How to generate a random integer array satisfying complex constraints

constraints = (2.3 x1 + 4.1 x2 + 2.2 x3 + 3.5 x4 + 1.8 x5 <= 
     50) && (0 <= x1 <= 5 && 0 <= x2 && 1 <= x3 <= 5 && 1 <= x4 && 
     0 <= x5) && (2 <= x3 + x4 <= 5);

vars = {x1, x2, x3, x4, x5};

Use constraints and vars to define an ImplicitRegion:

ir = ImplicitRegion[constraints, Evaluate@vars];

Use RandomPoint[region, n] to draw n random points from region:

sample = RandomPoint[ir, 3]

{{0.569525,0.655619,1.19525,2.5036,15.6858},{0.666914,5.79318,1.71149,2.1047,7.46705},{3.56459,3.98713,3.0594,1.16909,7.3919}}

Verify that each point satisfies the constraints:

constraints /. Thread[vars -> #] & /@ sample

{True, True, True}

Update: "to generate integer points in the region"

Two approaches:

1. Generate all integer tuples inside the cuboid that contains ir and select the ones that satisfy the constraints; then sample from the resulting list of tuples.

feasibleQ = Function[Evaluate@vars, Evaluate@constraints];

integertuples = Tuples @ MapThread[Range[Ceiling@#, Floor@#2] &, 
    Transpose @ RegionBounds[ir]];

integerfeasible = Select[Apply[feasibleQ]]@integertuples

Length@integerfeasible

5319

RandomChoice[integerfeasible, 3]

{{4, 1, 1, 1, 12}, {0, 4, 2, 3, 9}, {0, 0, 3, 1, 3}}

feasibleQ @@@ %

{True, True, True}

2. Rationalize the constraints and use Solve:

solutions = vars /. Solve[Rationalize@constraints, vars, Integers];

Length@solutions

5319

feasibleQ = Function[Evaluate@vars, Evaluate@constraints];

And @@ (feasibleQ @@@ solutions)

True

RandomChoice[solutions, 3]

{{0, 5, 4, 1, 8}, {3, 1, 2, 1, 4}, {2, 0, 1, 2, 1}}

Jumping off of @kglr's answer (pre-edit of the integer method), one can use FindInstance with RandomSeeding->Automatic to accomplish exactly the OP's updated task. I am not sure if you can get more efficient than this, as it takes negligible time on my PC for the production of a list of 10 Integer sets using RandomSeeding->Automatic. I suppose it would depend on your precise use case if this is efficient enough.

vars /. FindInstance[vars ∈ ir, vars, Integers, 10, RandomSeeding -> Automatic]

{{1, 4, 1, 1, 9}, {0, 3, 2, 1, 8}, {3, 0, 3, 1, 0}, {4, 1, 1, 3, 9}, {5, 0, 2, 3, 1}, {4, 1, 1, 1, 4}, {0, 0, 2, 3, 0}, {5, 3, 2, 3, 0}, {2, 1, 1, 2, 15}, {5, 0, 1, 2, 8}}

Here, I use 10 for the number of produced sets as not defining it, i.e., leaving it as 1, only produces the same solution each time.

vars /. FindInstance[vars ∈ ir, vars, Integers, RandomSeeding -> Automatic]

{{0, 0, 1, 1, 0}}

kglr's method is faster:

(vars /. FindInstance[vars ∈ ir, vars, Integers, 10, 
 RandomSeeding -> Automatic]) // feasibleQ @@@ # & // AbsoluteTiming

(integertuples = 
Tuples@MapThread[{Ceiling@#, Floor@#2} &, 
  Transpose@RegionBounds[ir]];
feasibleQ = Function[Evaluate@vars, Evaluate@constraints];
integerfeasible = Select[Apply[feasibleQ]]@integertuples;
RandomChoice[integerfeasible, 10]) // feasibleQ @@@ # & // AbsoluteTiming

{0.350308, {True, True, True, True, True, True, True, True, True, True}}
{0.0725979, {True, True, True, True, True, True, True, True, True, True}}

Both methods yield appropriate results, however.

With an updated definition of integertuples

integertuples = Tuples@MapThread[Range[Ceiling@#, Floor@#2] &, Transpose@RegionBounds[ir]];

Timing increases slightly

{0.215071, {True, True, True, True, True, True, True, True, True, True}}

However, if one is timing these appropriately by not computing integertuples again each time upon evaluation, one gets

RandomChoice[oldintegerfeasible, 10] // feasibleQ @@@ # & // AbsoluteTiming
RandomChoice[integerfeasible, 10] // feasibleQ @@@ # & // AbsoluteTiming

{0.0000818, {True, True, True, True, True, True, True, True, True, True}}
{0.0000852, {True, True, True, True, True, True, True, True, True, True}}

Illustrating comparable timings between versions of kglr's integertuples (definition shown above is oldintegertuples with oldintegerfeasible in comparison to kglr's correction using Range, which gives all solutions held within the cuboid.)