Linear Programming / Least Square / Quadratic Programming - Will they always find the solution?

Dear Mathcentral community,

I need to investigate if a solution to the constrained problem A*x = b exists.

The constraints are that x is bounded by a lower and upper vector and that B*x < c

My idea is to just select any arbitrary objective function and to let Matlab solve the constrained optimization problem using e.g. Linear programming, quadratic programming or the least square optimization.

My question now is: Will these algorithms always find one constrained solution if one solution exists that fullfills the constraints?

And is there a fast way to check if a solution exists? At the moment I have just set the OptimalityTolerance to the max value, such that the optimization stops as soon as possible, but I am not sure if there is a faster way. E.g. with rank([A,b]) == rank(A) one can check if a solution exists, but this is only valid for unconstrained problems.

I am quite sure that linear programming will always return a solution if a solution exists, but I am not so sure about the other ones. I would say that, as the area for these problems needs to be convex in the first place, the algorithm will always find a solution if existend, but I want to be sure about that.

Thank you guys very much in advance!

Greetings Sebastian