0Constrained optimization helps find the best solution while following rules or limits. It is useful when you want to maximize or minimize something but must stay within certain boundaries.
from scipy.optimize import minimize result = minimize( fun, # function to minimize x0, # initial guess method='SLSQP', # optimization method constraints=constraints, # constraints as dict or list bounds=bounds # variable bounds )
fun is the function you want to minimize.
constraints can be equality or inequality conditions.
from scipy.optimize import minimize def fun(x): return x[0]**2 + x[1]**2 constraints = [{'type': 'eq', 'fun': lambda x: x[0] + x[1] - 1}] x0 = [0, 0] result = minimize(fun, x0, method='SLSQP', constraints=constraints) print(result.x)
from scipy.optimize import minimize fun = lambda x: (x[0]-1)**2 + (x[1]-2.5)**2 constraints = [ {'type': 'ineq', 'fun': lambda x: x[0] - 1}, {'type': 'ineq', 'fun': lambda x: 2 - x[0]}, {'type': 'ineq', 'fun': lambda x: x[1] - 0}, {'type': 'ineq', 'fun': lambda x: 2 - x[1]} ] x0 = [2, 0] result = minimize(fun, x0, method='SLSQP', constraints=constraints) print(result.x)
This program finds the values of x and y that minimize the distance to point (3, -1) while following two rules: x + 2y must equal 2, and x must be at least 1 more than y.
from scipy.optimize import minimize # Objective function: minimize (x-3)^2 + (y+1)^2 # Constraints: x + 2y = 2 (equality), x - y >= 1 (inequality) def objective(x): return (x[0] - 3)**2 + (x[1] + 1)**2 constraints = [ {'type': 'eq', 'fun': lambda x: x[0] + 2*x[1] - 2}, {'type': 'ineq', 'fun': lambda x: x[0] - x[1] - 1} ] x0 = [0, 0] # initial guess result = minimize(objective, x0, method='SLSQP', constraints=constraints) print('Optimal values:', result.x) print('Objective function value:', result.fun)
Use 'SLSQP' method for problems with constraints.
Constraints are dictionaries with 'type' ('eq' or 'ineq') and a function 'fun' that returns zero or positive values.
Initial guess x0 can affect the solution found.
Constrained optimization finds the best solution within limits.
Use scipy.optimize.minimize with constraints and bounds.
Define constraints as functions returning zero (equalities) or positive (inequalities).
scipy.optimize.minimize?minimize function in scipy can include constraints and bounds to ensure the solution respects these limits.scipy.optimize.minimize?fun(x) >= 0 means the constraint holds.lambda x: x[0] - 1 means x[0] - 1 >= 0 or x[0] >= 1, correct for the constraint.1 - x[0] >= 0 or x[0] <= 1, which is a different constraint.from scipy.optimize import minimize
fun = lambda x: (x[0]-2)**2 + (x[1]-3)**2
cons = ({'type': 'eq', 'fun': lambda x: x[0] + x[1] - 5})
res = minimize(fun, [0, 0], constraints=cons)
print(res.x.round(2))x[0] + x[1] = 5.(x0-2)^2 + (x1-3)^2 with x0+x1=5. Using substitution: x1=5-x0.(x0-2)^2 + (5-x0-3)^2 = (x0-2)^2 + (2 - x0)^2.(x0-2)^2 + (2-x0)^2 = 2*(x0-2)^2.x0=2, then x1=3, which satisfies the constraint.from scipy.optimize import minimize
fun = lambda x: x[0]**2 + x[1]**2
cons = [{'type': 'ineq', 'fun': lambda x: x[0] + x[1] - 1}]
res = minimize(fun, [0, 0], constraints=cons)
print(res.x)x[0] + x[1] - 1 >= 0, or x[0] + x[1] >= 1.f(x) = (x[0]-1)^2 + (x[1]-2)^2 with constraints x[0] >= 0, x[1] >= 0, and x[0] + 2*x[1] = 3. Which setup correctly applies these constraints using scipy.optimize.minimize?x[0] >= 0 and x[1] >= 0 are bounds: [(0, None), (0, None)] means lower bound 0, no upper bound.x[0] + 2*x[1] = 3 is expressed as fun(x) = x[0] + 2*x[1] - 3 = 0 with type 'eq'.Pass the test and this pattern moves unknown strong while earning +1 Grip toward your certificate.