# Variables Reduction in Sequential Resource Allocation Problems
## 3 Optimal Control via Virtual Resource Price
In the presence of such virtual price \(A\in\mathbb{R}\), each the decision-maker would examine the virtual charges for the resource consumption and chose an activity accordingly, by obtaining  
\[\mathcal{X}_{t}^{e,z}(A)\text{, a minimizer of }\xi\mapsto C_{t}^{e,z}(\xi)+ AE_{t}^{e,z}(\xi)\text{ over }\xi\in\Xi. \tag{12}\]  
Obviously, such mapping \(A\mapsto\mathcal{X}_{t}^{e,z}(A)\) represents an optimal activity depending on the current state \((e,z)\in I\times\mathbb{R}^{d}\), time \(t=0,\ldots,T-1\), given the resource price \(A\in\mathbb{R}\). To some degree, this mapping can be interpreted as the willingness to save resources by following a less profitable strategy in response to an increased value of the resource. For ease of understanding, let us postpone the discussion concerning the existence of the minimizer in (12) and the properties of the relation \(A\mapsto\mathcal{X}_{t}^{e,z}(A)\) which are crucial for the targeted results. The important assumption for now is that for each state \((e,z)\) there exists a bounded interval \(\mathcal{A}^{e,z}\subset\mathbb{R}\) such that the one-parameter family of activities  
\[\xi\in\{\mathcal{X}_{t}^{e,z}(A)\in\Xi\mid A\in\mathcal{A}^{e,z}\}\subset\Xi \tag{13}\]  
contains only the \"best candidates\" (for the purpose of minimization in (11)). They can be used in the Bellman recursion, replacing the minimization in (11) by (16) as follows  
\[v_{T}(e,z)=C_{T}(e,z), \tag{14}\] \[\tilde{v}_{t+1}(e,z)=\left\{ \begin{array}{ll}\mathbb{E}(v_{t+1} (e,Z_{t+1})\,|\,Z_{t}=z)&e\in I\\ +\infty&e\
ot\in I\end{array} \right.z\in\mathbb{R}^{d},\] (15) \[v_{t}(e,z)=\min_{A\in\mathcal{A}^{e,z}}\left(C_{t}^{e,z}( \mathcal{X}_{t}^{e,z}(A))+\tilde{v}_{t+1}(e+E_{t}^{e,z}(\mathcal{X}_{t}^{e,z }(A)),z)\right). \tag{16}\]  
Please note that now the minimization in (16) must be performed merely over a curve (13) instead of over the whole space \(\Xi\) as in (11). In practical applications, such reduction can provide a reasonable (numerical) approach to a virtually unsolvable problem. The main question here is whether  
\[\begin{array}{l}\text{the value functions }(V_{t})_{t=0}^{T}\text{ from (\ref{eq  
while the terminal costs are determined by  
\[C_{T}:I\times\mathbb{R}^{d}\to\mathbb{R},\ \ (e,z)\mapsto C_{T}(e,z). \tag{19}\]  
All our considerations rely on additional assumptions on the functions (18) which we formulate next. To ensure that the minimization in (11) is well-defined, let us agree that there exists an _idle_ activity which does not consume any resources:  
\[\left. \begin{array}{l}\text{for each }(e,z)\in I\times\mathbb{R}^{d}\text{ there }\\ \text{exits }\xi\in\Xi\text{ such that }E_{t}^{e,z}(\xi)=0.\end{array} \right\} \tag{20}\]  
To ease our argumentation, we suppose that  
\[\left. \begin{array}{l}\text{for each }(e,z)\in I\times\mathbb{R}^{d}\text{ there is an interval }\mathcal{A}^{e,z}\subset\mathbb{R}\text{ such that }\\ \text{the minimizer }\mathcal{X}_{t}^{e,z}(A)\in\Xi\text{ to }\eqref{eq:12}\text{ exists for each }A\in\mathcal{A}^{e,z}.\end{array} \right\} \tag{21}\]  
Furthermore, let us propose a mild technical assumption  
\[\left. \begin{array}{l}\text{for each }z\in\mathbb{R}^{d},\xi^{\prime}\in\Xi \text{ and }e,e^{\prime}\in I\text{ with }e<e^{\prime}\text{ there exists }\\ \xi\in\Xi\text{ reaching the same level }e+E_{t}^{e,z}(\xi)=e^{\prime}+E_{t}^{e^{\prime},z}( \xi^{\prime})\text{ of }\\ \text{resource consumption at no greater costs }C_{t}^{e,z}(\xi)\leq C_{t}^{e^{\prime},z}( \xi^{\prime}).\end{array} \right\} \tag{22}\]  
To determine the desired minimizer \(\xi^{*}=\mathcal{X}_{t}^{e,z}(A^{*})\) to (11), we rely on the following natural technical assumption  
\[\left. \begin{array}{l}\text{for each }(e,z)\in I\times\mathbb{R}^{d}\text{ and }t=0,\ldots,T-1,\\ \text{the function }A\mapsto E_{t}^{e,z}(\mathcal{X}_{t}^{e,z}(A))\text{ is continuous on }\\ \mathcal{A}^{e,z}\text{, strictly decreasing, and possesses a root.}\end{array} \right\} \tag{23}\]  
Finally, we require some convexity properties in the sense that  
\[\left.