#math1010##tex2html_wrap_inline9386# is a hyper
elliptic paraboloid with the minimum at point #tex2html_wrap_inline9388# in the N-D
space.
We first consider the special case where the QP problem is only
subject to equality constraints and we assume #tex2html_wrap_inline9390#, i.e.,
the number of constraints is smaller than the number of unknowns
in #tex2html_wrap_inline9392#. Then the solution #tex2html_wrap_inline9394# has to satisfy
#math1011##tex2html_wrap_inline9396#, i.e., it has to be on #tex2html_wrap_inline9398# hyper
planes in the N-D space.
The Lagrangian function of the QP problem is:
#math1012# #tex2html_wrap_indisplay20525#
(225)
To find the optimal solution, we first equate the derivatives
of the Lagrangian with respect to both #tex2html_wrap_inline9400# and
#math1013##tex2html_wrap_inline9402# to zero:
#math1014#
| #tex2html_wrap_indisplay20529# |
#tex2html_wrap_indisplay20530# |
#tex2html_wrap_indisplay20531# |
|
| #tex2html_wrap_indisplay20532# |
#tex2html_wrap_indisplay20533# |
#tex2html_wrap_indisplay20534# |
(226) |
These two equations can be combined and expressed in matrix form as:
#math1015# #tex2html_wrap_indisplay20536#
(227)
We then solve this system of #tex2html_wrap_inline9404# equations to get the optimal
solution #tex2html_wrap_inline9406# and the corresponding #math1016##tex2html_wrap_inline9408#.
<#5412#>Example<#5412#>
#math1017# #tex2html_wrap_indisplay20541#
where
#math1018# #tex2html_wrap_indisplay20543#
#math1019# #tex2html_wrap_indisplay20545#
Solving this equation we get the solution #math1020##tex2html_wrap_inline9410# and
#tex2html_wrap_inline9412#, at which the function #math1021##tex2html_wrap_inline9414# is
minimized subject to #tex2html_wrap_inline9416#.
If #tex2html_wrap_inline9418#, i.e., the number of equality constraints is the same
as the number of variables, then the variable #tex2html_wrap_inline9420# is uniquely
determined by the linear system #math1022##tex2html_wrap_inline9422#, as the intersect
of #tex2html_wrap_inline9424# hyper planes, independent of the objective function
#tex2html_wrap_inline9426#. Further if #tex2html_wrap_inline9428#, i.e., the system #math1023##tex2html_wrap_inline9430#
is over constrained, and its solution does not exist in general.
It is therefore more interesting to consider QP problems subject to
both equality and inequality constraints:
#math1024# #tex2html_wrap_indisplay20558#