- #1

drawar

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## Homework Statement

Given 2 problems:

(P1) min min(##x_1,x_2##)

s.t ##x_1, x_2 \geq 0##

(P2) min t

s.t ##t \leq x_1##

##t \leq x_2##

##x_1, x_2 \geq 0##

(i) Is the mapping f(##x_1,x_2##)=min(##x_1,x_2##) convex?

(ii) What are the objectives of (P1) and (P2)?

## Homework Equations

## The Attempt at a Solution

Assume f is indeed convex, then I have to prove the following:

For every ##(x_1,x_2), (y_1,y_2) \in \mathrm{R}^2##, and ##0 \leq \lambda \leq 1## ,

##f(\lambda(x_1,x_2)+(1-\lambda)(y_1,y_2)) \leq \lambda f(x_1,x_2) + (1-\lambda) f(y_1,y_2)## .

But how can I proceed from here?

Any help would be much appreciated, thanks!