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The actual-to-predicted ratio approaches one at small radii away from stationarity

Statement

Let $m$ be the model of Quadratic trust-region models and the actual-to-predicted reduction ratio, with $g \neq 0$ , symmetric $B$ , and radius $Δ > 0$ . Suppose $∥ B ∥_{2} \leq M_{B}$ for a declared $M_{B} \geq 0$ , and $f$ is $C^{1}$ on an open set containing the segment from $x$ to $x + p$ . Suppose the gradient of $f$ is $L$ -Lipschitz on that segment, meaning $∥ \nabla f (u) - \nabla f (v) ∥_{2} \leq L ∥ u - v ∥_{2}$ there. Let $0 < θ \leq 1$ , let $p$ be feasible, and suppose $m (0) - m (p) \geq θ [m (0) - m (p_{C})]$ for the Cauchy step $p_{C}$ of The Cauchy step for a quadratic trust-region model. Interpret $∥ g ∥_{2} / (2 M_{B}) = + \infty$ if $M_{B} = 0$ , and assume $Δ \leq ∥ g ∥_{2} / (2 M_{B})$ . Then the predicted reduction is positive and $∣ ρ (p) - 1 ∣ \leq \frac{2 (L + M_{B}) Δ}{θ ∥ g ∥_{2}} .$ Consequently, for $0 < η_{2} < 1$ , if $L + M_{B} > 0$ and $Δ \leq \frac{θ (1 - η_{2}) ∥ g ∥_{2}}{2 (L + M_{B})},$ then $ρ (p) \geq η_{2}$ . If $L + M_{B} = 0$ , then $ρ (p) = 1$ .

Facts & Assumptions

Given: The model, the smoothness and Hessian-bound assumptions in the Statement, and a feasible step satisfying the stated fraction of Cauchy decrease.

[F1]

The Cauchy step for a quadratic trust-region model: if $c : = g^{T} B g \leq 0$ , then $p_{C} = - (Δ / ∥ g ∥_{2}) g$ ; if $c > 0$ , then $p_{C} = - \min {∥ g ∥_{2}^{2} / c, Δ / ∥ g ∥_{2}} g$ .

[F2]

The Cauchy step gives a quantitative model decrease: $m (0) - m (p_{C}) \geq \frac{1}{2} ∥ g ∥_{2} \min {Δ, ∥ g ∥_{2} / M_{B}},$ with the zero-matrix convention from that lemma.

[F3]

Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1: $∥ B v ∥_{2} \leq ∥ B ∥_{2} ∥ v ∥_{2}$ .

[F4]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ u^{T} v ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ .

[F5]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : if a scalar function is continuous on a closed interval and differentiable in its interior, its endpoint difference equals its derivative at an interior point times the interval length.

[A1]

$∥ B ∥_{2} \leq M_{B}$ .

[A2]

The gradient Lipschitz inequality in the Statement holds along the segment $x + t p$ , $0 \leq t \leq 1$ .

[A3]

$m (0) - m (p) \geq θ [m (0) - m (p_{C})]$ with $0 < θ \leq 1$ .

Proof

Given: The hypotheses in the Statement.

1.1givenA1A3F1F2F3F4algebra

Put $a : = ∥ g ∥_{2} > 0$ and $c : = g^{T} B g$ . By [F3] and [F4], $c \leq ∣ c ∣ \leq a ∥ B g ∥_{2} \leq M_{B} a^{2}$ . If $c \leq 0$ , [F1] gives $∥ p_{C} ∥_{2} = Δ$ . If $c > 0$ , then $M_{B} > 0$ and $a^{2} / c \geq 1 / M_{B}$ ; since $Δ / a \leq 1 / (2 M_{B})$ , [F1] again gives $∥ p_{C} ∥_{2} = Δ$ . Put $M : = ∥ B ∥_{2}$ . If $M > 0$ , [A1] gives $a / M \geq a / M_{B}$ ; if $M = 0$ , use the zero-matrix convention in [F2]. Hence [F2] and [A3] imply $m (0) - m (p) \geq θ [m (0) - m (p_{C})] \geq \frac{1}{2} θ a Δ > 0.$

2.1givenA1A2F4F5step 1.1algebra∎

Define $ϕ (t) : = f (x + t p)$ and $h (t) : = ϕ (t) - ϕ (0) - t ϕ^{'} (0)$ . For $0 < t \leq 1$ , the gradient Lipschitz bound and [F4] give $∣ h^{'} (t) ∣ = ∣ ⟨ \nabla f (x + t p) - \nabla f (x), p ⟩ ∣ \leq L t ∥ p ∥_{2}^{2} .$ By [F5] on $h$ over $[0, 1]$ , $∣ h (1) ∣ \leq L ∥ p ∥_{2}^{2}$ . From the model formula and [A1], $∣ ared (p) - pred (p) ∣ \leq ∣ h (1) ∣ + \frac{1}{2} ∣ p^{T} B p ∣ \leq (L + M_{B}) ∥ p ∥_{2}^{2} .$ Since $∥ p ∥_{2} \leq Δ$ , step 1.1 yields $∣ ρ (p) - 1 ∣ \leq \frac{(L + M_{B}) Δ^{2}}{\frac{1}{2} θ a Δ} = \frac{2 (L + M_{B}) Δ}{θ a} .$ If $L + M_{B} = 0$ , the numerator vanishes and $ρ (p) = 1$ ; otherwise the stated threshold gives $∣ ρ (p) - 1 ∣ \leq 1 - η_{2}$ , hence $ρ (p) \geq η_{2}$ .

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, second edition (standard reference, not scraped)
Nick Gould, Continuous Optimization Algorithms, Part 3.2: Trust-Region Methods (standard reference, not scraped)