# How to add multiple objectives within an objective

(Arvind Krishna) #1

Hi,
I want to my objective to be:
`objective<-Minimize(norm2(a-b[1,]) + norm2(a-b[2,]) + ..... + norm2(a-b[n,]))`

where
a=Variable(2)
b is n x 2 matrix

How can I define this objective?

(Mark L. Stone) #2

You can use for loops to build up an objective. Or you can use `sum`. In this case you can use `sum` with `norms` to get an efficient vectorized calculation.

``````cvx_begin
variable a(1,2)
minimize(sum(norms(repmat(a,n,1) - b,2,2)))
cvx_end
``````

I leave you to check that this is correct. You can add constraints as appropriate.

help norms

norms Computation of multiple vector norms.
norms( X ) provides a means to compute the norms of multiple vectors
packed into a matrix or N-D array. This is useful for performing
max-of-norms or sum-of-norms calculations.

``````All of the vector norms, including the false "-inf" norm, supported
by NORM() have been implemented in the norms() command.
norms(X,P)           = sum(abs(X).^P).^(1/P)
norms(X)             = norms(X,2).
norms(X,inf)         = max(abs(X)).
norms(X,-inf)        = min(abs(X)).
If X is a vector, these computations are completely identical to
their NORM equivalents. If X is a matrix, a row vector is returned
of the norms of each column of X. If X is an N-D matrix, the norms
are computed along the first non-singleton dimension.

norms( X, [], DIM ) or norms( X, 2, DIM ) computes Euclidean norms
along the dimension DIM. norms( X, P, DIM ) computes its norms
along the dimension DIM.

Disciplined convex programming information:
norms is convex, except when P<1, so an error will result if these
non-convex "norms" are used within CVX expressions. norms is
nonmonotonic, so its input must be affine.``````