# Optimization of two functions

• A
hello, in the task of finding the optimal phase covariant cloning machine, i have to maximize two functions of six variables :f1=a.C+b.D and f2=a.B+c.D , they are many constraints, but I've already used them to get to those expressions in the first place, the variables are real scalars and vary between -1 and 1. of course i'm not searching for numerical values for the six variables, in fact i've already got a transformation (set of variables) which i know is optimal, all i'm looking for is a way to prove that it is actually optimal and i know there can be others sets of variables which provide optimality.

mfb
Mentor
The functions are clearly maximized if all variables become 1. You can also flip the signs of a, B and C together, or the signs of b, c, and D together, or both groups together, without changing the result. Proof: looked at the expression. But you can also use some inequalities to prove that those solutions maximize both f1 and f2 in a formal way.

The functions are clearly maximized if all variables become 1..
my fault, the 6 variables along with two others that i've managed to simplify in the expressions verifies a orthonormalization conditition :
a^2+b^2+c^2+d^2=1 and aA+bB+cC+dD=0 . and here are the ramaining constraints : Ac+Bd=Ab+Cd=0 and ac+AC+bd+BD=ab+AB+cd+CD=0. the set that i want to proof it's optimality and which verify the above conditions is {a=1,b=c=d=0, A=D=0}.

mfb
Mentor
Okay, so a=B=C=1, and everything else zero? Then f1=f2=1.

From looking at it: a=-b=C=D=1/sqrt(2) and everything else zero lead to f1=1 as well, but then f2=0. How exactly do you want to maximize the functions? The sum of both? Both individually? Something else?

There are 6 constraints on 8 variables, unless you need a mathematical proof most algorithms should quickly find all local maxima.

well, B and C aren't equal to one, they are parameters which satisfy A^2+B^2=1 (first constraint). and for the way to maximize the functions, well i guess in anyway, as the two variables B and C (which can be seen as Cos and Sin) are not specified, the symmetrical case correspond to maximizing the Sum and minimizing the difference. to be specific i need to show that this set of variables with the constraint A^2+B^2=1, verify the optimality condition (df1=0, df2=0), so i was thinking that maybe i could write this conditions in a more explicit form, as a function of the variables, i've tried the la grange multipliers, but i'v got confuse with putting the second optimality condition (df2=0) as a constraint for the first one.

mfb
Mentor
A^2+B^2=1 (first constraint)
Where does that come from now?

Where does that come from now?
i'm really sorry, i've forget one costraint, the constraint is A^2+B^2+C^2+D^2=1, and since A=D=0 ...

mfb
Mentor
Then you have 7 constraints on 8 parameters. That is a one-dimensional problem, fixing one parameter fixes all others (up to some discrete changes like flipping signs).

yes it is, i managed (finally) to write the lagrange's equations, which are satisfied for the specific set of variables i've mentioned ! thanks for your time and you help 