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September 19th, 2017, 06:14 AM   #1
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Convexity of non-linear problem

Problem in English: ''Find out if the following non-linear optimization problem is convex.''

Tried solving det(H-lambda*I)=0 and see for which interval x will give positive lambda roots.

Not sure how I'm supposed to approach this.. Tried using pq-formula but got wrong roots for lambda, or try using Sylvester's criterion.
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Last edited by skipjack; September 19th, 2017 at 12:09 PM.
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September 20th, 2017, 06:00 AM   #2
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$\displaystyle x_1^2+ x_2^2= 2$ is a circle with center at (0, 0) and radius $\displaystyle \sqrt{2}$ so $\displaystyle x_1^2+ x_2^2\le 2$ is that circle and its interior, a convex set. Completing the square in $\displaystyle 4x_1^2+ 3x_1x_2+ x_2^2\le 4$ gives $\displaystyle 4(x_1+ \frac{3}{8}x_2)^2+ \frac{25}{64}x_2^3\le 4$, an ellipse with its axes at an angle to the coordinate axes and its interior, also a convex set. The intersection of two convex sets is a convex set.
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