I have a problem on the form: find x ∈ R n , y ∈ R m such that: A x ≥ a B y ≥ b f ( x ) ≥ 0 f i ( x ) ⋅ y j = 0 x i , y j ≥ 0In the later A and B are matrices, a ∈ R n , b ∈ R m , f = ( f 1 … f n ) is a linear vectorial function of x and x = ( x 1 … x n ), y = ( y 1 … y m )There is a solver or a method that could solve that kind of inequalities?

Question

I have a problem on the form: find   x  ∈        R    n    ,  y  ∈        R    m   such that:  A  x  ≥  a  B  y  ≥  b  f  (  x  )  ≥  0      f    i    (  x  )  ⋅      y    j    =  0      x    i    ,      y    j    ≥  0In the later A and B are matrices,   a  ∈        R    n    ,  b  ∈        R    m  ,   f  =  (      f    1    …      f    n    ) is a linear vectorial function of   x and   x  =  (      x    1    …      x    n    ),   y  =  (      y    1    …      y    m    )There is a solver or a method that could solve that kind of inequalities?

Percyaehyq · Accepted Answer

One fairly generic way to find a point that satisfies all of the required inequalities is to model it as a nonlinear feasibility problem (optimization problem with trivial objective). One such formulation is below:                    (FP)                              min                      x            ∈                                      R              n                        ,                        y            ∈                                      R              m                                                                  0                            s.t.                                            A        x        ≥        a        ,                                  B        y        ≥        b        ,                                  f        (        x        )        ≥        0        ,                                            f          i                (        x        )        ⋅                  y          j                =        0        ,                                  x        ,        y        ≥        0        .            Any feasible solution to Problem (FP), if one exists, is also optimal. You can potentially throw Problem (FP) to fmincon in MATLAB to try and find a feasible solution.It is noteworthy that Problem (FP) is a (special instance of) nonlinear program with complementarity constraints (see, for instance, this article) since it contains equations of the form       f    i    (  x  )  ⋅      y    j    =  0 (whatever the notation means). Nonlinear programming solvers may face numerical difficulties on instances of problems in this general class, so you may have to use tailored solution techniques based on the form of your function   f.

linziboobeary1o8p · Accepted Answer

What do you mean by   A  x  ≥  a?   A  x is a vector and a is a scalar.

I have a problem on the form: find x ∈ <mi mathvariant="double-st

Answered question

Answer & Explanation

New Questions in Algebra I