LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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The joint relaxed objective controls Bregman and range errors

Statement

Let $1 \leq p < \infty$ . Let $X, Y, Z$ be real Banach spaces, $X_{0} \subseteq X$ a closed linear subspace, $D \subseteq X$ a domain containing $X_{0}$ and $U$ , and $x_{0} \in U \cap X_{0}$ . Let $F : D \to Y$ , $K \in L (Z, Y)$ , and $r : D \to Z$ satisfy $F (x) - F (x_{0}) = K r (x)$ on $D$ , with $r ∣_{U} : U \to \tilde{U}$ bijective and $r$ Lipschitz with constant $L_{r}$ on $D$ . Let $P : X \to X$ be a bounded linear projection such that $P ∣_{X_{0}} = 0$ , $(I - P) (X) \subseteq X_{0}$ , and $P (x_{0}) = P (x^{†}) = 0$ for an exact solution $x^{†} \in U \cap X_{0}$ ; put $y : = F (x^{†})$ . Assume the projected points $(I - P) r^{- 1} (\hat{r})$ belong to $D \cap X_{0}$ for all admitted $\hat{r} \in \tilde{U}$ .

Let $\overset{ˇ}{R} : X_{0} \to [0, + \infty]$ be proper and convex, choose $ξ^{†} \in \partial \overset{ˇ}{R} (x^{†}) \subseteq X_{0}^{*}$ , and suppose that for some $R_{0} \geq \overset{ˇ}{R} (x^{†})$ , $b \in [0, 1)$ , and continuous nondecreasing concave $ψ : [0, \infty) \to [0, \infty)$ with $ψ (0) = 0$ , the variational source inequality

$- ⟨ ξ^{†}, \overset{ˇ}{x} - x^{†} ⟩ \leq b D_{\overset{ˇ}{R}}^{ξ^{†}} (\overset{ˇ}{x}, x^{†}) + ψ (∥ F (\overset{ˇ}{x}) - F (x^{†}) ∥_{Y}^{p})$

holds whenever $\overset{ˇ}{x} \in D \cap X_{0}$ and $\overset{ˇ}{R} (\overset{ˇ}{x}) \leq R_{0}$ , where $D_{\overset{ˇ}{R}}^{ξ^{†}} (\overset{ˇ}{x}, x^{†}) : = \overset{ˇ}{R} (\overset{ˇ}{x}) - \overset{ˇ}{R} (x^{†}) - ⟨ ξ^{†}, \overset{ˇ}{x} - x^{†} ⟩$ . Set $R_{P} = 0$ in the joint objective. Let $Q : \tilde{U} \times \tilde{U} \to [0, \infty]$ satisfy $Q (a, b) = Q (b, a)$ and $Q (a, b) = 0$ exactly when $a = b$ , and assume there is a constant $C_{Q} \geq 0$ such that

$∥ P (r^{- 1} (a)) - P (r^{- 1} (b)) ∥_{X} \leq C_{Q} Q (a, b) (a, b \in \tilde{U}) .$

For $δ > 0$ , let $y^{δ} \in Y$ satisfy $∥ y^{δ} - y ∥_{Y} \leq δ$ . Let $α, β > 0$ , $C_{η} \geq 0$ , and let $({\hat{r}}_{δ}, x_{δ}) \in \tilde{U} \times U$ be an $η_{δ}$ -minimizer of the joint objective, meaning

$J_{α, β}^{δ} ({\hat{r}}_{δ}, x_{δ}) \leq J_{α, β}^{δ} (\hat{r}, x) + η_{δ} ((\hat{r}, x) \in \tilde{U} \times U), 0 \leq η_{δ} \leq C_{η} δ^{p} .$

Assume ${\overset{ˇ}{x}}_{δ} : = (I - P) r^{- 1} ({\hat{r}}_{δ})$ lies in the stated VSC sublevel. Define

$s_{δ} : = 2^{1 - p} ∥ K ({\hat{r}}_{δ} - r (x^{†})) ∥_{Y}^{p} + β Q (r (x_{δ}), {\hat{r}}_{δ})^{p} + ∥ P x_{δ} ∥_{X}^{p}, e_{δ} : = D_{\overset{ˇ}{R}}^{ξ^{†}} ({\overset{ˇ}{x}}_{δ}, x^{†}),$

and

$C_{*} : = 2^{2 p - 2} \max {1, ∥ K ∥^{p} L_{r}^{p}, \frac{∥ K ∥^{p} L_{r}^{p} C_{Q}^{p}}{β}}, d_{δ} : = (2 + C_{η}) δ^{p} .$

Then

$s_{δ} + (1 - b) α e_{δ} \leq d_{δ} + α ψ (C_{*} s_{δ}) .$

Statement references

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equations (2.10)–(2.14)

Facts & Assumptions

Given: The Banach spaces, range factorization, projection, regularizer, VSC, compatibility inequality, exact datum, noise bound, and approximate minimizer in the Statement.

[A1]

The candidate projected point ${\overset{ˇ}{x}}_{δ}$ belongs to $D \cap X_{0}$ and satisfies $\overset{ˇ}{R} ({\overset{ˇ}{x}}_{δ}) \leq R_{0}$ , so the VSC applies to it.

[A2]

The noisy datum and approximate-minimizer tolerance satisfy the bounds in the Statement.

[A3]

The projection-coupling bound holds for all range variables in the Statement.

[F1]

Differential range invariance for Banach-space maps: $F (x) - F (x_{0}) = K r (x)$ for every $x$ in the range-factorization domain.

[F2]

Joint and frozen-Newton objectives for range-invariant relaxations: with $R_{P} = 0$ , the joint objective is $∥ K \hat{r} + F (x_{0}) - y^{δ} ∥_{Y}^{p} + α \overset{ˇ}{R} ((I - P) r^{- 1} (\hat{r})) + β Q (r (x), \hat{r})^{p} + ∥ P x ∥_{X}^{p}$ .

[F3]

Banach-space convex subgradients and dual pairings: $ξ^{†} \in \partial \overset{ˇ}{R} (x^{†})$ means $\overset{ˇ}{R} (z) \geq \overset{ˇ}{R} (x^{†}) + ⟨ ξ^{†}, z - x^{†} ⟩$ for every $z \in X_{0}$ .

[F4]

Bregman distances for convex regularizers: $D_{\overset{ˇ}{R}}^{ξ^{†}} (z, x^{†}) = \overset{ˇ}{R} (z) - \overset{ˇ}{R} (x^{†}) - ⟨ ξ^{†}, z - x^{†} ⟩$ .

[F5]

Variational source conditions on a regularizer sublevel: on the named regularizer sublevel, $- ⟨ ξ^{†}, z - x^{†} ⟩ \leq b D_{\overset{ˇ}{R}}^{ξ^{†}} (z, x^{†}) + ψ (∥ F (z) - F (x^{†}) ∥_{Y}^{p})$ .

Proof

1.1A2F1F2givenalgebra

Compare the approximate-minimum inequality with $(\hat{r}, x) = (r (x^{†}), x^{†})$ . By [F1], $K r (x^{†}) + F (x_{0}) = F (x^{†}) = y$ , while $Q (r (x^{†}), r (x^{†})) = 0$ and $P x^{†} = 0$ . Since $r^{- 1} (r (x^{†})) = x^{†}$ and $P (x^{†}) = 0$ , the comparison regularizer is $\overset{ˇ}{R} (x^{†})$ , so its objective is at most $δ^{p} + α \overset{ˇ}{R} (x^{†})$ . Writing the estimated data residual as $K ({\hat{r}}_{δ} - r (x^{†})) + (y - y^{δ})$ , the triangle inequality gives $∥ K ({\hat{r}}_{δ} - r (x^{†})) ∥ \leq ∥ K {\hat{r}}_{δ} + F (x_{0}) - y^{δ} ∥ + δ$ . For $p = 1$ the power inequality is equality; for $p > 1$ , $f (t) = t^{p}$ has $f'' (t) = p (p - 1) t^{p - 2} \geq 0$ for $t > 0$ and is continuous at zero, so it is convex on $[0, \infty)$ ; midpoint convexity gives $(a + b)^{p} \leq 2^{p - 1} (a^{p} + b^{p})$ for $a, b \geq 0$ . Rearranging yields $2^{1 - p} ∥ K ({\hat{r}}_{δ} - r (x^{†})) ∥^{p} \leq ∥ K {\hat{r}}_{δ} + F (x_{0}) - y^{δ} ∥^{p} + δ^{p}$ . Combining this with the comparison and $η_{δ} \leq C_{η} δ^{p}$ gives $s_{δ} + α (\overset{ˇ}{R} ({\overset{ˇ}{x}}_{δ}) - \overset{ˇ}{R} (x^{†})) \leq d_{δ}$ .

2.1A1F3F4F5step 1.1algebra

By [F3], $ξ^{†}$ is a subgradient at $x^{†}$ , and [F4] gives $\overset{ˇ}{R} ({\overset{ˇ}{x}}_{δ}) - \overset{ˇ}{R} (x^{†}) = e_{δ} + ⟨ ξ^{†}, {\overset{ˇ}{x}}_{δ} - x^{†} ⟩$ . The VSC [F5] therefore implies $\overset{ˇ}{R} ({\overset{ˇ}{x}}_{δ}) - \overset{ˇ}{R} (x^{†}) \geq (1 - b) e_{δ} - ψ (∥ F ({\overset{ˇ}{x}}_{δ}) - F (x^{†}) ∥^{p})$ . Substituting this lower bound in step 1.1 yields $s_{δ} + (1 - b) α e_{δ} \leq d_{δ} + α ψ (∥ F ({\overset{ˇ}{x}}_{δ}) - F (x^{†}) ∥^{p})$ .

2.2A3F1step 1.1algebra

Put $u_{δ} = r^{- 1} ({\hat{r}}_{δ})$ , so ${\overset{ˇ}{x}}_{δ} = (I - P) u_{δ}$ . Since $r^{- 1} (r (x_{δ})) = x_{δ}$ , assumption [A3] and the triangle inequality give $∥ P u_{δ} ∥ \leq ∥ P x_{δ} ∥ + C_{Q} Q (r (x_{δ}), {\hat{r}}_{δ})$ . Lipschitz continuity of $r$ gives $∥ r ({\overset{ˇ}{x}}_{δ}) - {\hat{r}}_{δ} ∥ = ∥ r ((I - P) u_{δ}) - r (u_{δ}) ∥ \leq L_{r} ∥ P u_{δ} ∥$ . Applying [F1] at ${\overset{ˇ}{x}}_{δ}$ and $x^{†}$ , then applying $(a + b)^{p} \leq 2^{p - 1} (a^{p} + b^{p})$ twice, yields $∥ F ({\overset{ˇ}{x}}_{δ}) - F (x^{†}) ∥^{p} \leq 2^{p - 1} ∥ K ({\hat{r}}_{δ} - r (x^{†})) ∥^{p} + 2^{2 p - 2} ∥ K ∥^{p} L_{r}^{p} (∥ P x_{δ} ∥^{p} + C_{Q}^{p} Q (r (x_{δ}), {\hat{r}}_{δ})^{p}) \leq C_{*} s_{δ}$ . Monotonicity of $ψ$ gives $ψ (∥ F ({\overset{ˇ}{x}}_{δ}) - F (x^{†}) ∥^{p}) \leq ψ (C_{*} s_{δ})$ .

3.1step 2.1step 2.2algebra∎

Substitute the last bound from step 2.2 into the inequality from step 2.1. This gives $s_{δ} + (1 - b) α e_{δ} \leq d_{δ} + α ψ (C_{*} s_{δ})$ , as claimed.

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Sources

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equations (2.10)–(2.14) (standard reference, not scraped)