Maximize the sum of logistic curves subject to linear and piecewise linear constraints
Approximate the logistic curves using piecewise linear functions.
Further explanation of this example: 'Xpress Python Reference Manual'
''' Maximize the sum of logistic curves subject to linear and piecewise linear constraints Approximate the logistic curves using piecewise linear functions (c) 2020 Fair Isaac Corporation ''' import numpy as np import xpress as xp import matplotlib.pyplot as plt def logistic(x, K, r, c): return K / (1 + np.exp(-r * (x - c))) n_curves = 10 N = range(n_curves) U = 10 # upper bound of the variables # Create two numpy vectors of variables x = xp.vars(N, ub=U, name='x') y = xp.vars(N, name='y') # Create a problem and add these two vectors p = xp.problem(x, y) n_intervals = 100 # define the breakpoints of the piecewise linear terms breakpoints = np.array([(U / n_intervals) * i for i in range(n_intervals + 1)]) # compute the function values at breakpoints y_vals = [logistic(breakpoints, U, np.random.uniform(0.5, 3), U / 2) for _ in N] # Enable to visualize curves for i in N: plt.plot(breakpoints, y_vals[i]) y_vals = np.array(y_vals).flatten().tolist() x_vals = np.array() for i in N: x_vals = np.concatenate((x_vals, breakpoints)) x_vals = x_vals.tolist() # Set the starting indices for the flattened piecewise linear function definitions start = [i * (n_intervals + 1) for i in N] # Add piecewise linear functions p.addpwlcons(x, y, start, x_vals, y_vals) # Add a constraint that limits the weighted sum of x variables w = np.random.randint(1, 10, n_curves) p.addConstraint(xp.Dot(w, x) <= 10) # Maximize the sum of logistic functions p.setObjective(xp.Dot(np.ones(n_curves), y), sense=xp.maximize) p.write('test_logistic.mps', 'mps') p.solve()
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