Questions tagged [local-minimum]

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3
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1answer
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Radial unboundedness vs convexity

We have a simple problem, namely minimizing: $$f(x) = x_1^2 + x_2^2 - x_1.$$ The gradient is $$\nabla f(x) = \begin{bmatrix} 2x_1 - 1 \\ 2x_2 \end{bmatrix},$$ hence the unique stationary point is: $...
5
votes
1answer
383 views

Single KKT solution for a simple problem: proof of being minimizer

I have a very simple problem: $$ \begin{align*} \begin{array}{ll} \min\limits_{x_1,x_2} & -x_1x_2 \\ \text{s.t.} & x_1 + x_2 - 2 = 0. \end{array} \end{align*} $$ The KKT system gives me $x_1^*...