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Arnold Neumaier
Arnold Neumaier
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In case of an inequality constraint only and semidefinite $P$, your problem is convex, and there may be better alternatives (CVX mentioned in the answer by Victor Liu, or the methods of arXiv:1009.2065 (which has at thre end a reference to a an implementation).

If $1<k<\infty$, you can use standard nonlinear programming software (see, e.g. http://neos-server.org/neos/ ); in case of an equality constraint or an inequality constraint with indefinite $P$, you are very unlikely to do better with other codes.

Otherwise, if $k=1$ or $k=\infty$, you need to introduce an extra variable $z$ for an upper bound on the objective, and minimize $z$ subject to your constraints and a bunch of linear constraints equvalent to $\|Ax-b\|_k\le z$, and again use standard nonlinear programming software. Alternatively, you might use a nonsmooth optimization code; don't know what to recommend there., but a list of codes is at http://www.mat.univie.ac.at/~neum/glopt/software_l.html#nonsm

In case of an inequality constraint only and semidefinite $P$, your problem is convex, and there may be better alternatives (CVX mentioned in the answer by Victor Liu, or the methods of arXiv:1009.2065 (which has at thre end a reference to a an implementation).

If $1<k<\infty$, you can use standard nonlinear programming software (see, e.g. http://neos-server.org/neos/ ); in case of an equality constraint or an inequality constraint with indefinite $P$, you are very unlikely to do better with other codes.

Otherwise, if $k=1$ or $k=\infty$, you need to introduce an extra variable $z$ for an upper bound on the objective, and minimize $z$ subject to your constraints and a bunch of linear constraints equvalent to $\|Ax-b\|_k\le z$, and again use standard nonlinear programming software. Alternatively, you might use a nonsmooth optimization code; don't know what to recommend there.

In case of an inequality constraint only and semidefinite $P$, your problem is convex, and there may be better alternatives (CVX mentioned in the answer by Victor Liu, or the methods of arXiv:1009.2065 (which has at thre end a reference to a an implementation).

If $1<k<\infty$, you can use standard nonlinear programming software (see, e.g. http://neos-server.org/neos/ ); in case of an equality constraint or an inequality constraint with indefinite $P$, you are very unlikely to do better with other codes.

Otherwise, if $k=1$ or $k=\infty$, you need to introduce an extra variable $z$ for an upper bound on the objective, and minimize $z$ subject to your constraints and a bunch of linear constraints equvalent to $\|Ax-b\|_k\le z$, and again use standard nonlinear programming software. Alternatively, you might use a nonsmooth optimization code; don't know what to recommend there, but a list of codes is at http://www.mat.univie.ac.at/~neum/glopt/software_l.html#nonsm

Source Link
Arnold Neumaier
Arnold Neumaier
  • 11.4k
  • 21
  • 49

In case of an inequality constraint only and semidefinite $P$, your problem is convex, and there may be better alternatives (CVX mentioned in the answer by Victor Liu, or the methods of arXiv:1009.2065 (which has at thre end a reference to a an implementation).

If $1<k<\infty$, you can use standard nonlinear programming software (see, e.g. http://neos-server.org/neos/ ); in case of an equality constraint or an inequality constraint with indefinite $P$, you are very unlikely to do better with other codes.

Otherwise, if $k=1$ or $k=\infty$, you need to introduce an extra variable $z$ for an upper bound on the objective, and minimize $z$ subject to your constraints and a bunch of linear constraints equvalent to $\|Ax-b\|_k\le z$, and again use standard nonlinear programming software. Alternatively, you might use a nonsmooth optimization code; don't know what to recommend there.