The analytic center for a set of inequalities is the minimizing position of . In particular it is often used with linear inequalities. It gives a reasonable and easily computable for convex constraint function center of the region. The hessian at that point can give you a reasonable ellipse that approximates the region too (both interior and exterior approximation).

I wrote a program for linear inequalities. It is not particularly robust. First I get a feasible point using the LP solver in scipy. Then I give the appropriate gradients and Hessians to a newton conjugate gradient solver in scipy. It does return a reasonable center, but I had to fiddle with some epsilons to avoid logarithms exploding and to avoid the hessian being so big it overwhelms the gradient. Possibly a couple burn in steps of gradient descent might help, or getting a feasible point that isn’t optimal since the optimal points being on the boundary is a huge problem. If the newton solver comes back with only 1 or 2 iterations, it probably failed.

import numpy as np import scipy.optimize # Anaytic Center Finding def analytic_center_fun(x, A, b): q = np.dot(A,x)-b + 1e-8 val = np.sum(np.log(q)) grad = np.sum((1./q).reshape((-1,1)) * A, axis = 0) hess = np.sum( (-1./(q*q)).reshape((-1,1,1)) * A.reshape((-1,-1,1)) * A.reshape((-1,1,-1)), axis = 0) return val, grad, hess def analytic_center_hess(A, b): def hess_fun(x): q = b - np.dot(A,x) + 1e-8 hess = np.sum( (1./(q*q)).reshape((-1,1,1)) * np.expand_dims(A, axis=2) * np.expand_dims(A, axis=1) , axis = 0) return hess return hess_fun def analytic_center_grad(A, b): def grad_fun(x): q = b - np.dot(A,x) + 1e-8 grad = np.sum((1./q).reshape((-1,1)) * A, axis = 0) return grad return grad_fun def analytic_center_val(A, b): def val_fun(x): q = b - np.dot(A,x) + 1e-8 val = - np.sum(np.log(q)) return val return val_fun def get_feasible(A,b): #returns a feasible point for Ax < b # need to make the inequalities a little tighter so that the logarithms aren't freaking out res = scipy.optimize.linprog(-np.random.random(A.shape[1]),A_ub = A, b_ub = b - 1e-5, method='interior-point') if res.success == True: return res.x else: print("failure to find feasible point ", res) return "FAIL" def analytic_center(A,b): print("Calulating Center") x0 = get_feasible(A,b) print(x0) #xc = scipy.optimiatzew #xc, fopt, fcalls, gcalls, hcalls, warn # Had a problem where the hessian is too big at the boundary. Let it stop there # The stopping condition is based on delta x rather than grad? # Could maybe do some burn in with a couple gradient steps. res = scipy.optimize.fmin_ncg(f = analytic_center_val(A,b), x0 = x0, fprime = analytic_center_grad(A,b), fhess = analytic_center_hess(A,b) ) xc = res print(res) print(xc) return xc # Make a convenient box where I know where the cneter will be A = np.array([[1.0 , 0], [-1.0, 0], [0 , 1.0], [0. , -1.0]]) b = np.array([20,10,1,0]) #analytic_center(np.random.random((100,10)), np.random.random(100)) print(analytic_center_val(A,b)([0.5,0.5])) print(analytic_center_grad(A,b)([0.5,0.5])) print(analytic_center_hess(A,b)([0.5,0.5])) print(analytic_center_val(A,b)([0,0])) print(analytic_center_grad(A,b)([0,0])) print(analytic_center_hess(A,b)([0,0])) sol = analytic_center(A,b) print(analytic_center_val(A,b)(sol)) print(analytic_center_grad(A,b)(sol)) print(analytic_center_hess(A,b)(sol))