Linear optimization problem - global maxima?

given f:R^n ---%26gt; R


f(x)=%26lt;c,x%26gt;;


where %26lt;c,x%26gt; denotes the dot product of the vectors c and x;





Find a global maxima of this function on X= [0,1]^n





where R^n denotes RxRxR...xR n times;


and [0,1]^n denotes [0,1]x..x[0,1] n times.





thank you in advance.





i am thinking that it is easy to prove that f is convex; the set X is again convex.... so the problem has a global maxima, and by the fact that the function is convex. but, i don't know what to do next :)

Linear optimization problem - global maxima?
Actually the function is linear!!!





f(x)=c1*x1+...+cn*xn





Notice that if some ci is 0 then you can use any value of xi and f is the same.





Pick any (x1,...,xn) in [0,1]^n


For any ci=0 replace xi by 0 (or 1 or anything) and f does not change.


For any ci%26lt;0 replace xi by 0 and f increases by ci*xi so you have a better solution.


For any ci%26gt;0 replace xi by 1 and f increases by ci*(1-xi) so you have a better solution again.





This shows that this extremal solution you found is the best although not unique.