FRACTIONAL ORDER LOG BARRIER INTERIOR POINT ALGORITHM FOR POLYNOMIAL REGRESSION IN THE ℓ p -NORM

A = (\begin{matrix} 1 & a_{1} & {a_{1}}^{2} & \dots & {a_{1}}^{n - 1} \\ 1 & a_{2} & {a_{2}}^{2} & \dots & {a_{2}}^{n - 1} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & a_{m} & {a_{m}}^{2} & \dots & {a_{m}}^{n - 1} \end{matrix}) .

[{}_{x}^{C}D_{- κ +}^{α} {(x + κ)}^{p}] (x) = \frac{p}{Γ (1 - α)} \int_{- κ}^{x} {(x - τ)}^{- α} {(τ + κ)}^{p - 1} d τ = \frac{p}{Γ (1 - α)} \int_{0}^{\infty} {(\frac{x + κ}{1 + ξ})}^{- α} {(\frac{ξ (x + κ)}{1 + ξ})}^{p - 1} \frac{(x + κ)}{{(1 + ξ)}^{2}} d ξ = \frac{p {(x + κ)}^{p - α}}{Γ (1 - α)} \int_{0}^{\infty} ξ^{p - 1} {(1 + ξ)}^{α - p - 1} d ξ = \frac{p {(x + κ)}^{p - α}}{Γ (1 - α)} B (p, 1 - α) = \frac{p {(x + κ)}^{p - α}}{Γ (1 - α)} \frac{Γ (p) Γ (1 - α)}{Γ (p - α + 1)} = \frac{Γ (p + 1)}{Γ (p - α + 1)} {(x + κ)}^{p - α} .

\nabla^{α} ϕ + \nabla φ = [\begin{matrix} \nabla_{x} ϕ + \nabla_{x} φ \\ \nabla_{λ} ϕ + \nabla_{λ} φ \\ \nabla_{u}^{α} ϕ + \nabla_{u} φ \\ \nabla_{v}^{α} ϕ + \nabla_{v} φ \end{matrix}] = [\begin{matrix} A^{⊺} λ \\ A x + u - v - b \\ g_{α} - μ U^{- 1} 1_{m} + λ \\ g_{α} - μ V^{- 1} 1_{m} - λ \end{matrix}] = [\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \end{array}],

[\begin{matrix} 0 & A^{⊺} & 0 & 0 \\ A & 0_{m} & I_{m} & - I_{m} \\ 0 & U & D_{u} & U H_{α} \\ 0 & - V & V H_{α} & D_{v} \end{matrix}] [\begin{array}{r} Δ x \\ Δ λ \\ Δ u \\ Δ v \end{array}] = [\begin{array}{r} r_{1} \\ r_{2} \\ r_{3} \\ r_{4} \end{array}],

\begin{array}{rcl} x^{0} & = & {(A^{⊺} A)}^{- 1} A^{⊺} b, \\ r^{0} & = & b - A x^{0}, \\ λ^{0} & = & \frac{δ r^{0}}{{||r^{0}||}_{\infty}},with δ = 0.975, \\ u_{i}^{0} & = & \{\begin{cases} r_{i}^{0} + ϵ_{1}, & if r_{i}^{0} \geq 0, \\ ϵ_{1}, & if r_{i}^{0} < 0, \end{cases} (for i = 1, 2, \dots, m), \\ v_{i}^{0} & = & \{\begin{cases} ϵ_{1}, & if r_{i}^{0} \geq 0, \\ ϵ_{1} - r_{i}^{0}, & if r_{i}^{0} < 0, \end{cases} (for i = 1, 2, \dots, m), \end{array}

x^{20} = [\begin{array}{r} x_{0}^{20} \\ x_{1}^{20} \\ x_{2}^{20} \\ x_{3}^{20} \\ x_{4}^{20} \end{array}] = [\begin{array}{r} 5.3 E + 00 \\ - 3.4 E + 01 \\ 1.9 E + 02 \\ - 2.8 E + 02 \\ 1.2 E + 02 \end{array}] (n = 5; α = 0.5), x^{2} = [\begin{array}{r} x_{0}^{2} \\ x_{1}^{2} \\ x_{2}^{2} \\ x_{3}^{2} \\ x_{4}^{2} \\ x_{5}^{2} \\ x_{6}^{2} \\ x_{7}^{2} \\ x_{8}^{2} \\ x_{9}^{2} \end{array}] = [\begin{array}{r} 1.8 E + 00 \\ 1.7 E + 02 \\ - 3.7 E + 03 \\ 3.5 E + 04 \\ - 1.6 E + 05 \\ 4.6 E + 05 \\ - 7.6 E + 05 \\ 7.2 E + 00 \\ - 3.7 E + 05 \\ 7.9 E + 04 \end{array}] (n = 10; α = 0.4), x^{1} = [\begin{array}{r} x_{0}^{1} \\ x_{1}^{1} \\ x_{2}^{1} \\ x_{3}^{1} \\ x_{4}^{1} \\ x_{5}^{1} \\ x_{6}^{1} \\ x_{7}^{1} \\ x_{8}^{1} \\ x_{9}^{1} \\ x_{10}^{1} \\ x_{11}^{1} \\ x_{12}^{1} \\ x_{13}^{1} \\ x_{14}^{1} \end{array}] = [\begin{array}{r} 3.6 E + 00 \\ 2.6 E + 00 \\ - 1.9 E + 02 \\ 1.3 E + 04 \\ - 2.1 E + 05 \\ 1.4 E + 06 \\ - 5.6 E + 06 \\ 1.1 E + 07 \\ - 1.3 E + 07 \\ 5.9 E + 06 \\ - 2.6 E + 06 \\ 1.2 E + 07 \\ - 1.9 E + 07 \\ 1.2 E + 07 \\ - 2.8 E + 06 \end{array}] (n = 15; α = 0.4) .

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