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Analysis of a nonlinear optimization problem... 33Lemma 2. Let k ≥ 2 be an arbitrary, but fixed integer. Assume that the one-dimensional optimizationproblem1 − y kmax t(y k ) = 1 +y k + 2 − k − (k − 1)(2y k ) −1/(k−1) ,(11)subject to 0.25 ≤ y k ≤ 0.5has a unique optimal solution y k = y ∗ ∈ [0.25, 0.5] with t(y ∗ ) = t ∗ . Then the k–dimensional problem(10) also has a unique optimal solution, and it is given byy 1 = 1 2 , y i = 1 2 (2y∗ ) i−1k−1 , i = 2, . . . , k.Moreover, the optimum of the original problem also equals to t ∗ .For this modified form of the original problem the following statements hold:Lemma 3. [1] The optimal solution of (11) converges to the maximum of the functionin the interval [ 1 4 , 1 2] if k → ∞.1 − xf (x) := 1 +x + 1 + ln ( )1(12)x − ln (2)1Lemma 4. [1] The maximum of the function f(x) is 1 − “ ” ≈ 1.3871 in the interval [ 1W −2 −1e 3 +14 , 1 2 ],where W −1 (x) is a branch of the Lambert function.We solved (11) with a branch–and–bound global optimization algorithm using interval arithmeticcalculations ( [10, 13–15]) for the parameter values of k = 2, . . . , 10, 20, 50, 100, 1000. Ineach case, we have managed to verify the existence and uniqueness of y ∗ , and we have obtainedthe guaranteed interval enclosures of both y ∗ and t ∗ with high precision. The enclosureof y ∗ allowed us to verify the additional prerequisite y ∗ ≤ 0.5 as well, and thus, to constructthe interval enclosures of the unique solution of the original problem. The fourth column ofTable 1 contains the the optimum values for the solved problem instances up to 12 digits. Dueto the precision of the interval calculations there is a guarantee that the results are accurate inthe displayed digits.Table 1. Improved lower bounds for a variation of the 2-batched bin packing problem when only k + 1 item sizesare allowed known in advance.k p Equidistant points (y-s) Arbitrary points (y-s)1 2 1.3333... —2 3 1.3658... 1.36602540378...3 4 1.3738... 1.37393876913...4 5 1.3773... 1.37753136189...5 6 1.3793... 1.37958528769...6 7 1.38051 1.38091512540...7 8 1.38136 1.38184652163...8 9 1.38198 1.38253525895...9 10 1.38246 1.38306525702...10 11 1.38296 1.38348573275...20 21 1.38509 1.38534022765...50 51 1.38631 1.38642436208...100 101 1.38673 1.38678113846...1000 1001 1.38709 1.38710030535...