问题描述
1作为结果，我得到以下作为原始结果：
当然，我希望在线性约束 class LagrangianEqualityConstrainedQuadratic:
"""
Construct the lagrangian dual relaxation of an equality constrained quadratic function defined as:
1/2 x^T Q x + q^T x : A x = b = 0
"""
def __init__(self,Q,q,A):
self.Q = Q
self.q = q
self.A = A
self.last_rnorm = None
def _solve_sym_nonposdef(self,q):
# `min ||Qx - q||` is formally equivalent to solve the linear system:
# (Q^T Q) x = (Q^T q)^T x
Q,q = np.inner(Q,Q),Q.T.dot(q)
x = minres(Q,q)[0]
self.last_rnorm = np.linalg.norm(q - Q.dot(x)) # || q - Qx ||
return x
def function(self,lmbda):
"""
Compute the value of the lagrangian relaxation defined as:
L(x,lambda) = 1/2 x^T Q x + q^T x - lambda^T A x
L(x,lambda) = 1/2 x^T Q x + (q - lambda A)^T x
where lambda is constrained to be >= 0.
Taking the derivative of the Lagrangian wrt x and settings it to 0 gives:
Q x + (q - lambda A) = 0
so,the optimal solution of the Lagrangian relaxation is the solution of the linear system:
Q x = - (q - lambda A)
:param lmbda: the dual variable wrt evaluate the function
:return: the function value wrt lambda
"""
ql = self.q - lmbda.dot(self.A)
x = self._solve_sym_nonposdef(self.Q,-ql)
return 0.5 * x.dot(self.Q).dot(x) + ql.dot(x)
def jacobian(self,lmbda):
"""
With x optimal solution of the minimization problem,the jacobian
of the Lagrangian dual relaxation at lambda is:
[-A x]
However,we rather want to maximize the Lagrangian dual relaxation,hence we have to change the sign of gradient entries:
[A x]
:param lmbda: the dual variable wrt evaluate the gradient
:return: the gradient wrt lambda
"""
ql = self.q - lmbda.dot(self.A)
x = self._solve_sym_nonposdef(self.Q,-ql)
return self.A * x
上找到我的约束最小值，即沿着右图中的蓝线。正是黑星所在的位置。
我应该如何更改我的代码才能获得它？
根据：
我认为我是否应该强加解决方案 Ax=0 必须与 Q 矩阵的零空间正交（？）
解决方法
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