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from torch.autograd import Function
def _safe_norm(x, delta=1e-12):
return torch.sqrt(torch.dot(x, x) + x.new_tensor(delta))
def _newton_x(P, q, A, b, G, h, s, lam, nu, t, mu, rho, newton_max=30, newton_tol=1e-10, reg=1e-9):
n = P.shape[0]
I = torch.eye(n, device=P.device, dtype=P.dtype)
Hlin = P + rho * (A.T @ A) + rho * (G.T @ G)
rhs_lin = q + A.T @ lam - rho * (A.T @ b) + G.T @ nu + rho * (G.T @ (s - h))
x = torch.zeros(n, device=P.device, dtype=P.dtype)
for _ in range(newton_max):
r = _safe_norm(x)
u = x / r
c = r - 1.0 + t
alpha = mu + rho * c
F = Hlin @ x + rhs_lin + alpha * u
if torch.linalg.norm(F) <= newton_tol * (1.0 + torch.linalg.norm(rhs_lin)):
break
M = Hlin + rho * (u[:, None] @ u[None, :]) + alpha * ((1.0 / r) * I - (1.0 / (r**3)) * (x[:, None] @ x[None, :]))
M = M + reg * I
dx = torch.linalg.solve(M, -F)
step = 1.0
x_new = x + step * dx
if not torch.isfinite(x_new).all():
step = 0.5
x_new = x + step * dx
x = x_new
r = _safe_norm(x)
u = x / r
c = r - 1.0 + t
alpha = mu + rho * c
M = Hlin + rho * (u[:, None] @ u[None, :]) + alpha * ((1.0 / r) * I - (1.0 / (r**3)) * (x[:, None] @ x[None, :]))
M = M + reg * I
return x, M, u
def alt_diff_qp_ball(P, q, A, b, G, h, rho=50.0, eps=1e-8, max_iter=2000):
if rho <= 0:
raise ValueError("rho must be > 0")
rho = torch.tensor(float(rho), device=q.device, dtype=q.dtype)
n = P.shape[0]
m = b.shape[0]
p = h.shape[0]
x = torch.zeros(n, device=q.device, dtype=q.dtype)
s = torch.zeros(p, device=q.device, dtype=q.dtype)
t = torch.zeros((), device=q.device, dtype=q.dtype) # scalar slack
lam = torch.zeros(m, device=q.device, dtype=q.dtype)
nu = torch.zeros(p, device=q.device, dtype=q.dtype)
mu = torch.zeros((), device=q.device, dtype=q.dtype) # scalar multiplier
Jx = torch.zeros((n, n), device=q.device, dtype=q.dtype)
Js = torch.zeros((p, n), device=q.device, dtype=q.dtype)
Jt = torch.zeros((1, n), device=q.device, dtype=q.dtype)
Jlam = torch.zeros((m, n), device=q.device, dtype=q.dtype)
Jnu = torch.zeros((p, n), device=q.device, dtype=q.dtype)
Jmu = torch.zeros((1, n), device=q.device, dtype=q.dtype)
I = torch.eye(n, device=q.device, dtype=q.dtype)
for _ in range(max_iter):
x, M, u = _newton_x(P, q, A, b, G, h, s, lam, nu, t, mu, rho)
term = I
if m:
term = term + A.T @ Jlam
if p:
term = term + G.T @ Jnu + rho * (G.T @ Js)
term = term + (u[:, None] @ Jmu) + rho * (u[:, None] @ Jt)
Jx = torch.linalg.solve(M, -term)
# s-step
if p:
pre = -(nu / rho) - (G @ x - h)
s_new = torch.relu(pre)
gate_s = (s_new > 0).to(q.dtype)
Js = gate_s[:, None] * (-(1.0 / rho) * Jnu - (G @ Jx))
s = s_new
# t-step (scalar)
r = _safe_norm(x)
g = 1.0 - r - (mu / rho)
t_new = torch.relu(g)
gate_t = (t_new > 0).to(q.dtype) # scalar 0/1
Jt = gate_t * (-(u[None, :] @ Jx) - (1.0 / rho) * Jmu)
t = t_new
# dual updates
if m:
lam = lam + rho * (A @ x - b)
Jlam = Jlam + rho * (A @ Jx)
if p:
nu = nu + rho * (G @ x + s - h)
Jnu = Jnu + rho * ((G @ Jx) + Js)
mu = mu + rho * (r - 1.0 + t)
Jmu = Jmu + rho * ((u[None, :] @ Jx) + Jt)
# stopping (primal residuals)
r_eq = torch.linalg.norm(A @ x - b) if m else x.new_tensor(0.0)
r_in = torch.linalg.norm(G @ x + s - h) if p else x.new_tensor(0.0)
r_ball = torch.abs(r - 1.0 + t)
if (r_eq + r_in + r_ball) <= eps:
break
return x, Jx
def AltDiffLayer(eps=1e-8, max_iter=2000, rho=50.0):
class L(Function):
@staticmethod
def forward(ctx, P, q, G=None, h=None, A=None, b=None):
def bat(x, d):
if x is None: return None
return x.unsqueeze(0) if x.dim() == d else x
P = bat(P, 2); q = bat(q, 1)
B, n = q.shape
A = bat(A, 2) if A is not None else None
b = bat(b, 1) if b is not None else None
G = bat(G, 2) if G is not None else None
h = bat(h, 1) if h is not None else None
if A is None or A.numel() == 0: A = torch.empty((1, 0, n), device=q.device, dtype=q.dtype)
if b is None or b.numel() == 0: b = torch.empty((1, 0), device=q.device, dtype=q.dtype)
if G is None or G.numel() == 0: G = torch.empty((1, 0, n), device=q.device, dtype=q.dtype)
if h is None or h.numel() == 0: h = torch.empty((1, 0), device=q.device, dtype=q.dtype)
if A.shape[0] == 1 and B > 1: A = A.expand(B, -1, -1)
if b.shape[0] == 1 and B > 1: b = b.expand(B, -1)
if G.shape[0] == 1 and B > 1: G = G.expand(B, -1, -1)
if h.shape[0] == 1 and B > 1: h = h.expand(B, -1)
P_det = P.detach(); q_det = q.detach()
A_det = A.detach(); b_det = b.detach(); G_det = G.detach(); h_det = h.detach()
xs, Js = [], []
for i in range(B):
xi, Ji = alt_diff_qp_ball(P_det[i], q_det[i], A_det[i], b_det[i], G_det[i], h_det[i],
rho=rho, eps=eps, max_iter=max_iter)
xs.append(xi); Js.append(Ji)
X = torch.stack(xs, 0)
J = torch.stack(Js, 0)
ctx.save_for_backward(J)
ctx.unbatched = (q.dim() == 1 and P.dim() == 2)
return X[0] if ctx.unbatched else X
@staticmethod
def backward(ctx, grad_output):
(J,) = ctx.saved_tensors
g = grad_output.unsqueeze(0) if grad_output.dim() == 1 else grad_output
grad_q = torch.zeros_like(g)
for i in range(g.shape[0]):
grad_q[i] = J[i].T.mv(g[i])
if grad_output.dim() == 1:
grad_q = grad_q[0]
return None, grad_q, None, None, None, None
return L.apply
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