Function of a matrix of variables

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Christopher Morrow
Christopher Morrow 2021-3-30
编辑： Matt J 2021-3-30
I am trying to find minimum and maximum values of f=(1/2*X'*K*X)-(X'*F), where X is a 3x1[X1 X2 X3]' matix of varibles, K is a 3x3 matrix of real numbers, and F is a 3x1 Matrix of real numbers. I am using the conjugate gradient method of linear finite elemental analysis and wish to find the points at which the function "f" have max and min points. Any and all help will be appreciated.

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Matt J
Matt J 2021-3-30
编辑：Matt J 2021-3-30
Assuming X are the unknowns, then the function is quadratic. In the absence of constraints, a quadratic function cannot have both finite maximizing and minimizing points unless K=0. If K is positive definite, the minimizing point is given analytically by setting the gradient to zero, leading to,
X=K\F
There are no maximizing points.

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