Formal proof of convexity for the level cost function in the Lagrange-multiplier bit-width optimization

Prove that, in the per-level bit-width optimization for the nested multilevel Monte Carlo (MLMC) framework described in Section 6.1, the level objective function f_ell(λ) = sqrt(V_ell * C̃_ell(d(λ))) + sqrt(V^{Δ}_ell(d(λ)) * C_ell) is convex with respect to the Lagrange multiplier λ, where for each λ the vector of real-valued bit-widths d(λ) = (d_{1,ell}, ..., d_{m_ell,ell}) solves the stationarity conditions ∂V^{Δ}_ell/∂d_{i,ell} + λ * ∂C̃_ell/∂d_{i,ell} = 0 for all i. Here C̃_ell(d) is the fixed-point cost model defined by the per-variable quadratic form, V^{Δ}_ell(d) is the variance bound used for the correction term, and V_ell and C_ell are the full-precision variance and cost for level ell. Establishing convexity would rigorously justify the observed single optimum in the λ-based optimization procedure.

Background

The paper proposes a per-level bit-width optimization using a Lagrange multiplier to trade off the correction-term variance and the low-precision computation cost on FPGAs. The level objective to be minimized is the sum sqrt(V_ell * C̃ell(d)) + sqrt(V^{Δ}_ell(d) * C_ell), where C̃_ell(d) is a quadratic cost model in the bit-widths and V^{Δ}_ell(d) is an analytical upper bound on the variance due to rounding and approximate random numbers. For a given λ, the optimal bit-widths d(λ) are obtained by solving uncoupled nonlinear equations ∂V^{Δ}_ell/∂d{i,ell} + λ * ∂C̃ell/∂d{i,ell} = 0.

The authors empirically plot the level objective as a function of λ and observe a convex shape, indicating a unique optimum. However, they explicitly state that they do not provide a formal proof of convexity. A rigorous proof would validate the optimization strategy and the use of golden-section search to locate the optimum λ.