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• Theorem 5.3, p. 320 — from ‖x₀ − x*‖ ≤ μ/(2M), Newton's method is well defined and ‖x_{k+1} − x*‖ ≤ (M/μ)‖x_k − x*‖²Open

  Oct 2026

• §5.3.2, proof of Theorem 5.3, p. 321 — ∇²f(x_k) ⪰ ∇²f(x*) − M‖x_k − x*‖Iₙ ⪰ (μ − M‖x_k − x*‖)Iₙ ⪰ (μ/2)IₙOpen

  Oct 2026

• §5.3.2, proof of Theorem 5.3, p. 321 — ∇²f(x_k)(x_{k+1} − x*) = ∫₀¹ [∇²f(x_k) − ∇²f(x* + s(x_k − x*))](x_k − x*) dsOpen

  Oct 2026

• §5.3.2, proof of Theorem 5.3, p. 321 — ∫₀¹ ‖∇²f(x_k) − ∇²f(x* + s(x_k − x*))‖ ds ≤ (M/2)‖x_k − x*‖Open

  Oct 2026

• §5.3.2, proof of Theorem 5.3, p. 321 — ∫₀¹ ∇²f(x + sh) h ds = ∇f(x + h) − ∇f(x)Open

  Oct 2026

• Theorem 4.4, p. 305 — mirror prox with η = ρ/β on a convex β-smooth f satisfies f((1/t)Σ_{s=1}^t y_{s+1}) − f(x*) ≤ βR²/(ρt)Open

  Oct 2026

• §4.5, proof of Theorem 4.4, p. 307 — per-step bound f(y_{t+1}) − f(x) ≤ (D_Φ(x, x_t) − D_Φ(x, x_{t+1}))/η for η = ρ/βOpen

  Oct 2026

• §4.5, proof of Theorem 4.4, p. 307 — third term: (∇f(y_{t+1}) − ∇f(x_t))⊤(y_{t+1} − x_{t+1}) ≤ (β/2)‖y_{t+1} − x_t‖² + (β/2)‖y_{t+1} − x_{t+1}‖²Open

  Oct 2026

• (4.9), proof of Theorem 4.4, p. 307 — second term: η∇f(x_t)⊤(y_{t+1} − x_{t+1}) ≤ D_Φ(x_{t+1}, x_t) − (ρ/2)‖x_{t+1} − y_{t+1}‖² − (ρ/2)‖y_{t+1} − x_t‖²Open

  Oct 2026

• §4.5, proof of Theorem 4.4, p. 306 — first term: η∇f(y_{t+1})⊤(x_{t+1} − x) ≤ D_Φ(x, x_t) − D_Φ(x, x_{t+1}) − D_Φ(x_{t+1}, x_t)Open

  Oct 2026

• Lemma 4.1, pp. 298–299 — (∇Φ(Π^Φ_X(y)) − ∇Φ(y))⊤(Π^Φ_X(y) − x) ≤ 0 and D_Φ(x, Π^Φ_X(y)) + D_Φ(Π^Φ_X(y), y) ≤ D_Φ(x, y)Open

  Oct 2026

• Feasible point of the nested subproblem NLDS(t,k)Definition

  Oct 2026

• §5.3.2, p. 320 — C² function with gradient and Hessian maps, Lipschitz Hessian, and Newton's method x_{k+1} = x_k − [∇²f(x_k)]⁻¹∇f(x_k)Definition

  Oct 2026

• Ch. 4 preamble and §§4.1, 4.5, pp. 297–305 — Bregman divergence, mirror maps, ρ-strong convexity and β-smoothness w.r.t. ‖·‖, Bregman projection, mirror proxDefinition

  Oct 2026

• Initial cut set (Step 0)Definition

  Oct 2026

• Theorem 4.2, pp. 299–300 — mirror descent with η = (R/L)√(2ρ/t) satisfies f((1/t)Σ x_s) − f(x*) ≤ RL√(2/(ρt))Open

  Oct 2026

• §4.2, proof of Theorem 4.2, p. 300 — Σ_{s=1}^t (f(x_s) − f(x)) ≤ D_Φ(x, x₁)/η + ηL²t/(2ρ)Open

  Oct 2026

• §4.2, proof of Theorem 4.2, p. 300 — D_Φ(x_s, y_{s+1}) − D_Φ(x_{s+1}, y_{s+1}) ≤ (ηL)²/(2ρ)Open

  Oct 2026

• Lemma 4.1, pp. 298–299 — the Bregman projection satisfies D_Φ(x, Π(y)) + D_Φ(Π(y), y) ≤ D_Φ(x, y)Open

  Oct 2026

• §4.2, proof of Theorem 4.2, p. 300 — per-step inequality f(x_s) − f(x) ≤ (1/η)(D_Φ(x,x_s) + D_Φ(x_s,y_{s+1}) − D_Φ(x,x_{s+1}) − D_Φ(x_{s+1},y_{s+1}))Open

  Oct 2026

• Theorem 3.18, p. 290 — Nesterov's method on an α-strongly convex β-smooth f: f(y_t) − f(x*) ≤ ((α + β)/2)‖x₁ − x*‖² exp(−(t − 1)/√κ)Open

  Oct 2026

• Proof of Theorem 3.18, p. 293 — v_s − x_s = √κ(x_s − y_s)Open

  Oct 2026

• Eq. (3.20), p. 292 — Φ*_{s+1} ≥ (1 − 1/√κ)Φ*_s + (1 − 1/√κ)∇f(x_s)⊤(x_s − y_s) + f(x_s)/√κ − ‖∇f(x_s)‖²/(2β)Open

  Oct 2026

• Eq. (3.22), p. 292 — Φ*_{s+1} + (α/2)‖x_s − v_{s+1}‖² = (1 − 1/√κ)Φ*_s + (α/2)(1 − 1/√κ)‖x_s − v_s‖² + f(x_s)/√κOpen

  Oct 2026

• Proof of Theorem 3.18, p. 292 — Φ_s(x) = Φ*_s + (α/2)‖x − v_s‖² with v_s given by (3.21)Open

  Oct 2026

• Proof of Theorem 3.18, p. 291 — f(y_t) − f(x*) ≤ ((α + β)/2)‖x₁ − x*‖²(1 − 1/√κ)^{t−1}Open

  Oct 2026

• Eq. (3.18), p. 291 — Φ_{s+1}(x) ≤ f(x) + (1 − 1/√κ)^s (Φ₁(x) − f(x))Open

  Oct 2026

• Eq. (3.19), p. 291 — f(y_s) ≤ min_{x ∈ ℝⁿ} Φ_s(x)Open

  Oct 2026

• Theorem 3.19, p. 294 — Nesterov's accelerated gradient descent on a convex β-smooth f satisfies f(y_t) − f(x*) ≤ 2β‖x₁ − x*‖²/t²Open

  Oct 2026

• Proof of Theorem 3.19, p. 295 — λ_{t−1} ≥ t/2 for t ≥ 2Open

  Oct 2026

• Proof of Theorem 3.19, p. 295 — summing from s = 1 to t − 1, δ_t ≤ (β/(2λ²_{t−1}))‖u₁‖²Open

  Oct 2026

• Eq. (3.25), pp. 294–295 — λ²_sδ_{s+1} − λ²_{s−1}δ_s ≤ (β/2)(‖λ_sx_s − (λ_s − 1)y_s − x*‖² − ‖λ_sy_{s+1} − (λ_s − 1)y_s − x*‖²)Open

  Oct 2026

• Eq. (3.26), p. 295 — λ_{s+1}x_{s+1} − (λ_{s+1} − 1)y_{s+1} = λ_sy_{s+1} − (λ_s − 1)y_sOpen

  Oct 2026

• Proof of Theorem 3.19, p. 294 — the step sequence satisfies λ²_{s−1} = λ²_s − λ_sOpen

  Oct 2026

• Eq. (3.24), p. 294 — f(y_{s+1}) − f(x*) ≤ β(x_s − y_{s+1})⊤(x_s − x*) − (β/2)‖x_s − y_{s+1}‖²Open

  Oct 2026

• Eq. (3.23), p. 294 — f(y_{s+1}) − f(y_s) ≤ β(x_s − y_{s+1})⊤(x_s − y_s) − (β/2)‖x_s − y_{s+1}‖²Open

  Oct 2026

• Lemma 3.6, p. 270, unconstrained (X = ℝⁿ) — f(x − ∇f(x)/β) − f(y) ≤ ∇f(x)⊤(x − y) − ‖∇f(x)‖²/(2β)Open

  Oct 2026

• Theorem 3.14, p. 282 — some β-smooth convex f forces min_{s≤t} f(x_s) − f(x*) ≥ (3β/32)‖x₁ − x*‖²/(t + 1)² under (3.15)Open

  Oct 2026

• Proof of Theorem 3.14, p. 283 — f*_t − f*_{2t+1} = (β/8)(1/(t+1) − 1/(2t+2)) ≥ (3β/32)‖x*_{2t+1}‖²/(t+1)²Open

  Oct 2026

• Proof of Theorem 3.14, p. 283 — ‖x*_k‖² = Σᵢ (i/(k+1))² ≤ (k+1)/3Open

  Oct 2026

• Proof of Theorem 3.14, p. 282 — x*_k(i) = 1 − i/(k+1) solves A_k x = e₁, minimizes f_k, and f*_k = −(β/8)(1 − 1/(k+1))Open

  Oct 2026

• Proof of Theorem 3.14, p. 282 — under (3.15) the query x_s lies in Span(e₁, …, e_{s−1}), so f(x_s) = f_s(x_s)Open

  Oct 2026

• Proof of Theorem 3.14, p. 282 — A_k is symmetric and 0 ⪯ A_k ⪯ 4Iₙ via the quadratic-form identityOpen

  Oct 2026

• Cut sets of the nested L-shaped methodDefinition

  Oct 2026

• Ch. 4 preamble, §4.1, (4.2)–(4.3), (4.6) — Bregman divergence, mirror maps, Bregman projection, mirror descent and dual averaging runsDefinition

  Oct 2026

• §3.7.1, pp. 290–292 — β-smoothness, Nesterov's accelerated gradient descent for κ = β/α, the functions Φ_s (3.17), the centres v_s (3.21)Definition

  Oct 2026

• Theorem 1 — strong duality $I=J$, a dual optimizer $(\lambda^*,\varphi_{\lambda^*})$ and complementary slacknessOpen

  Oct 2026

• Remark 1, (9) — $I=\inf_{\lambda\ge0}\{\lambda\delta+E_\mu[\sup_y\{f(y)-\lambda c(X,y)\}]\}$Open

  Oct 2026

• Proposition 7 — $\inf_{(\lambda,\varphi)\in\Lambda(S_\pi\times S_\pi)}J(\lambda,\varphi)\le I$Open

  Oct 2026

• Proposition 5 — strong duality and a primal optimizer on compact $S$ with continuous costOpen

  Oct 2026