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September 19th, 2017, 05:14 AM  #1 
Senior Member Joined: Dec 2011 Posts: 112 Thanks: 2  Convexity of nonlinear problem
Problem in English: ''Find out if the following nonlinear optimization problem is convex.'' Tried solving det(Hlambda*I)=0 and see for which interval x will give positive lambda roots. Not sure how I'm supposed to approach this.. Tried using pqformula but got wrong roots for lambda, or try using Sylvester's criterion. Last edited by skipjack; September 19th, 2017 at 11:09 AM. 
September 20th, 2017, 05:00 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,966 Thanks: 807 
$\displaystyle x_1^2+ x_2^2= 2$ 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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convexity, nonlinear, problem 
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