DOMINO: Data-driven Optimization of bi-level Mixed-Integer NOnlinear Problems

Abstract

The Data-driven Optimization of bi-level Mixed-Integer NOnlinear problems (DOMINO) framework is presented for addressing the optimization of bi-level mixed-integer nonlinear programming problems. In this framework, bi-level optimization problems are approximated as single-level optimization problems by collecting samples of the upper-level objective and solving the lower-level problem to global optimality at those sampling points. This process is done through the integration of the DOMINO framework with a grey-box optimization solver to perform design of experiments on the upper-level objective, and to consecutively approximate and optimize bi-level mixed-integer nonlinear programming problems that are challenging to solve using exact methods. The performance of DOMINO is assessed through solving numerous bi-level benchmark problems, a land allocation problem in Food-Energy-Water Nexus, and through employing different data-driven optimization methodologies, including both local and global methods. Although this data-driven approach cannot provide a theoretical guarantee to global optimality, we present an algorithmic advancement that can guarantee feasibility to large-scale bi-level optimization problems when the lower-level problem is solved to global optimality at convergence.

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Acknowledgements

The authors would like to acknowledge the funding and support provided by the U.S. National Institutes of Health Superfund Research Program (NIH P42-ES027704), the National Science Foundation projects INFEWS (1739977) and PAROC (CBET-1705423), the U.S. Department of Energy project RAPID SYNOPSIS (DE-EE0007888-09-03), the Texas A&M University Superfund Research Center and the Texas A&M Energy Institute. Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing. The manuscript contents are solely the responsibility of the grantee and do not necessarily represent the official views of the NIH. Further, NIH does not endorse the purchase of any commercial products or services mentioned in the publication.

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Appendix

Appendix

Best found solutions for benchmark problems 18, 46 and 47

Problem 18 (“wk_2015_01”):

\(x^{*} = 9.999776\), \(y^{*} = 9.9998\), \(f_{best} = 4.5443471 \cdot 10^{-7}\), \(F_{best} = 99.9955201008\).

Lower Level Relative Gap: 0 (Retrieved from CPLEX version 12.8.0.0)

Problem 46 (“wk_2015_04”):

\(x^{*}_{1} = 0, x^{*}_{2} = 0, y^{*}_{1} = 0, y^{*}_{2} = 0, y^{*}_{3} = 0, y^{*}_{4} = 0, f_{best} = 0, F_{best} = 0\).

Lower Level Relative Gap: \(1 \cdot 10^{-9}\) (Retrieved from ANTIGONE version 1.1)

Problem 47 (“wk_2015_06”):

\(x^{*}_{1} = 0.000984369218350, x^{*}_{2} = -0.001021751016379, x^{*}_{3} = 1.663984077237546, x^{*}_{4} = -0.076938496530056, y^{*}_{1} = -1.0187598163, y^{*}_{2} = 1.0574476104, y^{*}_{3} = -0.0004531744, y^{*}_{4} = 0, f_{best} = -5, F_{best} = 0.0000045078\).

Lower Level Relative Gap: \(1.76 \cdot 10^{-7}\) (Retrieved from BARON version 18.11.12)

Notation for the food-energy-water nexus case study

\(e\) :

efficiency

\(energy\) :

energy

\(max\) :

maximum

\(min\) :

minimum

\(profit\) :

profit

\(total\) :

total

\(trans\) :

transportation

\(H_{2}O\) :

water

List of land processes considered in the food-energy-water nexus case study

  • Energy Land Processes

    1. 1.

      Solar Energy

    2. 2.

      Wind Energy

  • Agricultural Processes

    1. 3.

      Fruit Production

    2. 4.

      Vegetable Production

    3. 5.

      Livestock Grazing

Agricultural developer’s problem

The chosen land allocation problem considers a piece of land which will be processed by an agricultural developer over 4 seasons in a climate similar to that of Texas, U.S. and is divided into 8 equal (1 km2) plots. The nomenclature for this problem is provided in Table 9. On each piece of land, a subset of agricultural and energy land processes can occur, where fruit production, vegetable production, and livestock grazing are representatives of agricultural processes defined by the subset \(T_{A}\), whereas solar energy and wind energy are representatives of energy land processes, defined by the subset \(T_{E}\). Two important properties regarding these subsets are given in Eqs. 4 and 5.

$$\begin{aligned} T_{A} \cup T_{E}= & {} {T_{L}} \end{aligned}$$
(4)
$$\begin{aligned} T_{A} \cap T_{E}= & {} \emptyset \end{aligned}$$
(5)
Table 9 Nomenclature for the Food-Energy-Water Nexus case study
Table 10 Land properties for the case study. These limit the processes that can occur on each plot over 4 seasons, defined by the binary variable \(y_{i,j,k}\). The water availability is defined by the binary variable \(y^{H_{2}0}_{j}\). 1 indicates existence and 0 indicates absence of that property

The agricultural producer will be subject to various constraints regarding the properties of the land, the properties of the agricultural and energy production processes while making an optimal decision towards its own objective. First, the land characteristics will affect the selection of any process that can occur in each land plot. If good soil is not available in a plot section, agricultural processes are restricted to not to take place in that land section for all seasons. If the adequate sun is not available in a plot section, solar energy will not be implemented in that land section for all seasons. Finally, if a plot section does not have access to the adequate amount of wind, wind energy production will not be implemented in that land section for all seasons. These characteristics are summarized in Table 10. Based on this information, constraints regarding water transportation can be defined for the problem such as water must be transported to the land if there is no water on a plot and an agricultural process is selected to occur on that plot:

$$\begin{aligned} y^{trans,H_{2}O}_{i,j,k} \le y_{i,j,k} + y^{H_{2}0}_{j} \qquad \forall i \in T_{A},j,k \end{aligned}$$
(6)

No water will be transported, if water is already available on the plot:

$$\begin{aligned} y^{trans,H_{2}O}_{i,j,k} \le 1 - y^{H_{2}0}_{j} \qquad \forall i \in T_{A},j,k \end{aligned}$$
(7)

No water should be transported, if there is no water on the plot and no agricultural process is selected to occur on that plot:

$$\begin{aligned} y^{trans,H_{2}O}_{i,j,k} \ge y_{i,j,k} - y^{H_{2}0}_{j} \qquad \forall i \in T_{A},j,k \end{aligned}$$
(8)

In addition to the land properties, there are other constraints that further influence the selection of land processes and restrict the feasible space for this case study. The constraints regarding the selection of land processes is imposed such that at least one land process must be allocated on each plot.

$$\begin{aligned} \sum _{i \in I}^{} y_{i,j,k} \ge 1 \qquad \forall j,k \end{aligned}$$
(9)

Furthermore, it is not practical to have solar panels and agricultural production on the same plot. Thus, at most one out of solar energy, fruit, vegetables and livestock can be allocated in one plot:

$$\begin{aligned} \sum _{i \ne 2, {i \in T_{L}}}^{} y_{i,j,k} \le 1 \qquad \forall j,k \end{aligned}$$
(10)

Wind energy will occupy minimal space on the land plot, compared to solar energy production systems, hence both wind energy and either fruit or vegetable production can be allocated on the same plot:

$$\begin{aligned} \sum _{i =2}^{4} y_{i,j,k} \le 2 \qquad \forall j,k \end{aligned}$$
(11)

Moreover, only one energy process is allowed on a plot:

$$\begin{aligned} \sum _{i \in T_{E}}^{} y_{i,j,k} \le 1 \qquad \forall j,k \end{aligned}$$
(12)

If an energy process is selected in a plot, the type of energy production will stay the same throughout the year, since it is too expensive to move equipment over seasons:

$$\begin{aligned} \begin{array}{c} y_{i,j,k+1} \ge y_{i,j,k} \qquad \forall i \in T_{E},j,k \le card(k)-1 \\ \end{array} \end{aligned}$$
(13)

Second, the seasonal differences must be considered, as these can impact the energy demand, water transportation cost, water availability for irrigation and efficiency of energy production processes. For example, in seasons with rainfall, such as winter, spring and fall, the transportation cost for water will be less and less water will be required for irrigation. On the other hand, the solar systems will have lower efficiency due to the reduced amount of sunshine throughout these seasons. A similar analysis is also done for the summer, where there is going to be greater demand for energy and water, and higher transportation costs for water will be in effect. However, the solar systems will have greater efficiency since there will be plenty of sunshine during summer. Hence, both spatial and time scenarios are considered and their respective parameters are included in the model equations (for the parameters please see Tables 1114).

The land processes will be quantified on the amount of energy produced or agricultural yield, if an energy or an agricultural process is selected, respectively. It is important to note that, if an energy process is selected for a given plot in a given season, a fixed amount of energy can be produced from these technologies:

$$\begin{aligned} \begin{array}{c} EP_{1,j,k} = P^{e}_{1,k} \cdot 50 \cdot y_{1,j,k} \qquad \forall j,k \\ EP_{2,j,k} = P^{e}_{2,k} \cdot 1000 \cdot y_{2,j,k} \qquad \forall j,k \\ \end{array} \end{aligned}$$
(14)

Likewise, the yield for agricultural processes can be calculated as a function of water and energy consumption. The parameter \(P^{e}_{i,k}\) is used to take in consideration the changes in efficiency of land processes over different seasons.

$$\begin{aligned} Y_{i,j,k} = P^{e}_{i,k} \big (M^{energy}_{i} \cdot EC_{i,j,k} + M^{H_{2}O}_{i} \cdot W_{i,j,k}\big ) \qquad \forall i \in T_{A},j,k \end{aligned}$$
(15)

The amount of energy consumption and water consumption (from an already existing source) by agricultural processes, which are used to calculate the yield in Eq. 15, are bounded. Note that the lower bound on the water consumption depends on seasonal effects (dry seasons versus seasons with rainfall), hence multiplied by its respective parameter, \(D^{H_{2}O}_{k}\).

$$\begin{aligned} \begin{array}{lll} L^{energy}_{i} \cdot y_{i,j,k} \le EC_{i,j,k} \le U^{energy}_{i} \cdot y_{i,j,k} \qquad \forall i \in T_{A},j,k \\ D^{H_{2}O}_{k} \cdot L^{H_{2}O}_{i} \cdot y_{i,j,k} \le W_{i,j,k} \le U^{H_{2}O}_{i} \cdot y_{i,j,k} \qquad \forall i \in T_{A},j,k \\ \end{array} \end{aligned}$$
(16)

In addition to the box-constraints, it is important to supply adequate amount of water to each plot in each season for the agricultural land processes. Thus, the amount of water consumption (source-based and transportation-based) is set to be at least 200 times greater than the energy consumption in each plot and in each season:

$$\begin{aligned} \sum _{i \in T_{A}}W_{i,j,k} + D^{H_{2}O}_{k}\cdot \sum _{i \in T_{A}} W^{trans}_{i,j,k} \ge 200 \cdot \sum _{i \in T_{A}} EC_{i,j,k} \qquad \forall j,k \end{aligned}$$
(17)

The amount of water transported for agricultural processes is also bounded and affected by the seasonal differences:

$$\begin{aligned} \begin{array}{l} D^{H_{2}O}_{k} \cdot L^{H_{2}O}_{i} \cdot y^{trans,H_{2}O}_{i,j,k} \le W^{trans}_{i,j,k} \le U^{H_{2}O}_{i} \cdot y^{trans,H_{2}O}_{i,j,k} \qquad \forall i \in T_{A},j,k \\ \end{array} \end{aligned}$$
(18)

As described previously in Sect. 3.2, the objective of the agricultural developer is to maximize its profit. The profit calculation for all land processes includes the money made from energy production and the yield from the agricultural processes, if an energy or an agricultural process is selected, respectively. For energy producing land processes profit is given as:

$$\begin{aligned} G^{profit}_{i,j,k} = M^{profit}_{i} \cdot P^{profit}_{i,k} \cdot EP_{i,j,k} + \acute{S}_{i,j,k} \qquad \forall i \in T_{E},j,k \end{aligned}$$
(19)

For agricultural processes, the profit is given as:

$$\begin{aligned} G^{profit}_{i,j,k} = M^{profit}_{i} \cdot Y_{i,j,k} + \acute{S}_{i,j,k} \qquad \forall i \in T_{A},j,k \end{aligned}$$
(20)

The profit calculations also considers the relevant subsidies (\({\acute{S}_{i,j,k}}\)) offered by the government agencies for developing different processes on the land, where these subsidies should only be considered in the profit when their respective land process is activated.

$$\begin{aligned} {\acute{S}_{i,j,k}} = S_{i} \cdot y_{i,j,k} \qquad \forall i,j,k \end{aligned}$$
(21)

To avoid this bilinear term that appears in the profit equation, the variable \({\acute{S}_{i,j,k}}\) and its Big-M formulation is introduced in Eqs. 2124, where \(BM\) is the Big-M parameter.

$$\begin{aligned} {S}_{i}\le & {} BM \cdot \sum _{j} \sum _{k} y_{i,j,k}\qquad \forall i \end{aligned}$$
(22)
$$\begin{aligned} \acute{S}_{i,j,k}\le & {} BM \cdot y_{i,j,k} \qquad \forall i,j,k \end{aligned}$$
(23)
$$\begin{aligned} \acute{S}_{i,j,k}\le & {} S_{i}\qquad \forall i,j,k \end{aligned}$$
(24)

Moreover, the agricultural developer is interested in maximizing the total profit, which is a function of the total energy production, total yield from agricultural production and total water consumption. The total energy, \(E^{total}\), is defined as the difference between total energy produced from energy land processes and total energy consumed by the agricultural processes in all plots throughout the 4 seasons.

$$\begin{aligned} E^{total} = \sum _{i \in T_{E}}\sum _{j}^{}\sum _{k}^{} EP_{i,j,k} - \sum _{i \in T_{A}}\sum _{j}^{}\sum _{k}^{} EC_{i,j,k} \end{aligned}$$
(25)

Similarly, the total yield, \(Y^{total}\), is the summation of yield of all agricultural processes over all plots and 4 seasons.

$$\begin{aligned} Y^{total} = \sum _{i \in T_{A}}\sum _{j}^{}\sum _{k}^{} Y_{i,j,k} \end{aligned}$$
(26)

The total water consumption, \(W^{total}\), includes both the amount of water consumed from a natural source (i.e. water already existing as in the land properties, given in Table 10) and from a transported source. The transported total water also considers seasonal demand, defined by the parameter \(D^{H_{2}O}_{k}\).

$$\begin{aligned} W^{total} = \sum _{i \in T_{A}}\sum _{j}^{}\sum _{k}^{} W_{i,j,k} + \sum _{k}^{}D^{H_{2}O}_{k}\sum _{i \in T_{A}}\sum _{j}^{} W^{trans}_{i,j,k} \end{aligned}$$
(27)

The total profit, \(G^{profit,total}\), is calculated by subtracting the total water transportation cost throughout all plots, all seasons and all agricultural land processes from the cumulative profit from all land processes. The cost of water transportation is assumed to be $10/kg of water. In addition, the cost of transportation is impacted by seasonal differences, as explained previously, hence the formulation includes the \(C^{H_{2}O,trans}_{k}\) parameter to account for such effects. The objective function of the LLP is given as:

$$\begin{aligned} G^{profit,total} = \sum _{i}\sum _{j}^{}\sum _{k}^{} G^{profit}_{i,j,k} - 0.01 \cdot \sum _{k}^{}C^{H_{2}O,trans}_{k}\sum _{i \in T_{A}}\sum _{j}^{} W^{trans}_{i,j,k} \end{aligned}$$
(28)

Finally, the continuous variables defined in Eqs. 2528 are bounded and their respective values are obtained through minimizing and maximizing each variable as the sole objective to the land allocation problem.

$$\begin{aligned} \begin{array}{l} 0 \le W^{total} \le 2.46 \cdot 10^{9} \\ 0 \le Y^{total} \le 13860 \\ 0 \le E^{total} \le 21945 \\ G^{profit,total} \ge 0 \\ \end{array} \end{aligned}$$
(29)

The variables defined in Eqs. 2528 as well as their respective bounds, provided in Eq. 29, are used to enumerate the upper-level objective function of the government regulators. The ULP is discussed in detail in the following section.

Government regulators’ problem

As shown in Eq. 3, the objective of the government regulators is to minimize the nexus stress. However, the mathematical quantification of the nexus, which will take in consideration of the trade-offs between food, energy and water, has not yet been fully established. Recently, Avraamidou et al. [13] has introduced a methodology to develop a FEW-N metric, which brings relevant decision elements and their respective quantification together through rth order averaging. In this work, we adopt this idea through a similar methodology where a single geometric metric, i.e. the area of a triangle, is used to represent the FEW-N metric as the government regulators’ objective. An illustration of the FEW-N metric is provided in Fig. 7.

Fig. 7
figure7

FEW-N metric represented as the area of a triangle. Shaded area demonstrates an example solution to FEW-N

In Fig. 7, the corners of the triangle represent the scaled quantities of each FEW-N element, where their respective values lie between 0 and 1. In this case, a value of 1 represents the best possible scenario and 0 represents the worst. The objective of the government regulators is to maximize the best possible scenario for each element, namely minimizing the total water consumed and maximizing the total energy and food produced, which essentially translates into maximizing the area of the triangle. The explicit formulation of this objective is provided in Eq. 30.

$$\begin{aligned} FEW_{metric}= & {} \Bigg [\frac{E^{total} - E^{min}}{E^{max} - E^{min}}\cdot \Bigg (1-\frac{W^{total} - W^{min}}{W^{max}-W^{min}}\Bigg ) + \frac{E^{total} - E^{min}}{E^{max} - E^{min}}\cdot \frac{Y^{total} - Y^{min}}{Y^{max} - Y^{min}} \nonumber \\&+ \Bigg (1-\frac{W^{total} - W^{min}}{W^{max}-W^{min}}\Bigg ) \cdot \frac{Y^{total} - Y^{min}}{Y^{max} - Y^{min}} \Bigg ] \cdot \frac{{\sin }120^\circ }{2} \end{aligned}$$
(30)

Note that \(E^{total}\), \(Y^{total}\), and \(W^{total}\) is obtained through solving the agricultural producer’s problem, explicitly defined in Eqs. 2527, respectively.

In this case study, the government is offering subsidies (\(S_{i}\)) to the land developers for each nexus element, as much as their budget (\(B_{i}\)) allows.

$$\begin{aligned} 0 \le S_{i} \le B_{i} \qquad \forall i \end{aligned}$$
(31)

These subsidies further motivate the land owner to properly allocate and utilize the land to maximize their own profit (Eqs. 1920). The upper bound on the total governmental budget is set to be $250M where this is allocated equally among all land processes. Essentially, the goal of the government agency is to decide on the amount of subsidies to be offered to the agricultural producer in such a way that the objective function defined in Eq. 30 is maximized.

Parameters

Parameter values are tabulated in Tables 1114, where 4 seasons (autumn, winter, spring, and summer) are considered for the FEW-N case study with production starting in autumn and ending after summer. These parameters are used as multipliers to capture seasonal differences among technological efficiencies, water demand and transportation costs. The efficiency of the solar energy production process is lower in autumn and winter whereas it is higher in the summer. Likewise, the efficiency of agricultural processes is lower in winter as shown in Table 11.

Table 11 Parameter values for \(P^{e}_{i,k}\)

The profit from energy production during winter and summer should be higher since there would be higher demand for energy in very cold and hot weathers. Hence, higher multipliers are assigned for both energy production land processes, which are summarized in Table 12.

Table 12 Parameter values for \(P^{profit}_{i,k}\)
Table 13 Parameter values for \(D^{H_{2}O}_{k}\) and \(C^{trans,H_{2}O}_{k}\)
Table 14 Parameter values for \(L^{H_{2}O}_{i}\), \(U^{H_{2}O}_{i}\), \(L^{energy}_{i}\), \(U^{energy}_{i}\), \(M^{energy}_{i}\), \(M^{H_{2}O}_{i}\) and \(M^{profit}_{i}\)

Table 13 summarizes the multipliers for the minimum amount of water required as well as the cost of transporting water over 4 seasons. Both the required amount of water and the cost of transportation is expected to be higher in summertime due to elevated temperatures and higher demand for water in agricultural production. Finally, Table 14 summarizes other parameters used in the FEW-N case study.

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Beykal, B., Avraamidou, S., Pistikopoulos, I.P.E. et al. DOMINO: Data-driven Optimization of bi-level Mixed-Integer NOnlinear Problems. J Glob Optim 78, 1–36 (2020). https://doi.org/10.1007/s10898-020-00890-3

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Keywords

  • Data-driven modeling
  • Bi-level optimization
  • Global optimization
  • Grey-box optimization
  • Food-energy-water nexus