Finance Notes_Part_22

# Finance Notes_Part_22 - -→ one knows that ξ ≤ 1 4 for...

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Example (model improvement based on LP) C = 1 . 11 Audet and Vicente (SIOPT 2008) Unconstrained optimization 65/109

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Example (model improvement based on LP) C = 1 . 01 Audet and Vicente (SIOPT 2008) Unconstrained optimization 65/109
Example (model improvement based on LP) C = 1 . 001 Audet and Vicente (SIOPT 2008) Unconstrained optimization 65/109

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Geometry constants The geometry constant can also be given by C ( Y ) = cond ( scaled interpolation matrix M ) -→ leading to model-improvement algorithms based on pivotal factorizations (LU/QR or Newton polynomials). These algorithms yield: k M - 1 k ≤ C ( n ) ε growth ξ where ε growth is the growth factor of the factorization. ξ > 0 is a (imposed) lower bound on the absolute value of the pivots.
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Unformatted text preview: -→ one knows that ξ ≤ 1 / 4 for quadratics and ξ ≤ 1 for linears. Audet and Vicente (SIOPT 2008) Unconstrained optimization 66/109 Underdetermined polynomial models Consider a underdetermined quadratic polynomial model built with less than ( n + 1)( n + 2) / 2 points. Theorem If Y is C ( Y ) –poised for linear interpolation or regression then k∇ f ( y )- ∇ m ( y ) k ≤ C ( Y ) [ C ( f ) + k H k ] Δ ∀ y ∈ B ( x ; Δ) where H is the Hessian of the model.-→ Thus, one should minimize the norm of H . Audet and Vicente (SIOPT 2008) Unconstrained optimization 67/109...
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Finance Notes_Part_22 - -→ one knows that ξ ≤ 1 4 for...

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