Theorem 10.72 (1): clause (a). Under the displayed g-regularity, smoothness, domain, and
Bregman-potential hypotheses, if the non-Euclidean proximal-gradient trajectory uses either the
constant stepsize rule L_k = L_f or backtracking procedure B5, then the objective sequence
F(x^k) is nonincreasing.
Helper for Theorem 10.72: a finite linear term is convex as an extended-real-valued function.
Helper for Theorem 10.72: the source linearized objective
u ↦ (⟨∇f(x^n), u⟩ + g(u)) / L_n appearing in (10.u400), before the Bregman term is added
back.
Instances For
Helper for Theorem 10.72: adding the finite linearization term
u ↦ ⟪∇f(x^n), u⟫ / L_n does not change the effective domain of the scaled penalty, so the source
linearized objective has the same effective domain as g.
Helper for Theorem 10.72: the source linearized objective is proper and convex, so it can be
used as the ψ_n input of Theorem 9.12 on the exact textbook route.
Helper for Theorem 10.72: convexity of the smooth term gives the supporting-hyperplane
inequality for the finite-valued restriction x ↦ (f x).toReal at a differentiability point.
Helper for Theorem 10.72: the accepted upper-model inequality together with the minimizing
property of the realized step bounds the successor objective by the textbook model at any
comparator u. This is the source (10.94) and (10.u399) package.
Helper for Theorem 10.72: convexity of f replaces the local linear model at a comparator by
the true objective value, leaving only the Bregman penalty. This is the source passage from
m(u, x^n) to f(u).
Helper for Theorem 10.72: the optimal value is a lower bound for every objective value along the non-Euclidean proximal-gradient trajectory.
Helper for Theorem 10.72: every optimizer has finite nonsmooth value, so it belongs to
effective_domain g.
Helper for Theorem 10.72: the realized next iterate minimizes exactly the Chapter 9 objective
ψ_n(·) + B_ω(·, x^n) used in the source proof of clause (b).
Helper for Theorem 10.72: the source three-point identity for the Chapter 10 iterates can be
rewritten directly in add form, so the Chapter 9 optimality inequality can be normalized without
passing through an EReal subtraction chain.
Helper for Theorem 10.72: apply the source second-prox segment argument to the linearized
objective (m(·, x^n) + g) / L_n. The trajectory already provides successor membership in
dom(∂ω), so no generic exact-sum-rule qualification is needed, and the result has the plus form
of equation (10.96) in scaled_bregman_objective language.
Helper for Theorem 10.72: expose the source second-prox segment comparison for
(m(·, x^n) + g) / L_n in the chapter's scaled_bregman_objective notation.
Helper for Theorem 10.72: any trial curvature Lbar ≥ L_f is accepted by the B5 upper-model
test at the current non-Euclidean proximal-gradient iterate.
Helper for Theorem 10.72: under B5, the chosen curvature sits between the previous trial
curvature and max {η L_f, L_prev}.
Helper for Theorem 10.72: if α = max {η, s / L_f} with L_f > 0, then
α L_f = max {η L_f, s}.
Helper for Theorem 10.72: the non-Euclidean sublinear-rate owner forces the rate constant
α to be positive.
Helper for Theorem 10.72: every admissible constant/B5 stepsize satisfies the uniform bound
L_n ≤ α L_f.
Helper for Theorem 10.72: on effective_domain g, the composite objective is a finite real
sum of the finite f- and g-values.
Helper for Theorem 10.72: on effective_domain g, the objective value itself is the cast of
its toReal value.
Helper for Theorem 10.72: every positive non-Euclidean proximal-gradient iterate lies in
effective_domain g.
Helper for Theorem 10.72: every positive-index objective gap is finite, so its real value
casts back to the displayed EReal gap F(x^n) - F_opt.
Helper for Theorem 10.72: every positive-index objective gap is nonnegative as a real number.
Helper for Theorem 10.72: rewrite the stabilized Chapter 9 comparator directly into the
textbook finite-EReal one-step inequality
F(x^(n+1)) + L_n B_ω(x*, x^(n+1)) ≤ F(x*) + L_n B_ω(x*, x^n).
Helper for Theorem 10.72: route correction for clause (b). After the Chapter 9 comparator has
been stabilized, convert it to the textbook one-step gap drop
F(x^(n+1)) - F_opt ≤ L_n (B_ω(x*, x^n) - B_ω(x*, x^(n+1))).
Helper for Theorem 10.72: dividing the one-step drop by the source denominator α L_f
produces the textbook normalized gap inequality (10.100).
Helper for Theorem 10.72: summing the normalized one-step inequalities telescopes the Bregman
terms exactly as in (10.102).
Theorem 10.72 (2): clause (b). Under the same assumptions as clause (1), every positive
iterate satisfies the non-Euclidean sublinear objective-gap estimate
F(x^k) - F_opt ≤ α L_f B[ω] x* x^0 / k for every optimizer x* ∈ X^*.