Now I present the constrained optimization procedure to compute the sharp lower bound under MTR. I pay particular attention to the special case where a$_{k}$$_{+1}$ − a$_{k}$ = δ for each integer k at the optimum. In this case, the lower bound reduces to and computation of (B.1) poses a simple one-dimensional optimization problem. Let and where Step 4. Repeat Step 3 for J = K + 1 , . . . , 2 K. 25 sup 0 ≤ y ≤ δ ∞ ∑ k = −∞ max ( F$_{1}$ ( y + ( k + 1) δ ) − F$_{0}$ ( y + kδ ) , 0) , (B.1) V ( δ ) = sup 0 ≤ y ≤ δ ∞ ∑ k = −∞ max ( F$_{1}$ ( y + ( k + 1) δ ) − F$_{0}$ ( y + kδ ) , 0) , V$_{K}$ ( δ ) = max y ∈{ y $^{∗}$+ kδ } ∞ k = −∞ K ∑ k = − K max ( F$_{1}$ ( y + ( k + 1) δ ) − F$_{0}$ ( y + kδ ) , 0) , sup { a$_{k}$ } J k = − $_{J}$∈S J,K δ ($_{̂}$ y ) J ∑ k = − J max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } , (B.2) S J,K δ ( y ) =      { a$_{k}$ } J k = − J ; a$_{J}$ ≤ y + Kδ, a$_{−}$$_{J}$ ≥ y − Kδ, 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ, δ < a$_{k}$$_{+2}$ − a$_{k}$ for each integer k      , ̂ y = arg max y ∈{ y $^{∗}$+ kδ } ∞ k = −∞ K ∑ k = − K max ( F$_{1}$ ( y + ( k + 1) δ ) − F$_{0}$ ( y + kδ ) , 0) . Step 1. Compute V ( δ ) . Step 2. To further reduce computational costs, set K to be a nonnegative integer satisfying | V ( δ ) − V$_{K}$ ( δ ) | < ε for small ε > 0 . 24 Step 3. For J = K , solve the following optimization problem: where y ∗ ∈ arg max 0 ≤ y ≤ δ ∑ ∞ k = $_{−∞}$max ( F$_{1}$ ( y + ( k + 1) δ ) − F$_{0}$ ( y + kδ ) , 0) and K is a nonnegative integer. It is not straightforward to solve the problem (B.2) numerically in Step 3; the function max { x, 0 } is non- differentiable. Furthermore in practice, marginal distribution functions are often estimated in a complicated $^{24}$I put ε = 10 − 5 for the implementation in Section 4 and Section 5. $^{25}$By Lemma B.1, I considered J = K, K + 1 , . . . , 2 K for the sequence { a$_{k}$ } J k = − J and compared the values of local maxima achieved by { a$_{k}$ } J k = − J with V$_{K}$ ( δ ) 74