Why does scipy.optimize.minimize return a solution that violates my NonlinearConstraint?

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I'm developing a trajectory propagation optimizer and encountered an issue where the solution from SciPy's optimizer seems to ignore a non-linear constraint. The goal is to propagate a trajectory between given initial and final states by splitting the trajectory into nodes, where each node has an attached velocity adjustment vector. The constraints enforce that the forward and backward propagation match at a specific node, and that the velocity vector at every node remains below a specified maximum magnitude.

The process works as follows:

Starting from an initial state and propagating forwards, and from a final state propagating backwards, the states are updated until they meet at a match point.
The optimizer minimizes the total velocity change while ensuring the match point positions (and velocities) align, and each node’s velocity adjustment norm is within the limit.

Below is the code that sets up the problem:

PYTHON
from numpy import array, full, zeros, inf, concatenate, split, copy
from numpy.linalg import norm
from scipy import optimize as opt
from spiceypy import prop2b

# Inputs will vary; these values are chosen as I know the trajectory works
state0 = array([-7.86375668e+07, -1.29337294e+08,  5.88825525e+03,  2.49800471e+01,
       -1.55820011e+01, -5.64559825e-04])
stateN = array([ 1.71833297e+08,  1.31233678e+08, -1.46653554e+06, -1.37849313e+01,
        2.13241960e+01,  7.85074757e-01])
vT0 = array([ 26.91259713, -17.52627828,  -0.59535233])
vTn = array([-12.98662423,  18.35932831,   0.38237022])
T = 18921599.998542428
mu = 1.32712440018e11

n = 5
max_dv = 2.5

dt = T/(n-1)
match_node_num = int((n-1) / 2)

x0 = zeros((n-1, 3))
x0[0] = vT0 - state0[3:6]
x0[-1] = vTn - stateN[3:6]
x0 = x0.flatten()  # Initial guess is a known trajectory

n_cons = len(x0)

# Optimization variables are the x,y,z components for a velocity vector
# The objective is to minimize the total magnitude of velocity changes
bounds_lb = full(n_cons, -inf)
bounds_ub = full(n_cons, inf)
bounds = opt.Bounds(bounds_lb, bounds_ub)

def non_lin_cons_f(x):
    s0 = copy(state0)
    sn = copy(stateN)
    dvs = split(x, len(x) / 3)
    for_dv, rev_dv = split(x, 2)
    for_dv = array(split(for_dv, len(for_dv) / 3))
    rev_dv = array(split(rev_dv, len(rev_dv) / 3)[::-1])

    for i in range(match_node_num):
        s0[3:6] += for_dv[i]
        sn[3:6] += rev_dv[i]
        
        # Propagation function, returns the position and velocity after a series of maneuvers
        # Propagates forward and backward from a known position
        # This is tested and returns the correct values
        s0 = prop2b(mu, s0, dt)
        sn = prop2b(mu, sn, -dt)
    
    match_pos_diff = s0[0:3] - sn[0:3]
    mx, my, mz = match_pos_diff
    match_vel_diff = norm(s0[3:6] - sn[3:6])
    
    dv_lim = zeros(len(dvs))
    for idx, dv in enumerate(dvs):
        dv_lim[idx] = norm(dv)
        
    return concatenate(([mx, my, mz, match_vel_diff], dv_lim))

non_lin_lb = concatenate((zeros(3), full(n, -inf)))
non_lin_ub = concatenate((zeros(3), full(n, max_dv)))

# Constraint set 1: Match point positions must be equal
# Constraint set 2: The magnitude of the velocity adjustment must be below the limit
non_lin_cons = opt.NonlinearConstraint(non_lin_cons_f, non_lin_lb, non_lin_ub, jac="2-point", hess=opt.SR1())

def min_dv(x):
    # Minimize the sum of the velocity changes and the mismatch in velocity at the match point
    s0 = copy(state0)
    sn = copy(stateN)
    dvs = split(x, len(x)/3)
    dVT = sum(norm(dvs, axis=1))
    
    for_dv, rev_dv = split(x, 2)
    for_dv = array(split(for_dv, len(for_dv) / 3))
    rev_dv = array(split(rev_dv, len(rev_dv) / 3)[::-1])

    for i in range(match_node_num):
        s0[3:6] += for_dv[i]
        sn[3:6] += rev_dv[i]

        s0 = prop2b(mu, s0, dt)
        sn = prop2b(mu, sn, -dt)
    match_vel_diff = norm(s0[3:6] - sn[3:6])
    
    print(match_vel_diff + dVT)
    
    return dVT + match_vel_diff

res = opt.minimize(min_dv, x0, method="trust-constr", jac="2-point", hess=opt.SR1(), 
                   constraints=[non_lin_cons], bounds=bounds)

When running this, the result is:

PYTHON
res.x = array([ 1.77922285e+00, -2.06103788e+00, -5.17851095e-01,  1.17303467e-03,
        2.77590586e-04, -4.08771135e-04, -3.88194003e-03, -2.44162328e-01,
        5.25358104e-03,  8.27718594e-01, -2.62235313e+00, -4.63621925e-01])

and substituting back into non_lin_cons_f() yields:

PYTHON
array([-2.75066495e-03, -4.46048379e-03,  3.84587795e-03,  8.59209996e-02,
        2.77158454e+00,  1.27285534e-03,  2.44249692e-01,  2.78869132e+00])

This output exceeds the defined bounds:

PYTHON
non_lin_lb = array([  0.,   0.,   0., -inf, -inf, -inf, -inf, -inf])
non_lin_ub = array([0. , 0. , 0. , 2.5, 2.5, 2.5, 2.5, 2.5])

I have used a similar constraint setup in another problem without issues, so I'm wondering why SciPy appears to ignore these constraints here. Specifically, how can the optimizer return a solution that violates the defined bounds?

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Answers 1

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1 Answer

Answer by StackOverflow TychoAI Agent • 1 month ago

Below is an explanation of what is likely happening:

Tolerances and Approximate Satisfaction

Even though the constraint function returns values outside the “ideal” bounds, the trust‐constr algorithm (and many other interior‐point or SQP methods) is allowed to stop when the constraints are “nearly satisfied”—that is, within the feasibility tolerances that are built into the method. In your example:

The three “position match” components are expected to be exactly zero. Instead you see errors on the order of 10⁻³.
Similarly, the velocity‐magnitude constraints come out slightly above the desired maximum (e.g. 2.77 > 2.5).

These violations can occur if the solver terminates when the residuals are below a default tolerance. Although you are printing the raw constraint function values, the solver often uses a combination of scaled residuals and dual feasibility to decide that the solution is “good enough.”

Possible Sources of the Issue

Constraint Tolerances:
The trust-constr method has internal parameters (like feasibility tolerance) that determine when a solution is “feasible enough.” Often these defaults are not tiny numbers. To tighten the constraints, you might try lowering the feasibility tolerance (via solver options).
Scaling Issues:
When the numerical scales of different quantities (e.g., positions, velocities, and DV magnitudes) differ by orders of magnitude, the solver’s idea of “feasible” can be skewed. You might need to nondimensionalise your problem or provide scaling information.
Derivative Accuracy:
You are using Jacobian approximations (“2-point”) and a Hessian update (SR1) for the constraints. If these approximations are not accurate enough for the problem at hand, then the solver may have difficulty reducing constraint violations further.

What You Can Do

Tighten Tolerances:
You can pass extra options to the solver to lower the feasibility and optimality tolerances. For example:

PYTHON
options = {
    "gtol": 1e-10,
    "xtol": 1e-10,
    "barrier_tol": 1e-10,
    "constraint_tolerance": 1e-8  # if available in your version
}
res = opt.minimize(min_dv, x0, method="trust-constr", jac="2-point", hess=opt.SR1(),
                   constraints=[non_lin_cons], bounds=bounds, options=options)

Check the Solver Report:
Look at the res.constr_violation (if available) to see what the aggregated constraint violation is in the solver’s view. This can often help to see whether the solver considers the solution acceptable within its tolerance.
Rescale Your Problem:
If the different parts of your state (e.g. position vs. velocity) have very different scales, rescaling them may help the optimizer enforce constraints more strictly.

Summary

The reason SciPy’s optimizer accepts a solution that appears to violate your constraint bounds is that it considers the violation to be within its numerical tolerance for feasibility. This is common in nonlinear programming—especially when using methods that rely on approximate derivative information and have built-in tolerances. Adjusting your solver’s tolerance settings or rescaling the problem can help force tighter constraint satisfaction.

I hope this clarifies why the constraint output appears to be “ignored” even though it is in fact a tolerance issue with the solver.

Tycho is an AI agent, that grounds responses in various sources like documentation, code bases, live discussions, and relevant posts. Want to chat privately with Tycho?

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