Minimize linear objective under LMI constraints
[copt,xopt] = mincx(lmisys,c,options,xinit,target)
solves the convex program
where x denotes the vector of scalar decision variables.
The system of LMIs is described by
c must be of the same length as x.
This length corresponds to the number of decision variables returned
by the function
linear objectives expressed in terms of the matrix variables, the
c vector is easily derived with
mincx returns the global minimum
the objective cTx,
as well as the minimizing value
xopt of the vector
of decision variables. The corresponding values of the matrix variables
is derived from
The remaining arguments are optional. The vector
an initial guess of the minimizer
xopt. It is ignored
when infeasible, but may speed up computations otherwise. Note that
be of the same length as
c. As for
it sets some target for the objective value. The code terminates as
soon as this target is achieved, that is, as soon as some feasible x such
that cTx ≤
target with the
The optional argument
options gives access
to certain control parameters of the optimization code. In
this is a five-entry vector organized as follows:
options(1) sets the desired relative
accuracy on the optimal value
lopt (default = 10–2).
options(2) sets the maximum number
of iterations allowed to be performed by the optimization procedure
(100 by default).
options(3) sets the feasibility
radius. Its purpose and usage are as for
options(4) helps speed up termination.
If set to an integer value J > 0, the code terminates
when the objective cTx has
not decreased by more than the desired relative accuracy during the
last J iterations.
options(5) = 1 turns off the trace
of execution of the optimization procedure. Resetting
zero (default value) turns it back on.
option(i) to zero is equivalent to
setting the corresponding control parameter to its default value.
feasp for more detail.
In LMI optimization, the computational overhead per iteration mostly comes from solving a least-squares problem of the form
where x is the vector of decision variables. Two methods are used to solve this problem: Cholesky factorization of ATA (default), and QR factorization of A when the normal equation becomes ill conditioned (when close to the solution typically). The message
* switching to QR
is displayed when the solver has to switch to the QR mode.
Since QR factorization is incrementally more expensive in most
problems, it is sometimes desirable to prevent switching to QR. This
is done by setting
options(4) = 1. While not guaranteed
to produce the optimal value, this generally achieves a good trade-off
between speed and accuracy.
QR-based linear algebra (see above) is not only expensive in terms of computational overhead, but also in terms of memory requirement. As a result, the amount of memory required by QR may exceed your swap space for large problems with numerous LMI constraints. In such case, MATLAB® issues the error
??? Error using ==> pds Out of memory. Type HELP MEMORY for your options.
You should then ask your system manager to increase your swap
space or, if no additional swap space is available, set
= 1. This will prevent switching to QR and
terminate when Cholesky fails due to numerical instabilities.
mincx implements Nesterov and
Nemirovski's Projective Method as described in
Nesterov, Yu, and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994.
Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, Baltimore, Maryland, pp. 840-844.
The optimization is performed by the C-MEX file