Contents

quadmax()

Usage:

quadmax(A,b [,eq:eqmat] [,gte:gtemat] [,ckbounds:F]), square REAL matrix
  A, REAL vector b with nrows(b) = nrows(A), REAL matrices eqmat and
  gtemat with ncols(eqmat) = ncols(gtemat) = nrows(A) + 1,

Keywords: analysis

quadmax(A,b) finds the x that maximizes x'Ax + b'x; if the problem is
unbounded then quadmax() returns an error.  A is a p by p REAL matrix and
b is a p by 1 REAL vector.

quadmax(A,b,eq:eqmat) does the same, except that eqmat is a q by p+1
REAL matrix that specifies q linear constraints on z.  eqmat is the
partitioned matrix [Q y], where Q is q by p and y is q by 1 and the
solution is constrained to satisfy Qx = y.  If the constrained problem
is unbounded, then quadmax() returns an error.  If the constraints cannot
be met, then quadmax() returns NULL.

quadmax(A,b,gte:gtemat) does the same except that gtemat is a g by p+1
REAL matrix specifying linear inequality constraints on the solution.
gtemat is the partitioned matrix [G z], where G is g by p and z is g by
1 and the solution is constrained to satisfy Gx >= z elementwise.  If
the constrained problem is unbounded, then quadmax() returns an error.  If
constraints cannot be met, then quadmax() returns NULL.

You can use both eq:eqmat and gte:gtemat to specify both equality and
inequality constraints.

For example, in a three variable mixture problem you might have the
equality constraint that the sum of the x's is 1 and each element of x
is at least .05.  Then you can use quadmax(A,b,eq:eqmat,gte:gtemat)
                                     [ 1 0 0 .05 ]
where eqmat = [1 1 1 1] and gtemat = [ 0 1 0 .05 ]
                                     [ 0 0 1 .05 ]

quadmax(A,b [,eq:eqmat] [,gte:gtemat], ckbounds:F) does the same but
there is no test for the solution being unbounded.  This can
substantially decrease the computational time when you know the problem
is bounded.  This is the case, for example, in a problem where the range
of the x's is totally bounded by inequality constraints, or in a problem
where x'Ax is known to have a unique maximum (A has all negative
eigenvalues).  However, ckbounds:F should not be used when you do not
know there is a bounded solution.

The algorithm used by quadmax() can be described as intelligent brute
force and will probably be overwhelmed by too many constraints.  When
you are sure the problem has a bounded solution, be sure to use keyword
phrase ckbounds:F.