Complex Quadratic Programming ProblemI have a quadratic programming problem as follows: min x ‖ A x + b ‖ 2 with no constraints. If A,x, and b are real, It would be a straight forward QP problem that can be also written as follows: min x 1 2 x T J x + c T xwhere J = A T A and c = A T b. Such problems can be handled by Matlab. However, in my case, A, x, and b are all complex, and therefore can't be solved by Matlab (My only tool at the moment). Of course, the target function (the norm) is real.My question: Any Idea on how to reform such a problem to get a real QP problem so I can use Matlab to solve it?

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Complex Quadratic Programming ProblemI have a quadratic programming problem as follows:      min    x    ‖  A  x  +  b      ‖    2  with no constraints. If A,x, and b are real, It would be a straight forward QP problem that can be also written as follows:      min    x          1    2        x    T    J  x  +      c    T    xwhere   J  =      A    T    A and   c  =      A    T    b. Such problems can be handled by Matlab. However, in my case, A, x, and b are all complex, and therefore can&#039;t be solved by Matlab (My only tool at the moment). Of course, the target function (the norm) is real.My question: Any Idea on how to reform such a problem to get a real QP problem so I can use Matlab to solve it?

ysik92ASw · Accepted Answer

Step 1Because the problem involves complex variables, I find it more aesthetic to use z as a complex vector, rather than x. The problem is the same as minimizing  f  (  z  )  :=  (  A  z  +  b      )    †      (  A  z  +  b  )  =      z    †      (      A    †    A  )    z  +            (            b    †    A  z  +      z    †        A    †    b            )        +      b    †    b    ,where   (  _      )    †   denotes the Hermitian conjugate operator.You have two choices here. First, you can translate everything into real variables. Let   z  =  x  +  y  i, where x and y are real vectors. Similarly, write   A  =  B  +  C  i and   b  =  c  +  d  i, where B, C, c, and d are real matrices. Thus, if w is the double-sized vector       [                            x                                      y                      ]  , then you can minimize  ϕ  (  w  )  :=  f  (  x  +  y  i  )  =  ‖  E  w  +  h      ‖    2    2  instead, where  E  :=      [                            B                          −          C                                      C                          B                      ]  and  h  :=      [                            c                                      d                      ]      .Step 2This is a real quadratic programming, so you should know how to deal with it.Alternatively, you can use some complex differentiation. Note that            ∂      f              ∂      z        =      z    †    (      A    †    A  )  +      b    †    A    .Hence, if   z  =            z      ~       is an optimal vector, then                    z        ~              †    (      A    †    A  )  +      b    †    A  =  0    ,or  (      A    †    A      )    †              z      ~        +      A    †    b  =  0    ,whence      A    †    A              z      ~        +      A    †    b  =  0    .This shows that            z      ~        =  −  (      A    †    A      )          pinv              A    †    b  +  k  =  −      A          pinv        b  +  k    ,where   k  ∈  ker  ⁡  (  A  ). Here,   (  _      )    pinv   denotes the pseudoinverse operator.

Complex Quadratic Programming Problem. I have a quadratic programming problem as follows: min_x ∥Ax+b∥_2

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