TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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A verified inexact gradient residual bounds central-path distance and objective error

Statement

Consider the conic model with proper cone $K$ , equality $A x = b$ , and barrier objective $Φ_{t} (x) = t c^{T} x + F (h - G x)$ as in Log-barrier central paths and primal-dual Newton residuals, where $F$ is a $ν$ -log-homogeneous self-concordant barrier on $int K$ with parameter $ν > 0$ . Parameterize the affine feasible space by $x = x_{0} + P z$ , where $A x_{0} = b$ and the columns of $P$ are an orthonormal basis of $\ker A$ . Let $φ_{t} (z) = Φ_{t} (x_{0} + P z)$ on an open convex region containing an exact central minimizer $z_{*}$ and a strict feasible candidate $\tilde{z}$ . Assume $D^{2} φ_{t} (z) ⪰ μ I$ on that region for a known $μ > 0$ , and the minimum is attained in its interior. If $η, δ \geq 0$ and a computed gradient $\hat{g}$ has independently verified bounds $∥ \hat{g} ∥_{2} \leq η$ and $∥ \hat{g} - \nabla φ_{t} (\tilde{z}) ∥_{2} \leq δ$ , then $∥ \tilde{x} - x_{*} (t) ∥_{2} \leq \frac{η + δ}{μ}, c^{T} \tilde{x} - p^{*} \leq \frac{ν}{t} + ∥ c ∥_{2} \frac{η + δ}{μ},$ provided $\tilde{x}$ is primal feasible. A tolerance or software stopping flag alone supplies neither verified bound.

Facts & Assumptions

Given: The affine parameterization, uniform restricted Hessian lower bound, attained central minimizer, primal feasibility, and two verified gradient bounds.

[F1]

Under a Hessian lower bound $D^{2} φ ⪰ μ I$ , a verified computed-gradient residual $η$ and discrepancy $δ$ imply distance at most $(η + δ) / μ$ from the exact minimizer (An inexact gradient residual certifies distance under strong convexity).

[F2]

For a $ν$ -log-homogeneous barrier, $- \nabla F (s) \in K^{*}$ and $D F (s) [s] = - ν$ (The negative gradient of a log-homogeneous cone barrier is dual feasible).

[F3]

When the primal value is finite, an exact central point has objective gap at most $ν / t$ (Exact central points have barrier gap m/t or nu/t).

[F4]

Euclidean Cauchy--Schwarz gives $∣ a^{T} b ∣ \leq ∥ a ∥_{2} ∥ b ∥_{2}$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F5]

At an exact conic barrier minimizer, $t c - G^{*} \nabla F (s) + A^{T} w = 0$ for some equality multiplier $w$ (Log-barrier central paths and primal-dual Newton residuals).

Proof

1.1F1givenalgebra

The linear map $z \mapsto x_{0} + P z$ preserves Euclidean distances because $P^{T} P = I$ . The restriction $φ_{t}$ is $C^{2}$ on the stated open convex coordinate region, its exact minimizer is attained in the interior, and the assumed Hessian lower bound is precisely the strong-convexity hypothesis in [F1]. Applying [F1] gives $∥ \tilde{z} - z_{*} ∥_{2} \leq (η + δ) / μ$ , hence the same bound for $\tilde{x} - x_{*} (t)$ .

1.2F2F5givenalgebra

Let $x_{*} = x_{0} + P z_{*}$ and $s_{*} = h - G x_{*}$ . By [F5], $t c - G^{*} \nabla F (s_{*}) + A^{T} w = 0$ for some $w$ . Set ${\hat{z}}_{*} = - \nabla F (s_{*}) / t \in K^{*}$ by [F2] and $y_{*} = - w / t$ . The stationarity equation gives $A^{T} y_{*} - G^{*} {\hat{z}}_{*} = c$ , so $(y_{*}, {\hat{z}}_{*})$ is dual feasible. The conic gap identity in Conic, LP, and convex QP primal-dual models then gives a finite dual lower bound on every primal feasible objective. Since $x_{*}$ is primal feasible with finite objective, this proves that $p^{*}$ is finite, establishing the finiteness hypothesis needed for [F3].

2.1F3F4step 1.1step 1.2givenalgebra∎

Since $P^{T} P = I$ , $∣ c^{T} (\tilde{x} - x_{*} (t)) ∣ = ∣ (P^{T} c)^{T} (\tilde{z} - z_{*}) ∣ \leq ∥ c ∥_{2} (η + δ) / μ$ by [F4] and $∥ P^{T} c ∥_{2} \leq ∥ c ∥_{2}$ . Add the exact-center gap [F3]. Primal feasibility of $\tilde{x}$ makes $p^{*} \leq c^{T} \tilde{x}$ , giving the stated candidate bound.

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Sources

Stephen Boyd and Lieven Vandenberghe, Convex Optimization (standard reference, not scraped)
Hamza Fawzi, Topics in Convex Optimisation, Lecture 15: Path-following methods (standard reference, not scraped)