Sentence examples for be a gap function from inspiring English sources

Definition 4.1 A function g : K → R ∪ is said to be a gap function for the problem (SVVLI) if (i) g ( x ) ≤ 0, ∀ x ∈ K ;   (ii) g ( x ˆ ) = 0 if and only if x ˆ solves (SVVLI).  .

Definition 4.2 A function g : K → R ∪ is said to be a gap function for the problem (MVVLI) if (i) g ( x ) ≥ 0, ∀ x ∈ K ;   (ii) g ( x ˆ ) = 0 if and only if x ˆ solves (MVVLI).  .

A function (p: kappatomathbb{R}) is said to be a gap function for SVMQVIP (1.1), if it satisfies the following properties: (i) (p(x geq0), (forall xinkappa);   (ii) (p(x^) = 0), (x^inkappa), if and only if (x^) solves SVMQVIP (1.1).  .

A function (G: Htomathbb{R}) is said to be a gap function for the RGVIP (2.1), if it satisfies the following properties: (i) (G x geq0), (forall xin H);   (ii) (G(x^) = 0), if and only if (x^in H) solves the RGVIP (2.1).  .

A function (gamma: mathbb{R}^{n} rightarrowoverline{mathbb{R}}) is said to be a gap function for the problem ((mathit{QVI})) if it satisfies the following properties: (i) (gamma y) geq0), (forall y in K x));   (ii) (gamma(x) = 0) if and only if x solves the problem ((mathit{QVI})).  .

A function ϕ : K → R is said to be a gap function for the variational inequality (VI) if it satisfies the following properties: (i) ϕ ( x ) ≤ 0, ∀ x ∈ K ;   (ii) ϕ ( x 0 ) = 0 if and only if x 0 solves (VI).  .

The function (psi_{alpha}), with (alpha> 0), defined by (8) is a gap function for (GVVI).

The function (phi_{alpha}), with (alpha> 0), defined by (9) is a gap function for (VVI).

Proof By virtue of Lemma 2.3, ϕ is a gap function of (GVQEP).

Proof ϕ is a gap function of (GVQEP) owing to Lemma 2.3.

By Theorem 4.2 in [10], N is a gap function of (MVVI).