Homework 1
Convex Optimization
1. Consider convex functions fi : Rn →R, i = 1,··· ,k. Prove that the set
{x | fi(x) ≤ 0}
is convex.
2. Consider a convex function f : Rn →R. Prove that the set
{(x,t) | f(x) ≤ t}
is convex.
3. (a) If M1,M2 ∈S2 are positive definite, prove that M1 + M2 is positive definite.
(b) Prove that the set of all n × n positive definite symmetric matrices is convex.
4. Prove that the following set is convex: !
5. Find a necessary and sufficient condition under which the following quadratic function is convex:
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