How to set linear constraints on elements of a long list

Question

I would like to minimize over quite large set of $a_i$ some function $f(a_1,a_2,\dots,a_n)$ subject to a set of $k$ linear constraints: $$ i=1\dots n: \quad a_i\ge0;\quad j=1\dots k:\quad\sum_{i=1}^n m_{ji}a_i\ge b_j,\tag1 $$ where the matrix $m$ and the vector $b$ are given. This is very similar to LinearProgramming, except for the art of the minimized function. Ideally the command should have 3 parameters: $f,m,b$ and probably the size of the vector $n$, though the latter can be read from the dimensions of the matrix $m$ and should return the minimized value of $f$ and the corresponding minimizing vector.

I can find the solutions for small $m$ with Minimize, but it is not realistic to type all the constraints by hand.

For test purpose the function $f$ can be considered just quadratic $f=a.M.a$.

Any help is appreciated.

Answer 1

You can specify linear inequality constraints succinctly using VectorGreaterEqual or using Element + NonNegativeReals:

constraints1 = Element[{m.a - b, a}, NonNegativeReals];

constraints2 = VectorGreaterEqual /@ {{m.a, b}, {a, 0}}; 

Example:

SeedRandom[1];
ClearAll[a, b, m]

n = k = 4;
m = RandomReal[4, {n, k}];
a = Array[Symbol["a" <> ToString[#]] &, n]
b = RandomReal[{-3, 3}, k];

obj = a.m.a;

sol1 = Minimize[{obj, constraints1}, a]

{0.553132, {a1 -> 0., a2 -> 0., a3 -> 0.807969, a4 -> 0.}} 

We get the same result using constraints2:

sol2 = Minimize[{obj, constraints2}, a]

{0.553132, {a1 -> 0., a2 -> 0., a3 -> 0.807969, a4 -> 0.}}