I have a relatively simple question, normally Porfolio Optimization using Lagrange method uses constraints of:
uT w = m
1T w = 1
then Lagrange function differentiate wrt w and solve.
but what if constraint are:
wT 1 = 1
wT u = 0.1
or simply
wT 1 = 1
do I still differentiate wrt w as before, or I now differeniate wrt wT? Also what do these optimization cases perform?
Regards
Sebastian
Porfolio Optimization using LaGrange methods
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 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: Porfolio Optimization using LaGrange methods
Hi
Its usually easier to think of the constraints not in vector form, to get the intuition straight. Let w=(w_1,...,w_n)' where I use ' for transpose and all vectors are column vectors. Also set 1 = (1,1,..,1)'. Then w'1 = 1'w = sum w_i. Similarly, if u = (u_1, ..,u_n)' then u'w = w'u  sum u_i*w_i. So, the contraints can be written either way, the problem is exactly the same.
Carol
Its usually easier to think of the constraints not in vector form, to get the intuition straight. Let w=(w_1,...,w_n)' where I use ' for transpose and all vectors are column vectors. Also set 1 = (1,1,..,1)'. Then w'1 = 1'w = sum w_i. Similarly, if u = (u_1, ..,u_n)' then u'w = w'u  sum u_i*w_i. So, the contraints can be written either way, the problem is exactly the same.
Carol
Re: Porfolio Optimization using LaGrange methods
Thanks Prof Alexander, but what if now the constraints are simply
wT 1 = 1
this time, no return constraint of uT w=0.1, just a budget equation to say sum of all weights adding up to 1.
Then in this case this is an unconstrained problem? If so then how to get lambda, mu and subsequently A, B and C
coefficients?
Sebastian
wT 1 = 1
this time, no return constraint of uT w=0.1, just a budget equation to say sum of all weights adding up to 1.
Then in this case this is an unconstrained problem? If so then how to get lambda, mu and subsequently A, B and C
coefficients?
Sebastian

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: Porfolio Optimization using LaGrange methods
Still a constrained problem but there is an analytic solution, see p243244 and solution is given in excel spread for example I.6.7
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