nlopt

import nlopt import numpy as np # Objective function to minimize: f(x) = sqrt(x[1]) # subject to x[1] >= (a*x[0] + b)**3 and x[1] >= 0 # for a1=2, b1=0, a2=-1, b2=1 def myfunc(x, grad): if grad.size > 0: grad[0] = 0.0 grad[1] = 0.5 / np.sqrt(x[1]) return np.sqrt(x[1]) def myconstraint(x, grad, a, b): if grad.size > 0: grad[0] = 3 * a * (a * x[0] + b)**2 grad[1] = -1.0 return (a * x[0] + b)**3 - x[1] # Problem dimension n = 2 # Create an optimizer object opt = nlopt.opt(nlopt.LD_MMA, n) # Set minimization objective opt.set_min_objective(myfunc) # Set bounds opt.set_lower_bounds([-float('inf'), 0.0]) # x[1] >= 0 # Add nonlinear inequality constraints (h(x) <= 0) # Constraint 1: x[1] >= (2*x[0])**3 => (2*x[0])**3 - x[1] <= 0 opt.add_inequality_constraint(lambda x, grad: myconstraint(x, grad, 2.0, 0.0), 1e-8) # Constraint 2: x[1] >= (-1*x[0] + 1)**3 => (-1*x[0] + 1)**3 - x[1] <= 0 opt.add_inequality_constraint(lambda x, grad: myconstraint(x, grad, -1.0, 1.0), 1e-8) # Set stopping criteria opt.set_xtol_rel(1e-4) opt.set_maxeval(1000) # Initial guess x0 = np.array([1.234, 5.678]) try: x_opt = opt.optimize(x0) minf = opt.last_optimum_value() result_code = opt.last_optimize_result() print(f"Optimized result: x = {x_opt}, f(x) = {minf}, return code: {result_code}") except nlopt.RunTimeError as e: print(f"NLopt failed: {e}") # Example of using a derivative-free algorithm opt_df = nlopt.opt(nlopt.LN_COBYLA, n) opt_df.set_min_objective(myfunc) # Note: grad argument will be empty for derivative-free opt_df.set_lower_bounds([-float('inf'), 0.0]) opt_df.add_inequality_constraint(lambda x, grad: myconstraint(x, grad, 2.0, 0.0), 1e-8) opt_df.add_inequality_constraint(lambda x, grad: myconstraint(x, grad, -1.0, 1.0), 1e-8) opt_df.set_xtol_rel(1e-4) try: x_opt_df = opt_df.optimize(x0) minf_df = opt_df.last_optimum_value() print(f"Derivative-free result: x = {x_opt_df}, f(x) = {minf_df}") except nlopt.RunTimeError as e: print(f"NLopt (derivative-free) failed: {e}")