generalized_stiefel_np#
CLASS generalized_stiefel_np(var_shape, B = lambda X:X )
This manifold class defines the generalized Stiefel manifold, i.e.
where \(B \in \mathbb{R}^{n\times n}\) is a positive-definite matrix.
Parameters:#
var_shape ( (int, int) ) – The shape of the variables of the manifold.
len(var_shape)must equal to \(2\).B = lambda X:X: (callable )– The function that maps \(X\) to \(BX\).
Attributes:#
A(x) (callable)
The constraint dissolving mapping \(\mathcal{A}(x)\). A(X) is set as 1.5 * X - X @ (X.T @ self.B(X) /2).
C(X) (callable)
Describe the constraints \(c\). C(X) returns X.T @ self.B(X) - self.Ip.
m2v(x) (callable)
Flatten the variable of the manifold.
v2m(x) (callable)
Recover flattened variables to its original shape as variable_shape.
Init_point(Xinit = None) (callable)
Generate the initial point.
tensor2array(x) (callable)
Transfer the variable of the manifold to the numpy Nd-array while keep its shape. Default settings are provided in the core.backbone_autograd.
array2tensor(x) (callable)
Transfer the numpy Nd-array to the variable of the manifold while keep its shape. Default settings are provided in the core.backbone_autograd.
JC(x, lambda) (callable)
The Jacobian of C(x).
JC_transpose(x, lambda) (callable)
The transpose of \(J_c(x)\), expressed by matrix-vector production.
JA(x, d) (callable)
The transposed Jacobian of \(\mathcal{A}(x)\).
JA_transpose(x, d) (callable)
The transpose (or adjoint) of JA(x), i.e. \(\lim_{t \to 0} \frac{1}{t}(J_A(x+td) -J_A(x)) \).
C_quad_penalty(x) (callable)
Returns the quadratical penalty term \(||c(x)||^2\).
hessA(X, U, D) (callable)
Returns the Hessian of \(\mathcal{A}(x)\) in a tensor-vector product form.
hess_feas(X, D) (callable)
Returns the hessian-vector product of \(\frac{1}{2} ||c(x)||^2\).
Feas_eval(X) (callable)
Returns the feasibility of \(x\), measured by value of \(||c(x)||\).
Post_process(X) (callable)
Return the post-processing for X to achieve a point with better feasibility.
generate_cdf_fun(obj_fun, beta) (callable)
Return the function value of the constraint dissolving function. obj_fun is a callable function that returns the value of \(f\) at \(x\). beta is a float object that refers to the penalty parameter in the constraint dissolving function.
generate_cdf_grad(obj_grad, beta) (callable)
Return the gradient of the constraint dissolving function. obj_grad is a callable function that returns the gradient of \(f\) at \(x\). beta is a float object that refers to the penalty parameter in the constraint dissolving function.
generate_cdf_hess(obj_grad, obj_hvp, beta) (callable)
Return the hessian of the constraint dissolving function. obj_grad is a callable function that returns the gradient of \(f\) at \(x\). obj_hvp is the hessian-vector product of \(f\) at \(x\), i.e., \(\nabla^2 h(x)[d]\). beta is a float object that refers to the penalty parameter in the constraint dissolving function.