Case

Assumption

Conclusion

Proof

I.

either or for some and in the latter case for at least one index among the components of s_k there occur one(s) with zero and other(s) with positive value

is not bounded from below on K

Theorem 4.1

II.

 for every

 takes on its minimum on K

Lemma 5.1

III.

For some vector the following hold:
a) ;
b) for every index either s_k >> 0 or s_k = 0;
c) s_k >> 0 is fulfilled for at least one index ;

the nonlinear part of can be shortened

Theorem 6.1

IV.

the linear part of can be contracted

Theorem 7.3