Helper for Lemma 14.4 InactiveBlockSupport helper: the current second block is always an
exact minimizer of the current x₂-subproblem.
Helper for Lemma 14.4 InactiveBlockSupport helper: every outer iterate of the two-block
trajectory remains in effective_domain F, and the full objective never exceeds its initial
value.
Helper for Lemma 14.4 InactiveBlockSupport helper: the intermediate half-step
x^{k+1/2} = (x₁^{k+1}, x₂^k) is finite and lies below the initial objective level.
Helper for Lemma 14.4 InactiveBlockSupport helper: a convex differentiable real-valued
function on Set.univ satisfies the first-order support inequality in fderiv form.
Helper for Lemma 14.4 InactiveBlockSupport helper: once a supporting affine lower bound is known for every point in a fixed target fiber, the same bound descends to the corresponding partial infimum.
Helper for Lemma 14.4 InactiveBlockSupport helper: if a convex extended-real-valued function
is touched from above at x0 by a convex differentiable real-valued majorant, then the gradient
of that majorant supports the convex function at every comparison point.
Helper for Lemma 14.4 InactiveBlockSupport helper: finiteness of the full objective at a pair forces finiteness of the first penalty term at the same point.
Helper for Lemma 14.4 InactiveBlockSupport helper: finiteness of the full objective at a pair forces finiteness of the second penalty term at the same point.
Helper for Lemma 14.4 InactiveBlockSupport helper: finiteness of the full objective at a pair puts the first penalty term in its effective domain.
Helper for Lemma 14.4 InactiveBlockSupport helper: finiteness of the full objective at a pair puts the second penalty term in its effective domain.
Helper for Lemma 14.4 InactiveBlockSupport helper: convexity of the frozen smooth slice gives its first-order support inequality at the current first block.
Helper for Lemma 14.4 InactiveBlockSupport helper: exact minimization of the current second
block identifies the first partial infimum φ₁(x₁^k) with the current objective value.
Helper for Lemma 14.4 InactiveBlockSupport helper: global optimality of xStar identifies the
fiber infimum φ₁(xStar.1) with the full objective value F xStar.
Helper for Lemma 14.4 InactiveBlockSupport helper: once the current iterate is finite, the
inactive marginal η₁(y₁) = inf_z₂ (f(y₁, z₂) + g₂(z₂)) is attained at the current second
block.
Helper for Lemma 14.4 InactiveBlockSupport helper: the optimizer's second block is one witness
for the inactive marginal at xStar.1.
Helper for Lemma 14.4 InactiveBlockSupport helper: reattaching the active penalty to the current inactive marginal recovers the full current objective value.
Helper for Lemma 14.4 InactiveBlockSupport helper: reattaching the active penalty to the
optimizer witness bounds the inactive marginal by the full objective at xStar.
Helper for Lemma 14.4 InactiveBlockSupport helper: the current second-block fiber is a convex
touching majorant of the first inactive marginal, so η₁ inherits the frozen-slice first-order
support inequality at xStar.1.
Helper for Lemma 14.4 InactiveBlockSupport helper: the prox-gradient candidate on the current
x₁-slice should already satisfy the full optimizer gap bound. This is the remaining x₁-side
structural blocker after the exact-step comparison has been isolated.
Lemma 14.4 InactiveBlockSupport helper: the prox-gradient candidate on the current
x₁-slice satisfies the optimizer gap bound before comparing the exact alternating-minimization
half-step against that candidate.
Helper for Lemma 14.4 InactiveBlockSupport helper (1): if x1 and x2 are generated by the
two-block
alternating minimization method and the initial pair (x1 0, x2 0) lies in effective_domain F,
then the half-step objective gap satisfies
F(x^{k+1/2}) - F(x^*) ≤ ‖G^1_{L₁}(x^k)‖ * ‖x^k - x^*‖, with
x^{k+1/2} = (x1 (k + 1), x2 k). The intended Chapter 14 proof route is the established
current-fiber partial-infimum support bridge, followed by support on the fixed xStar.1 fiber.
Helper for Lemma 14.4 InactiveBlockSupport helper: convexity of the frozen smooth slice gives its first-order support inequality at the current second block.
Helper for Lemma 14.4 InactiveBlockSupport helper: exact minimization of the half-step first
block identifies the second partial infimum φ₂(x₂^k) with the half-step objective value.
Helper for Lemma 14.4 InactiveBlockSupport helper: global optimality of xStar identifies the
fiber infimum φ₂(xStar.2) with the full objective value F xStar.
Helper for Lemma 14.4 InactiveBlockSupport helper: once the half-step is finite, the inactive
marginal η₂(y₂) = inf_z₁ (f(z₁, y₂) + g₁(z₁)) is attained at the updated first block.
Helper for Lemma 14.4 InactiveBlockSupport helper: the optimizer's first block is one witness
for the inactive marginal at xStar.2.
Helper for Lemma 14.4 InactiveBlockSupport helper: reattaching the inactive penalty to the current second marginal recovers the half-step objective value.
Helper for Lemma 14.4 InactiveBlockSupport helper: reattaching the inactive penalty to the
optimizer witness bounds the second inactive marginal by the full objective at xStar.
Helper for Lemma 14.4 InactiveBlockSupport helper: the half-step first-block fiber is a convex
touching majorant of the second inactive marginal, so η₂ inherits the frozen-slice first-order
support inequality at xStar.2.
Helper for Lemma 14.4 InactiveBlockSupport helper: the prox-gradient candidate on the current
x₂-slice should already satisfy the full optimizer gap bound. This is the remaining x₂-side
structural blocker after the exact-step comparison has been isolated.
Helper for Lemma 14.4 InactiveBlockSupport helper (2): if x1 and x2 are generated by the
two-block
alternating minimization method and the initial pair (x1 0, x2 0) lies in effective_domain F,
then the full-step objective gap satisfies
F(x^{k+1}) - F(x^*) ≤ ‖G^2_{L₂}(x^{k+1/2})‖ * ‖x^{k+1/2} - x^*‖, with
x^{k+1} = (x1 (k + 1), x2 (k + 1)) and x^{k+1/2} = (x1 (k + 1), x2 k). Its canonical bridge
is the symmetric partial-infimum support descent on the current half-step fiber.