Łojasiewicz inequalities with explicit exponents for smallest singular value functions.

Let F ( x ) : = ( f i j ( x ) ) i = 1 , … , p ; j = 1 , … , q , be a ( p × q )-real polynomial matrix and let f ( x ) be the smallest singular value function of F ( x ) . In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function f with an explicit exponent: For any x ̄ ∈ R n , there exist c > 0 and ϵ > 0 such that we have for all ‖ x − x ̄ ‖ < ϵ , inf { ‖ w ‖ : w ∈ ∂ f ( x ) } ≥ c | f ( x ) − f ( x ̄ ) | 1 − τ , where ∂ f ( x ) is the limiting subdifferential of f at x , d : = max i = 1 , … , p ; j = 1 , … , q deg f i j , ℛ ( n , d ) : = d ( 3 d − 3 ) n − 1 if d ≥ 2 and ℛ ( n , d ) : = 1 if d = 1 , and τ : = 1 ℛ ( n + p , 2 d + 2 ) . Then we establish some versions of Łojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function f ( x ) of the matrix F ( x ) . [ABSTRACT FROM AUTHOR]