Python线性编程

Question

I am trying to solve the following equation:

maximize x^{T}Ax where x is a 3 X 1 vector of the variables to be maximized and A is a 3 X 3 matrix of values.

So basically x^{T} = [a,b,c] which are the unknowns to be maximized and A could be something like

A = [ [29, 29, 79], [28, 28, 48], [9, 40, 0 ]]

Could someone show me how to represent this in the form of a maximization problem using PuLP or some other linear programming package in python?

Any help would be much appreciated. I am extremely new to this area and have not idea how to get started representing this formulation.

I have so far tried to use CVXPY to model this function. I have the following code but am seeing an error:

    [1] A = np.array([[29,29,79],[28,28,48],[9,40,0]])
    [2] x=Variable(3)
    [3] objective=Minimize(x.T*A*x)
     Warning: Forming a nonconvex expression (affine)*(affine).
  warnings.warn("Forming a nonconvex expression (affine)*(affine).")
    [4] constraints=[0<=x,x<=1,sum_entries(x)==1] #what I'm trying to say is each entry of x should be between 0 and 1 and all entries should add up to 1.
    [5] prob = Problem(objective, constraints)
    [6] prob.solve()
    DCPError: Problem does not follow DCP rules.

Answer 1

I don't believe PuLP supports quadratic programming (QP). Your model is quadratic and PuLP is only for linear programming models (LPs and MIPs). There are quite a few options to express QPs in Python. High performance commercial solvers often provide Python bindings, and otherwise you can look at for example CVXOPT . Notice that only convex QPs are "easy" to solve. If you have a non-convex QP things become much more difficult and you may need to look at a global solver (there not as many of those type of solvers). Non-convex QPs can be reformulated via the KKT conditions as a linear MIP model, although these models may not always perform very well.