# Variables Reduction in Sequential Resource Allocation Problems
## 6 The Variables Reduction Technique
Since this root is included in the minimization, with a finite (due to \(e\in I\)) value  
\[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]2  
the infimum is also finite  
\[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]3By the minimum property (27), for each \(A\in\mathcal{A}^{\varepsilon,z}\) it holds that  
\[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]4 \[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]5  
Furthermore, by definition of \(\mathcal{X}_{t}^{\varepsilon,z}(A)\), it holds on the other hand that  
\[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]6  
for each \(\xi\in\Xi\). Replacing \(\xi\) by \(\mathcal{X}_{t}^{\varepsilon,z}(A^{*})\), we deduce  
\[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]7  
which we combine with (70) to obtain  
\[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]8 \[\frac{\delta}{2}+V_{t}(e,z) \geq C_{t}^{e,z}(\xi)+\tilde{V}_{t+1}(e+E_{t}^{e,z}(\xi),z),\]9  
Introducing for \((\varepsilon,z)\in I\times\mathbb{R}^{d}\) the notations  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]0 \[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]1  
the inequality (71) is written as  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]2  
For a strictly increasing sequence \((A_{n}^{+})_{n\geq 1}\subset\mathcal{A}^{\varepsilon,z}\) converging to \(A^{*}\) define a sequence  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]3  
which is strictly decreasing, converges to zero due to (23), and satisfies  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]4  
because of (72). Similarly, for a strictly decreasing sequence \((A_{n}^{-})_{n\geq 1}\subset\mathcal{A}^{\varepsilon,z}\) converging to \(A^{*}\), define a sequence  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]5  
which is strictly increasing, converges to zero due to (23), and satisfies  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]6  
Now apply Lemma 4 with above sequences \((\Delta_{n}^{+})_{n\geq 1}\), \((A_{n}^{+})_{n\geq 1}\) and \((\Delta_{n}^{-})_{n\geq 1}\), \((A_{n}^{-})_{n\geq 1}\) using their properties (73) and (74) to deduce the assertion  
\[\frac{\delta}{2}+V_{t}(e^{\prime},z) \geq C_{t}^{e^{\prime},z}(\xi^{\prime})+\tilde{V}_{t+1}(e^{\prime} +E_{t}^{e^{\prime},z}(\xi^{\prime}),z).\]7  
\(\square\)  
**Lemma 4**. : _Let \(\tilde{V}:\mathbb{R}\mapsto\mathbb{R}\cup\{\infty\}\) be a convex function and \(\varepsilon\in\mathbb{R}\) such that \(\tilde{V}(\varepsilon)<\infty\). Suppose that there exist sequences \((\Delta_{n}^{+})_{n\geq 1}\subset]0,\infty[\) converging to zero and sequences \((A_{n}^{-})_{n\geq 1}\), \((A_{n}^{+})_{n\geq 1}\) converging to \(A^{*}\in\mathbb{R}\) such that_  
\[ \begin{array}{l}\tilde{V}(\varepsilon)+A_{n}^{+}\Delta_{n}^{+}\leq\tilde{V}( \varepsilon+\Delta_{n}^{+})\\ \tilde{V}(\varepsilon)+A_{n}^{-}\Delta_{n}^{-}\leq\tilde{V}(\varepsilon+ \Delta_{n}^{-})\end{array} ,\quad\text{for all $n\geq 1$}. \tag{75}\]  
_Then it holds that \(A^{*}\in\
abla\tilde{V}(\varepsilon)\)._  
Proof.