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Finance Notes_Part_39 - Limit of feasible poll centers...

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Unformatted text preview: Limit of feasible poll centers Theorem Let ˆ x ∈ Ω be the limit of unsuccessful feasible poll centers { x F k } on meshes that get infinitely fine. If f is Lipschitz near ˆ x , then f ◦ (ˆ x,v ) ≥ for all v ∈ T H Ω (ˆ x ) . Corollary In addition, if f is strictly differentiable near ˆ x , and if Ω is regular near ˆ x , then f (ˆ x,v ) ≥ for all v ∈ T Ω (ˆ x ) , i.e., ˆ x is a KKT point for min x ∈ Ω f ( x ) . Audet and Vicente (SIOPT 2008) Optimization under general constraints 89/109 Limit of infeasible poll centers Theorem Let ˆ x ∈ X be the limit of unsuccessful infeasible poll centers { x I k } on meshes that get infinitely fine. If h is Lipschitz near ˆ x , then h ◦ (ˆ x,v ) ≥ for all v ∈ T H X (ˆ x ) . Audet and Vicente (SIOPT 2008) Optimization under general constraints 90/109 Limit of infeasible poll centers Theorem Let ˆ x ∈ X be the limit of unsuccessful infeasible poll centers { x I k } on meshes that get infinitely fine. If h is Lipschitz near...
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