Minimizing mixed-norm subject to a nonconvex quartic constraint for small n and m












0














How can one solve this mixed-norm minimization problem subject to a quartic constraint for small N and M (ex: M=12 and N=4):




  • $textbf{c} = [c_{1},...,c_{N}]^T$

  • $minbiggl (|textbf{c}|_{1} + beta|textbf{c}|_{2} ^2biggr) subjectspace
    tospace sumlimits_ {m=1} ^M (|v_ {m} |^2-textbf{c}^Htextbf{a}_mtextbf{a}_m^Htextbf{c})
    ^2leqepsilon$


Thank you for advance for any suggestions.










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  • 1




    This is not a convex problem, so you should remove that tag
    – Johan Löfberg
    Jan 4 at 17:21






  • 2




    ...and it is not a quadratic constraint, it is quartic
    – Johan Löfberg
    Jan 4 at 17:22










  • Thank you for your comment. The problem is taken from this paper and there they mention that it is quadratic.
    – kogito
    Jan 4 at 17:50






  • 1




    Well that is obviously incorrect as you have a square of a quadratic. What they do is that they write the nonconvex quartic model as a rank-constrained (thus non-convex) semidefinite program with a convex quadratic constraint.
    – Johan Löfberg
    Jan 4 at 21:07
















0














How can one solve this mixed-norm minimization problem subject to a quartic constraint for small N and M (ex: M=12 and N=4):




  • $textbf{c} = [c_{1},...,c_{N}]^T$

  • $minbiggl (|textbf{c}|_{1} + beta|textbf{c}|_{2} ^2biggr) subjectspace
    tospace sumlimits_ {m=1} ^M (|v_ {m} |^2-textbf{c}^Htextbf{a}_mtextbf{a}_m^Htextbf{c})
    ^2leqepsilon$


Thank you for advance for any suggestions.










share|cite|improve this question









New contributor




kogito is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
















  • 1




    This is not a convex problem, so you should remove that tag
    – Johan Löfberg
    Jan 4 at 17:21






  • 2




    ...and it is not a quadratic constraint, it is quartic
    – Johan Löfberg
    Jan 4 at 17:22










  • Thank you for your comment. The problem is taken from this paper and there they mention that it is quadratic.
    – kogito
    Jan 4 at 17:50






  • 1




    Well that is obviously incorrect as you have a square of a quadratic. What they do is that they write the nonconvex quartic model as a rank-constrained (thus non-convex) semidefinite program with a convex quadratic constraint.
    – Johan Löfberg
    Jan 4 at 21:07














0












0








0







How can one solve this mixed-norm minimization problem subject to a quartic constraint for small N and M (ex: M=12 and N=4):




  • $textbf{c} = [c_{1},...,c_{N}]^T$

  • $minbiggl (|textbf{c}|_{1} + beta|textbf{c}|_{2} ^2biggr) subjectspace
    tospace sumlimits_ {m=1} ^M (|v_ {m} |^2-textbf{c}^Htextbf{a}_mtextbf{a}_m^Htextbf{c})
    ^2leqepsilon$


Thank you for advance for any suggestions.










share|cite|improve this question









New contributor




kogito is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











How can one solve this mixed-norm minimization problem subject to a quartic constraint for small N and M (ex: M=12 and N=4):




  • $textbf{c} = [c_{1},...,c_{N}]^T$

  • $minbiggl (|textbf{c}|_{1} + beta|textbf{c}|_{2} ^2biggr) subjectspace
    tospace sumlimits_ {m=1} ^M (|v_ {m} |^2-textbf{c}^Htextbf{a}_mtextbf{a}_m^Htextbf{c})
    ^2leqepsilon$


Thank you for advance for any suggestions.







optimization qcqp






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share|cite|improve this question









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Check out our Code of Conduct.









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edited Jan 4 at 21:35







kogito













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asked Jan 4 at 14:58









kogitokogito

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kogito is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






kogito is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    This is not a convex problem, so you should remove that tag
    – Johan Löfberg
    Jan 4 at 17:21






  • 2




    ...and it is not a quadratic constraint, it is quartic
    – Johan Löfberg
    Jan 4 at 17:22










  • Thank you for your comment. The problem is taken from this paper and there they mention that it is quadratic.
    – kogito
    Jan 4 at 17:50






  • 1




    Well that is obviously incorrect as you have a square of a quadratic. What they do is that they write the nonconvex quartic model as a rank-constrained (thus non-convex) semidefinite program with a convex quadratic constraint.
    – Johan Löfberg
    Jan 4 at 21:07














  • 1




    This is not a convex problem, so you should remove that tag
    – Johan Löfberg
    Jan 4 at 17:21






  • 2




    ...and it is not a quadratic constraint, it is quartic
    – Johan Löfberg
    Jan 4 at 17:22










  • Thank you for your comment. The problem is taken from this paper and there they mention that it is quadratic.
    – kogito
    Jan 4 at 17:50






  • 1




    Well that is obviously incorrect as you have a square of a quadratic. What they do is that they write the nonconvex quartic model as a rank-constrained (thus non-convex) semidefinite program with a convex quadratic constraint.
    – Johan Löfberg
    Jan 4 at 21:07








1




1




This is not a convex problem, so you should remove that tag
– Johan Löfberg
Jan 4 at 17:21




This is not a convex problem, so you should remove that tag
– Johan Löfberg
Jan 4 at 17:21




2




2




...and it is not a quadratic constraint, it is quartic
– Johan Löfberg
Jan 4 at 17:22




...and it is not a quadratic constraint, it is quartic
– Johan Löfberg
Jan 4 at 17:22












Thank you for your comment. The problem is taken from this paper and there they mention that it is quadratic.
– kogito
Jan 4 at 17:50




Thank you for your comment. The problem is taken from this paper and there they mention that it is quadratic.
– kogito
Jan 4 at 17:50




1




1




Well that is obviously incorrect as you have a square of a quadratic. What they do is that they write the nonconvex quartic model as a rank-constrained (thus non-convex) semidefinite program with a convex quadratic constraint.
– Johan Löfberg
Jan 4 at 21:07




Well that is obviously incorrect as you have a square of a quadratic. What they do is that they write the nonconvex quartic model as a rank-constrained (thus non-convex) semidefinite program with a convex quadratic constraint.
– Johan Löfberg
Jan 4 at 21:07










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