes, then ignore my response.
If you do decide this is the best approach, there's probably quite a few bugs in my code, but I'm willing to work with you. Just post an "issue" on github describing the problem and I'll get to it as soon as I can.

Best Answer

Related Solutions

MINLP problem with linear constraints and piecewise term

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