Constrained Minimization: Prove Local min => Global min

[kingwinner]

Note: A and Q are matrices and x is some point in R^d.

Attempted Proof:
("o" below means dot product)
Let M={x|Ax=c} and f(x)=(1/2)x o Qx - b o x
Suppose x₀ is a local min point.
I will use proof by contradiction. Suppose, on the contrary, that x₀ is NOT a global min point. Then there must exist a point x₁ s.t. f(x₁)<f(x₀).

Let x(s) = (1-s)x₀ + s x₁, s is any real number. (x: R->R^d, this is parametric equation of the straight line through x₀ and x₁)
Note that x(0)=x₀, x(1)=x₁

Let φ: R->R be defined by φ(s)=f(x(s)), s is any real number. Then φ has a local min at s=0.
=> φ'(0)=0 and φ''(0)≥0 (1st and 2nd order necessary conditions for a local min)

By chain rule, 0 = φ'(0) = x₁ o Qx₀ - x₀ o Qx₀ - b o (x₁-x₀)
and 0 ≤ φ''(0) = x₁ o Qx₁ - x₁ o Qx₀ - x₀ o Qx₁ + x₀ o Qx₀

and I'm stuck here...how can I arrive at a contradiction from here? I tried adding them to get φ'(0)+φ''(0)≥0, but it doesn't seem to help because it's a mess and doesn't seem to simplify to f(x₁)-f(x₀).

Any help would be very much appreciated!

 

[SlipEternal]

View attachment 1702

Note: A and Q are matrices and x is some point in R^d.

Attempted Proof:
("o" below means dot product)
Let M={x|Ax=c} and f(x)=(1/2)x o Qx - b o x
Suppose x₀ is a local min point.
I will use proof by contradiction. Suppose, on the contrary, that x₀ is NOT a global min point. Then there must exist a point x₁ s.t. f(x₁)<f(x₀).

Let x(s) = (1-s)x₀ + s x₁, s is any real number. (x: R->R^d, this is parametric equation of the straight line through x₀ and x₁)
Note that x(0)=x₀, x(1)=x₁

Let φ: R->R be defined by φ(s)=f(x(s)), s is any real number. Then φ has a local min at s=0.
=> φ'(0)=0 and φ''(0)≥0 (1st and 2nd order necessary conditions for a local min)

By chain rule, 0 = φ'(0) = x₁ o Qx₀ - x₀ o Qx₀ - b o (x₁-x₀)
and 0 ≤ φ''(0) = x₁ o Qx₁ - x₁ o Qx₀ - x₀ o Qx₁ + x₀ o Qx₀

and I'm stuck here...how can I arrive at a contradiction from here? I tried adding them to get φ'(0)+φ''(0)≥0, but it doesn't seem to help because it's a mess and doesn't seem to simplify to f(x₁)-f(x₀).

Any help would be very much appreciated!

Edit: Nevermind. I didn't read your problem carefully enough. I don't know. I would need to think about it more.