# Multicriteria Adjustable Regret Robust Optimization for Building Energy
Supply Design

Elisabeth Halser<EMAIL_ADDRESS>Elisabeth Finhold
<EMAIL_ADDRESS>Neele Leithäuser
<EMAIL_ADDRESS>Tobias Seidel
<EMAIL_ADDRESS>Karl-Heinz Küfer  karl-
<EMAIL_ADDRESS>
###### Abstract

Optimizing a building’s energy supply design is a task with multiple competing
criteria, where not only monetary but also, for example, an environmental
objective shall be taken into account. Moreover, when deciding which storages
and heating and cooling units to purchase (here-and-now-decisions), there is
uncertainty about future developments of prices for energy, e.g. electricity
and gas. This can be accounted for later by operating the units accordingly
(wait-and-see-decisions), once the uncertainty revealed itself. Therefore, the
problem can be modeled as an adjustable robust optimization problem. We
combine adjustable robustness and multicriteria optimization for the case of
building energy supply design and solve the resulting problem using a column
and constraint generation algorithm in combination with an
$\varepsilon$-constraint approach.

In the multicriteria adjustable robust problem, we simultaneously minimize
worst-case cost regret and carbon emissions. We take into account future price
uncertainties and consider the results in the light of information gap
decision theory to find a trade-off between security against price
fluctuations and over-conservatism. We present the model, a solution strategy
and discuss different application scenarios for a case study building.

###### keywords:

Building Energy Supply Design , Multicriteria Optimization , Regret Robustness
, Adjustable Robustness , Column and Constraint Generation

[label1]organization=Fraunhofer Institute for Industrial Mathematics
ITWM,addressline=Fraunhofer-Platz 1, city=Kaiserslautern, postcode=67663,
country=Germany

Novel multicriteria optimization model for building energy supply design

Consideration of cost regret with respect to price uncertainty as objective

Column and constraint generation algorithm and proof of convergence

Demonstration of the approach in a case study for an office building at an
early planning stage

##  1 Introduction

Buildings and their energy consumption are responsible for about 40% of global
greenhouse gas emissions (International Energy Agency, 2021) and for a
significant amount of many companies’ annual costs. In the following, we use
the term carbon emissions to consider all types of greenhouse gas emissions as
they can be converted to carbon equivalent emissions. The two possibly
competing criteria of minimizing costs and carbon emissions make the task of
designing a building’s energy supply an inherently multicriteria one for which
trade-offs have to be identified.

The existence of multiple criteria is not the only challenge in building
energy supply design. A second structural difficulty is that the problem
naturally comes with different decision stages. Initially, there is the first
stage device purchasing decision (_here-and-now_ decision). Throughout the
article, we use the term device to talk about all types of heating and cooling
units and storages as well as photovoltaic. Before the actual operation of the
purchased units, the costs for operating them, which are highly uncertain due
to potential price fluctuations, become known. Then, in a second decision
stage, there is the possibility to react to the then observed resource price
reality (_wait-and-see_ decisions) by controlling the heating and cooling
units. The reason for distinguishing between here-and-now and wait-and-see
variables is that at the beginning, the prices are uncertain. Because of the
two stages, where the second stage can be adjusted to the realized
uncertainty, we call the problem _adjustable_.

One way to deal with uncertainty is to consider the worst case of an assumed
set of possible scenarios, which we call uncertainty set. This approach is
called robust optimization. When considering the worst possible price
scenario, we call the problem structure multicriteria adjustable robust.

However, when solely focusing on the optimization of worst-case total costs,
it is likely that the purchasing decision is tailored for an unlikely extreme
scenario and is likely to underperform in most other, more moderate,
scenarios. Therefore, it is preferable to take a here-and-now decision which
will cause minimal regret when the actual resource prices are known. Regret
can be quantified as the difference between the minimal costs given a fixed
purchasing decision and the minimal costs that would have been possible, if
the prices had been known in advance. The concept for this kind of problem is
called regret robust optimization. Hence, in this study, we employ cost regret
alongside carbon emissions as an objective in our multicriteria adjustable
regret robust optimization.

Modeling building energy supply design as optimization problem is a large
field of research. Most of the models are linear programs (LP) or mixed
integer programs (MILP). Detailed modeling considerations are given by Wirtz
et al. (2021).

A good overview over the mathematical field of multicriteria optimization
(MCO) is for example given in the book by Ehrgott (2005). MCO problems have
more than one objective. If some of these objectives are contradicting, it is
not possible to minimize them at the same time. Therefore, compromises, also
called trade-offs, have to be found. The basic idea of MCO is to identify
solutions where no other solution outperforms them in every objective. Such
solutions are called efficient, and their image in objective space is known as
the Pareto front. One approach to approximate the Pareto front is the
$\varepsilon$-constraint method, which inspired our approach. Multicriteria
optimization has been applied to building energy supply design in different
articles, for example in the work of Richarz and Osterhage (2020) or Finke and
Bertsch (2023).

Robust optimization is an approach to deal with uncertainty in optimization
problems. A good overview is given by Ben-Tal et al. (2009) or more recently
by Bertsimas and Hertog (2022), where both books also cover adjustable
robustness. Robust optimization for energy supply design was for example done
by Moret et al. (2020) and for building renovation by Richarz et al. (2021).

Regret robustness aims for a solution that does not minimize the worst-case
outcome but the worst regret over all scenarios. Regret robustness ensures
that the optimal solution is not only determined by a potentially unlikely
extreme scenario, but rather by how much one would regret a decision when
comparing it to what would have been possible if the scenario had been known
in advance. It was applied to the energy problem by Yokoyama et al. for
continuous (Yokoyama et al., 2014) and discrete (Yokoyama et al., 2021) design
variables, where in these works the uncertainty comes from uncertain loads.

Robust multicriteria optimization allows for different definitions of
efficient solutions, of which some are given by Botte and Schöbel (2019). The
most common definition is that of point-based min-max robust efficiency, which
for building energy supply design was applied by Majewski et al. (2017).

Multicriteria adjustable robust optimization (MARO) is a relatively new topic,
that combines adjustable and multicriteria optimization and was studied in the
linear decision rule case by Chuong and Jeyakumar (2022) and for general cases
in (Halser et al., 2024a). There, different computational approaches are
compared, especially a weighted-sum scalarization and an
$\varepsilon$-constraint scalarization. Solutions which are found with these
approaches are called weighted sum MARO-efficient and constraint MARO-
efficient, respectively. Yokoyama and Ito (2001) used a weighted sum approach
to deal with regret robustness against uncertain loads, which correspond to
right-hand side uncertainty.

Inverse robustness (Berthold et al., 2024), also called information gap
decision theory, was already applied in the energy context for one (Javadi et
al., 2017) and multiple (Feng et al., 2023) objectives. It is a way of
answering the question of what size of uncertainty set the solution should be
robust against, which forms an additional objective on its own.

In this paper, we compute constraint MARO-efficient solutions for the problem
of multicriteria building energy supply design with the objectives carbon
emission and cost regret minimization. Constraint MARO-efficient solutions are
beneficial for the problem as they come with good bounds for the objectives,
which hold over all scenarios of the uncertainty set, and their solution in
the objective space can be interpreted similarly to a usual Pareto front
(Halser et al., 2024a). Both properties are especially valuable for decision
makers and make this approach valuable in practice. We will focus on
uncertainty sets that consist of possible future price fluctuations. Compared
to load uncertainties, price fluctuations are much more unpredictable and
therefore hold practical relevance. In this paper, we present a case study in
which we place the results in the context of inverse robustness by varying the
size of the uncertainty set. In this way, the decision makers can weigh up how
much price uncertainty they want to hedge against and what this might cost.

Yokoyama et al. (2014) Feng et al. (2023) Yang and Liu (2024) Fan et al.
(2014) Aghamohamadi and Mahmoudi (2019) our work multicriteria optimization X
$\surd$ws $\surd$ws X X $\surd$ adjustable $\surd$ X $\surd$ X $\surd$ $\surd$
price uncertainty X X X $\surd$ $\surd$ $\surd$ regret consideration $\surd$ X
X $\surd$ X $\surd$ inverse robust idea $\surd$e $\surd$o $\surd$e X $\surd$e
$\surd$e constraint generation $\surd$ X $\surd$ccg $\surd$ $\surd$ccg
$\surd$ccg energy system design $\surd$ X $\surd$ X X $\surd$ ws weighted sum
scalarization without aim to explore different trade-offs in objective space e
varying the sizes of the uncertainty set o maximizing the size of the
uncertainty set ccg column and constraint generation

Table 1: Classification of the contribution of this work.

Table 1 puts our work in the context of the already existing literature. There
is a great number of further articles which could be added to the table as
they deal with some of the topics. However, to the best of our knowledge,
there has not been a combination of all these points so far. To summarize our
contributions,

  * •

we compute constraint MARO-efficient solutions for the problem of minimizing
cost regret and carbon emissions of building energy supply design, thereby
combining regret robust optimization and multicriteria optimization.

  * •

we consider uncertain prices in the objective function (instead of uncertain
loads on the right-hand side as in (Yokoyama et al., 2014)) in the cost regret
optimization and show how the algorithmic ideas of Zeng and Zhao (2013) carry
over.

The remainder of this article is structured as follows. In Section 2, we give
a brief overview of our model, for which the details are given in A. In
Section 3 we explain how we apply the ideas of MARO to our problem. The
detailed formulation of the solution algorithm is given in B. In Section 4 we
discuss a case study and summarize our results in Section 5.

##  2 Model

As a basic model, yet without consideration of regret, we use a slightly
adapted model from (Halser et al., 2024b), which we summarize here, while the
detailed formulas are given in A. Unlike as in (Halser et al., 2024b), we do
not consider a third objective function of inconvenience but assume that all
predetermined loads have to be fulfilled.

The main goal in building energy supply design is to find a design
configuration $\mathbf{d}\in\mathbb{R}^{n_{d}},n_{d}\in\mathbb{N}$. For a list
of commercially available unit types and storages, the vector $\mathbf{d}$
contains the selected size. In order to find $\mathbf{d}$ and evaluate annual
costs and emissions, it is necessary to also find controls
$\mathbf{s}\in\mathbb{R}^{n_{s}},n_{s}\in\mathbb{N}$ for a representative
period of time. A suitable set of representative days $\mathcal{U}$, which can
be used to approximate annual costs and emissions, is found by k-Medoids
clustering as suggested in (Schütz et al., 2018). For all representative days
with all their time steps, the vector $\mathbf{s}$ represents the control
states for all unit types and the storage states. The decision variables
$\mathbf{d}$ and $\mathbf{s}$ have to fulfill the constraints that for all
representative days the loads are fulfilled and the technical constraints are
met. The objective expressions $costs^{p}(\mathbf{d},\mathbf{s})$ and
$co_{2}(\mathbf{d},\mathbf{s})$, which are to be minimized, represent the
annual costs and carbon emissions, respectively, caused by the design
configuration $\mathbf{d}$ with controls $\mathbf{s}$ under price vector $p$.
As we will need this split later, we introduce

$\displaystyle
co_{2}(\mathbf{d},\mathbf{s})=co_{2}^{inv}(\mathbf{d})+co_{2}^{op}(\mathbf{d},\mathbf{s}),$

where $co_{2}^{inv}(\mathbf{d})$ denotes the investment or life cycle carbon
emissions that are incorporated in producing, transporting and disposing the
devices, while $co_{2}^{op}(\mathbf{d},\mathbf{s})$ are the operational carbon
emissions. The detailed formulas are given in the appendix. In the model, all
variables are continuous and all expressions are linear.

As future price developments are very unpredictable, we introduce an
uncertainty set and consideration of regret in the model, which we will then
treat in a robust way. In the following, we use the notation

$\displaystyle p$ $\displaystyle=\begin{pmatrix}p^{e}\\\ p^{el}\\\ p^{w}\\\
p^{g}\\\ p^{h}\end{pmatrix},$

where $p^{e},p^{el},p^{h},p^{w},p^{g}\;[€{}/kWh]$ are the prices of buying
electricity, selling electricity, buying district heat, buying wood pellets
and buying gas. We define the cost regret

$\displaystyle costregret^{p}(\mathbf{d},\mathbf{s})$
$\displaystyle:=costs^{p}(\mathbf{d},\mathbf{s})-\min_{(\mathbf{d}^{*},\mathbf{s}^{*})}costs^{p}(\mathbf{d}^{*},\mathbf{s}^{*})$

as the difference between the minimal costs given purchasing decision
$\mathbf{d}$ and price scenario $p$ and the minimal costs that are possible
for this price scenario. Note that the decision variables of the minimization
have to satisfy the linear constraints $A\mathbf{d}^{*}+B\mathbf{s}^{*}\leq e$
for suitably chosen parameter matrices $A,B$ and a vector $e$ to represent the
constraints. The detailed formulation of these constraints is given in the
appendix. We obtain the model

$\displaystyle\begin{split}\min_{\mathbf{d}}\max_{p}\min_{\mathbf{s}}&\begin{pmatrix}costregret^{p}(\mathbf{d},\mathbf{s})\\\
co_{2}(\mathbf{d},\mathbf{s})\end{pmatrix}\\\ \text{s.t.
}&A\mathbf{d}+B\mathbf{s}\leq e\\\ &A\mathbf{d}^{*}+B\mathbf{s}^{*}\leq e\\\
&p\in\mathcal{P(\alpha)}.\end{split}$ ($P(\alpha)$)

$\alpha$ is a parameter for the size of the price uncertainty set
$\mathcal{P}(\alpha)$, for which different realizations like a box, polyhedron
or ellipsoid are possible choices.

The trade-off between $\alpha$ and solutions of ($P(\alpha)$) is exactly what
is addressed by inverse robustness or information gap decision theory.

##  3 Solution Strategy

The proposed problem formulation can be understood as a MARO problem. We
follow the approach of constraint MARO-efficiency from (Halser et al., 2024a),
which is the application of the $\varepsilon$-constraint method in a first-
scalarize-then-robustify manner to find solutions to ($P(\alpha)$). This is
done by restricting the annual carbon emissions to some value
$c\in\mathbb{R}$. The resulting optimization problem is

$\displaystyle\begin{split}\min_{\mathbf{d}}\max_{p}\min_{\mathbf{s}}\;&costregret^{p}(\mathbf{d},\mathbf{s})\\\
\text{s.t. }&co_{2}(\mathbf{d},\mathbf{s})\leq c\\\
&A\mathbf{d}+B\mathbf{s}\leq e\\\ &co_{2}(\mathbf{d}^{*},\mathbf{s}^{*})\leq
c\\\ &A\mathbf{d}^{*}+B\mathbf{s}^{*}\leq e\\\
&p\in\mathcal{P(\alpha)}.\end{split}$ ($P(\alpha,c)$)

The problem contains different stages, where we call the minimization of
$\mathbf{d}$ the first stage, the maximization the middle stage and the
minimization over the controls $\mathbf{s}$ the second stage problem. We use
this notation, where the middle stage is not counted, to be coherent with the
two-stage decision structure of the problem. To guarantee that the solution is
meaningful and the algorithm from B is applicable, we have to guarantee that
for every fixed $\mathbf{d}$ and all $p\in\mathcal{P}(\alpha)$ there is a
feasible second stage solution $\mathbf{s}$ to ($P(\alpha,c)$). This
assumption is called _relatively complete recourse_ assumption (Birge and
Louveaux, 2011). In our case, this means that it has to be guaranteed that all
loads can be fulfilled, while respecting all constraints, independent of the
first stage device selection. To ensure this, we restrict the investment
carbon emissions to $c$ by adding the constraint

$\displaystyle co_{2}^{inv}(\mathbf{d})\leq c$

to the first stage problem. Further, we add dummy generators to our model that
have neither investment costs nor carbon emissions, but enormous costs for
generating heating and cooling. These dummy generators can always step in for
providing the required energy. This guarantees relatively complete recourse
for all $c>0$.

We call solutions $\mathbf{d}$ to ($P(\alpha,c)$) constraint MARO-efficient
for ($P(\alpha)$). Moreover, for all price scenarios
$p\in\mathcal{P}(\alpha)$, there is the guarantee that there is a control
$\mathbf{s}$ such that for the optimal value $o$ of ($P(\alpha,c)$)

$\displaystyle costregret^{p}(\mathbf{d},\mathbf{s})$ $\displaystyle\leq o$
$\displaystyle co_{2}(\mathbf{d},\mathbf{s})$ $\displaystyle\leq c.$

To find solutions to ($P(\alpha,c)$), we essentially dualize the second stage
problem and then apply adaptive discretization (see e.g. (Blankenship and
Falk, 1976)). The detailed algorithm, including its convergence proof, is
given in B.

##  4 Case Study

###  4.1 Data Basis

For the case study, we consider a new office building in Kaiserslautern,
Germany, planned next to an existing office complex. In addition to the
offices, the building will also house data centers, that require significant
cooling. We use the data basis from (Halser et al., 2024b), where the
building, its heating and cooling loads, the weather conditions of the example
days, the unit properties, i.e. their coefficients of performance, and the
nominal price scenario are described in detail.

###  4.2 Uncertainty Set

Figure 1: Developments of gas and electricity prices for Germany from 1985 to
2022, including all taxes. Depicted are the available values for commercial
prices which are closest to the prices for companies with 4 GWh of annual
electricity as well as gas consumption. On the right side, the blue dots
correspond to the values from the left chart, while the gray dot is a value
from 2023. The gray boxes are visualizations of our choice of uncertainty set
shape.

To get a first impression of the uncertainty, we look at data from Eurostat
(2023), and observe the developments of the electricity and gas price from
Figure 1 to which we added the price from 2023 (gray dot), that was assumed in
(Halser et al., 2024b). We choose a box shape of the uncertainty set. As the
values from the last year are extreme for the known observations of the
uncertain parameters and future prices are unpredictable due to political and
economical uncertainties, this ensures that we are likely secured against
price developments of different kinds.

| $\hat{p}_{gas}$ | $\hat{p}_{el_{buy}}$ | $\hat{p}_{el_{sell}}$ | $\hat{p}_{dh}$ | $\hat{p}_{wp}$
---|---|---|---|---|---
Value [€/kWh] | 0.0934 | 0.2074 | 0.2074 | 0.095 | 0.0817
Table 2: Nominal prices for gas, buying electricity, selling electricity,
district heating and wood pellets.

We do not only consider uncertainty of gas and electricity prices but of all
prices, where we assume as nominal prices the values from Table 2. Then, we
can define a polyhedral box uncertainty set as

$\displaystyle\mathcal{P}_{box}(\alpha):=$
$\displaystyle\\{(p_{gas},p_{el_{buy}},p_{el_{sell}},p_{dh},p_{wp})^{T}\mid$
$\displaystyle
p_{gas}=\hat{p}_{gas}\cdot\beta_{gas},p_{el_{buy}}=\hat{p}_{el_{buy}}\cdot\beta_{el_{buy}},$
$\displaystyle
p_{el_{sell}}=\hat{p}_{el_{sell}}\cdot\beta_{el_{sell}},p_{dh}=\hat{p}_{dh}\cdot\beta_{dh},$
$\displaystyle p_{wp}=\hat{p}_{wp}\cdot\beta_{wp},$
$\displaystyle\beta_{gas},\beta_{el_{buy}},\beta_{el_{sell}},\beta_{dh},\beta_{wp}\in[1-\alpha,1+\alpha]\\}.$

###  4.3 Implementation and Computational Performance

All computations discussed in the following were performed on an Intel i7 with
32GB RAM and operating system Windows 11. The code was written in python 3.9,
using gurobipy 11.0.0. As solution algorithms, we implemented the constraint
generation (CG) and the column and constraint generation (C&CG) algorithm,
which are the two variants of Algorithm 1 that can be found in the appendix.
Remark 2 in the appendix suggests that using the C&CG algorithm is superior to
using the CG Algorithm. First numerical experiments confirm this suggestion.
Table 3 shows numerical results for the two algorithms for different model
sizes and parameters. In the table, the respective ratios are below one for
all comparisons, which means that the C&CG algorithm has less computational
time and fewer iterations for all considered scenarios. Therefore, in the
following, we use the C&CG algorithm. However, also for the C&CG algorithm
computation times increase rapidly with model size. Therefore, the problem
size that is tractable is limited. Furthermore, it can be seen that not only
model size but also $\alpha$ and the carbon limit influence the computational
complexity. For this and all following applications of Algorithm 1, the
termination accuracy $\varepsilon$ of the Algorithm was set to 100 €.

$\alpha$ 0.3 0.7 Carbon Limit [t CO2] 30 60 30 60 $|\mathcal{U}|=1,n=1$
#iterations CG 11 21 16 24 #iterations C&CG 1 3 2 3 #iterations ratio 0.09
0.14 0.12 0.12 time CG [s] 2.51 5.26 3.64 5.32 time C&CG [s] 0.51 1.30 0.76
1.19 time ratio 0.20 0.25 0.21 0.22 $|\mathcal{U}|=1,n=3$ #iterations CG 32 29
39 36 #iterations C&CG 1 3 2 3 #iterations ratio 0.03 0.10 0.05 0.08 time CG
[s] 18.73 17.61 44.65 35.48 time C&CG [s] 1.20 2.90 1.87 4.52 time ratio 0.06
0.16 0.04 0.13 $|\mathcal{U}|=3,n=1$ #iterations CG 69 62 46 55 #iterations
C&CG 2 1 2 2 #iterations ratio 0.03 0.02 0.04 0.04 time CG [s] 63.76 58.23
45.96 57.38 time C&CG [s] 3.33 1.58 2.88 2.78 time ratio 0.05 0.03 0.06 0.05
$|\mathcal{U}|=3,n=3$ #iterations CG 141 164 102 137 #iterations C&CG 2 1 3 2
#iterations ratio 0.01 0.01 0.03 0.01 time CG [s] 312.94 399.81 174.32 203.88
time C&CG [s] 8.82 3.27 14.89 5.28 time ratio 0.03 0.01 0.09 0.03

Table 3: Comparison of CG and C&CG algorithm for 16 different scenarios.
$|\mathcal{U}|$ is the number of considered type days and $n$ is the number of
time steps per day. The ratio is the ratio of the C&CG value to the CG value.

###  4.4 Results

In this section, we discuss possible trade-offs between cost regret, carbon
emissions and willingness to take risks, represented by $\alpha$, and their
implications to device selection. To account for model size limitations and
the chosen model formulation, we look at three different case study
configurations.

First, we consider an example with three representative days in a resolution
of three time steps of eight hours each and with annual carbon limits between
20 and 160 t CO2 in steps of 10 t CO2. Apart from the carbon limit, we also
vary the size of the uncertainty set, for which we test the $\alpha$-values of
0.25, 0.5 and 0.75.

The visualization of the results can be seen in Figure 2, where in the upper
plot for the different $\alpha$-values and carbon limits the cost regret of
the constraint MARO-efficient solutions is shown. The connecting lines between
the different carbon values are only for visual readability but not to
guarantee behavior for computations with carbon limits in between. In this
graphic, it is to be expected from the mathematical formulas that the regret
has to be monotonically decreasing for each $\alpha$ and that higher
$\alpha$-values cause higher regret, but this is the only structural knowledge
that is available. The plot can be used to find a preferred trade-off, and it
can be seen that for the two objectives as well as for the willingness to take
risks, represented by $\alpha$, there is significant potential for trade-off
finding.

In the lower plot, the influence of the set carbon limit on the selected
devices is visualized. The abbreviations used in the graphic are from Table 4,
where all possible converters with their respective dimension units are
listed. Although the developments are not necessarily monotone, some
tendencies can be observed. Higher carbon emission acceptance leads to smaller
heat storage and solar thermal, while the size of the cogeneration unit, the
water-water heat pump, district heating, compression chiller and photovoltaic
is increased. Note that photovoltaic and solar thermal compete for the area on
the roof. It is also important to notice that the behavior of the converter
sizes does not have to be monotone over increasing carbon limits, as, for
example, the value for the size of free cooling shows.

converter/storage abbreviation dimension unit air-water heat pump AWHP kW
output absorption chiller AdC kW output adsorption chiller AC kW output brine-
water heat pump BWHP kW output cogeneration unit CU kW output cold storage CS
kWh storage compression chiller CC kW output cooling dummy C-Dummy kW output
district heating DH kW output free cooling FC kW output gas boiler GB kW
output heating dummy H-Dummy kW output heat storage HS kWh storage pellet
boiler PB kW output photovoltaic PV $m^{2}$ reversible air-water heat pump
rev. AWHP kW output reversible brine-water heat pump rev. BWHP kW output
reversible water-water heat pump rev. WWHP kW output solar thermal ST $m^{2}$
water-water heat pump WWHP kW output

Table 4: Parameters of all considered converters and storages. Note that
$1m^{2}$ PV uses $2.975m^{2}$ roof area.

Figure 2: Overview of regret and device selection developments for different
$\alpha$ and carbon limit values for three representative days consisting of
three time steps, representing eight hours each. The minimal possible carbon
emission value is 17170 kg, if all loads have to be fulfilled without dummy
usage, as was computed by a single criteria optimization. In the lower plot,
the shaded areas are the range that is caused by the different $\alpha$-values
and the lines correspond to the mean of the three $\alpha$-values.

Second, as three days with three time step resolution might cause a very
limited case study result, we also performed computations for six days with 12
time steps of two hours each. To limit computational time, we only looked at
three different carbon limits (20, 40, 60 t CO2) in this case and again at the
three different $\alpha$-values of 0.25, 0.5 and 0.75. The resulting regret is
compared with that of the coarser computation from the first study in Table 5.
It can be seen that despite the difference in model resolution, the regret
values differ by at most 8%. This can be seen as an indicator that the results
from the coarser resolution are already quite meaningful. Therefore, it is
possible to select a desired compromise in Figure 2 and then compute the exact
device configuration in a second step, with a finer resolution.

$\alpha$ | carbon limit [t $CO_{2}$] | coarse regret [€] | fine regret [€] | deviation
---|---|---|---|---
0.25 | 20 | 132801 | 133537 | 0.01
40 | 92615 | 93410 | 0.01
60 | 58487 | 63813 | 0.08
0.5 | 20 | 252957 | 258994 | 0.02
40 | 191580 | 197533 | 0.03
60 | 133167 | 140059 | 0.05
0.75 | 20 | 374133 | 383486 | 0.02
40 | 288910 | 298527 | 0.03
60 | 206311 | 215559 | 0.04
Table 5: Comparison of the regret values of coarser optimization (based on
three days with three time steps) and finer optimization (based on six days
with twelve time steps). The deviation is the relative difference of the
coarser regret compared to the finer regret.

Third, we look at a modified version of the problem. So far, we have
considered the regret when comparing with all available designs and controls.
This approach would be the best choice to find trade-offs with a decision
maker who considers carbon emission reduction as an optional criterion and
will always ask how much money could have been saved by ignoring carbon
restrictions. In the following, we will only compare to designs and controls
that also respect the carbon limit. This addresses a decision maker who is
interested in guaranteeing good carbon limits. Formally, this is obtained by
adding the constraints

$\displaystyle co_{2}(\mathbf{d}^{*},\mathbf{s}^{*})$ $\displaystyle\leq c$
$\displaystyle co_{2}^{inv}(\mathbf{d}^{*})$ $\displaystyle\leq c$

to ($P(\alpha,c)$). The resulting problem structure is different from that of
the so far considered constraint MARO-efficiency. As computation times
increase in this case, we again only look at three days with three time steps
each and use the carbon and $\alpha$-values from the second study. The results
of this study can be seen in Figure 3. It can be observed that in this case
carbon emission reduction and regret reduction are no contradicting goals
anymore, but minimal carbon emissions are well in line with minimal regret.
Dealing with larger uncertainty sets, by considering larger $\alpha$-values,
still leads to larger regret. Considering the solutions in more detail, it can
be observed that the regret-minimal configuration for this plot is for all
$\alpha$-values based on solar thermal, absorption chiller, pellet boiler,
cold storage, heat storage, free cooling and a small compression chiller as
well as photovoltaic area. All other devices are not chosen in this case.

Figure 3: Overview of regret developments for different $\alpha$ and carbon
limit values for three representative days consisting of three time steps,
representing eight hours each. In contrast to Figure 2, the regret is computed
in comparison with only those designs and operations that also fulfill the
carbon limit.

##  5 Conclusion

We introduced a model and computed constraint MARO-efficient solutions for
building energy supply design with the objectives of minimizing cost regret
and carbon emissions under price uncertainty. Moreover, we visualized the
results similarly as it is usually done for MCO, offering intuitive access to
trade-off finding. For the computations, we successfully adapted an algorithm
for regret with respect to load uncertainty to regret with respect to price
uncertainty and proved the convergence in this case for the CG as well as for
the C&CG algorithm.

In the case study, we made the following observations.

  * •

There is a significant potential for trade-offs between cost regret, carbon
emissions and the size of the uncertainty set, which corresponds to
willingness to take risks.

  * •

Similar to the algorithm for load uncertainty, the problem size that is
tractable is limited.

  * •

The computational effort does not only depend on the model size, i.e. the
number of considered days and time steps, but also on the volume of the
uncertainty set and the value of the imposed carbon limit.

  * •

Minimizing carbon emissions and cost regret does not lead to contradicting
results if the comparison is only with devices and controls that also fulfill
the carbon limit. In this case, the most carbon saving is also the most regret
reducing design.

In further research, it would be beneficial to find possibilities to
accelerate the algorithm to be able to deal with larger models. Moreover, it
could be useful to combine the approach with more interactive decision aiding
approaches like Pareto navigation, which was already applied to the problem
without robustness in (Halser et al., 2024b). However, an additional challenge
for Pareto navigation is the nonconvexity of the robust problem. Furthermore,
it could be beneficial to consider additional objectives like, for example,
minimization of investment costs.

##  Appendix A Problem Formulations

In this section, we introduce the detailed terms of the problem formulation.

$\mathcal{M}$ set of all possible converters $\mathcal{M}^{d}$ set consisting
of heating and cooling dummy $\mathcal{M}^{d+}$ $\mathcal{M}$ with additional
dummy generators for heating and cooling $\mathcal{T}:=\\{0,...,n\\}$ set of
all time steps (equidistant) of the day, where we require that n divides 24
$\mathcal{T}^{+}:=\mathcal{T}\cup\\{n+1\\}$ set of all time steps of the day
with an additional step at the end $\mathcal{U}$ set of representative days,
consisting of finite number of cluster center days, which are found with a
$k$-Medoids algorithm $\mathbf{d}_{i}$ dimension of converter $i$ [-]
$\mathbf{d}^{hs}/\mathbf{d}^{cs}$ dimension of heat/cold storage [kWh storage
potential] $\mathbf{s}_{itk}$ dimension-scaled load of converter $i$ in time
step $t$ at day $k$ [-] $\mathbf{s}_{tk}^{h}/\mathbf{s}_{tk}^{c}$ heat/cold
storage charge in time step $t$ of day $k$ [kWh]
$c^{e}/c^{el}/c^{h}/c^{w}/c^{g}$ carbon equivalent emissions of buying
electricity/selling electricity/district heat/wood pellet/gas [kg/kWh]
$c^{hs}/c^{cs}$ depreciated carbon emission factor of heat/cold storage [$kg$]
$e_{itk}/k_{itk}/w_{itk}/$ $g_{itk}/t_{itk}$ maximum electric/heat/wood
pellet/gas/district heat consumption of converter $i$ in time step $t$ of day
$k$ [kWh] $el_{itk}/h_{itk}/c_{itk}$ maximum electric/heat/cold production of
converter $i$ in time step $t$ of day $k$ [kWh] $f:=\frac{24}{n}$ length of
the time steps $H_{tk}/C_{tk}$ heating/cooling loads in time step $t$ of day
$k$ [kWh] $p^{e}/p^{el}/p^{h}/p^{w}/p^{g}$ price of buying electricity/selling
electricity/district heat/wood pellet/gas [$€{}/kWh$] $p$ vector of all
resource prices (variable or fixed) $p_{i}^{d}/c_{i}^{d}$ size dependent
depreciation costs/carbon emission of converter $i$ [$€$]/[kg]
$p_{i}^{f}/c_{i}^{f}$ base depreciation costs/carbon emission of converter $i$
[$€$]/[kg] $p^{hs}/p^{cs}$ depreciated price factor of heat/cold storage [$€$]
$w_{k}$ weights of cluster day $k\in\mathcal{U}$

Table 6:  Notation for the LP. Indices are subscripts, while superscripts are
used for nomenclature.

Using the notation from Table 6 and defining

$\displaystyle p_{itk}$
$\displaystyle:=e_{itk}p^{e}-el_{itk}p^{el}+w_{itk}p^{w}+g_{itk}p^{g}+t_{itk}p^{h}$
$\displaystyle\forall i\in\mathcal{M}^{d+},t\in\mathcal{T},k\in\mathcal{U}$
$\displaystyle q_{itk}$
$\displaystyle:=e_{itk}c^{e}-el_{itk}c^{el}+w_{itk}c^{w}+g_{itk}c^{g}+t_{itk}c^{h}$
$\displaystyle\forall i\in\mathcal{M}^{d+},t\in\mathcal{T},k\in\mathcal{U}$
$\displaystyle p^{inv}$
$\displaystyle:=f\mathbf{d}^{hs}p^{hs}+f\mathbf{d}^{cs}p^{cs}+\sum_{i\in\mathcal{M}}\mathbf{d}_{i}p_{i}^{d}$
$\displaystyle c^{inv}$
$\displaystyle:=f\mathbf{d}^{hs}c_{hs}+f\mathbf{d}^{cs}c_{cs}+\sum_{i\in\mathcal{M}}\mathbf{d}_{i}c_{i}^{d}$
$\displaystyle\mathbf{d}$ $\displaystyle:=\begin{pmatrix}\mathbf{d}^{hs}\\\
\mathbf{d}^{cs}\\\ (\mathbf{d}_{i})_{i\in\mathcal{M}^{d+}}\end{pmatrix}$
$\displaystyle\mathbf{s}$
$\displaystyle:=\begin{pmatrix}(\mathbf{s}_{itk})_{i\in\mathcal{M}^{d+},t\in\mathcal{T},k\in\mathcal{U}}\\\
(\mathbf{s}_{tk}^{c})_{t\in\mathcal{T}^{+},k\in\mathcal{U}}\\\
(\mathbf{s}_{tk}^{h})_{t\in\mathcal{T}^{+},k\in\mathcal{U}}\end{pmatrix},$

we define the objective functions as

$\displaystyle costs^{p}(\mathbf{d},\mathbf{s})$
$\displaystyle:=f\sum_{\begin{subarray}{c}i\in\mathcal{M},t\in\mathcal{T},\\\
k\in\mathcal{U}\end{subarray}}w_{k}\mathbf{s}_{itk}p_{itk}+p^{inv}$
$\displaystyle co_{2}(\mathbf{d},\mathbf{s})$
$\displaystyle:=f\sum_{\begin{subarray}{c}i\in\mathcal{M},t\in\mathcal{T},\\\
k\in\mathcal{U}\end{subarray}}w_{k}\mathbf{s}_{itk}q_{itk}+c^{inv}.$

The carbon emission split in operational and investment emissions is then
given by

$\displaystyle co_{2}^{op}(\mathbf{d})$
$\displaystyle=f\sum_{\begin{subarray}{c}i\in\mathcal{M},t\in\mathcal{T},\\\
k\in\mathcal{U}\end{subarray}}w_{k}\mathbf{s}_{itk}q_{itk}$ $\displaystyle
co_{2}^{inv}(\mathbf{d},\mathbf{s})$ $\displaystyle=c^{inv}$

and analogously for costs. The detailed constraints are given as cooling and
heating load fulfilling

$\displaystyle\mathbf{s}_{tk}^{c}-\mathbf{s}_{t+1k}^{c}+\sum_{i\in\mathcal{M}^{d+}}\mathbf{s}_{itk}c_{itk}$
$\displaystyle=C_{tk}$ $\displaystyle\forall t\in\mathcal{T},k\in\mathcal{U}$
$\displaystyle\mathbf{s}_{tk}^{h}-\mathbf{s}_{t+1k}^{h}+\sum_{i\in\mathcal{M}^{d+}}\mathbf{s}_{itk}(h_{itk}-t_{itk})$
$\displaystyle=H_{tk}$ $\displaystyle\forall t\in\mathcal{T},k\in\mathcal{U},$

technical limits of the converters and storages

$\displaystyle\mathbf{s}_{itk}$ $\displaystyle\leq\mathbf{d}_{i}$
$\displaystyle\forall i\in\mathcal{M}^{d+},t\in\mathcal{T},k\in\mathcal{U}$
$\displaystyle\mathbf{s}_{tk}^{c}$ $\displaystyle\leq\mathbf{d}^{cs}$
$\displaystyle\forall t\in\mathcal{T},k\in\mathcal{U}$
$\displaystyle\mathbf{s}_{tk}^{h}$ $\displaystyle\leq\mathbf{d}^{hs}$
$\displaystyle\forall t\in\mathcal{T},k\in\mathcal{U}$

and start and end states of the storages for each day. These are chosen such
that the heat storage has to be empty at midnight and the cold storage has to
be full at that time

$\displaystyle\mathbf{s}_{0k}^{c}$ $\displaystyle=0$ $\displaystyle\forall
k\in\mathcal{U}$ $\displaystyle\mathbf{s}_{\\{n+1\\}k}^{c}$ $\displaystyle=0$
$\displaystyle\forall k\in\mathcal{U}$ $\displaystyle\mathbf{s}_{0k}^{h}$
$\displaystyle=\mathbf{d}^{hs}$ $\displaystyle\forall k\in\mathcal{U}$
$\displaystyle\mathbf{s}_{\\{n+1\\}k}^{h}$ $\displaystyle=\mathbf{d}^{hs}$
$\displaystyle\forall k\in\mathcal{U}.$

Moreover, there are the constraints that guarantee nonnegativity of dimensions
and controls and the fixation of the size of dummy generators

$\displaystyle p,\mathbf{s}$ $\displaystyle\geq 0$ $\displaystyle\mathbf{d}$
$\displaystyle\geq 0$ $\displaystyle\mathbf{d}_{i}$ $\displaystyle=M$
$\displaystyle\forall i\in\mathcal{M}^{d}$

for a sufficiently large constant $M$. There are also device specific
constraints. To avoid introducing too many indices, we give them only
verbally.

  * •

We model reversible heat pumps as two devices, where one represents the
heating and the other the cooling, and both devices have to have a fixed size
ratio. We ensure that only one of them has positive coefficient of performance
in every time step.

  * •

The roof area is an upper bound for the sum of the (weighted) dimensions of
photovoltaic and solar thermal.

  * •

An adsorption chiller needs to have a dimension smaller than the sum of the
dimensions of the district heating and the cogeneration unit.

##  Appendix B Solution Algorithm

###  B.1 Motivation

In the following, we motivate the solution algorithm. As this algorithm can
also be applied to other problems with the same structure, we use general
notation, that is independent of the problem-specific notation from before.
The minimax regret problem can be understood as a problem of the form

$\displaystyle\min_{x}\max_{p}\min_{y}\;$ $\displaystyle
c^{T}x+p^{T}Ay-\min_{x^{*},y^{*}}(c^{T}x^{*}+p^{T}Ay^{*})$ $\displaystyle
s.t.\;$ $\displaystyle Bx+Cy\geq d$ $\displaystyle Bx^{*}+Cy^{*}\geq d$
$\displaystyle Ep\geq f$ $\displaystyle Gx\geq h$ $\displaystyle
x,p,y,x^{*},y^{*}\geq 0$

with the variables
$x\in\mathbb{R}^{n_{x}},p\in\mathbb{R}^{n_{p}},y\in\mathbb{R}^{n_{y}},$ and
the parameters
$c\in\mathbb{R}^{n_{x}},d\in\mathbb{R}^{n_{d}},f\in\mathbb{R}^{n_{f}},h\in\mathbb{R}^{n_{h}},A\in\mathbb{R}^{n_{p}\times
n_{y}},B\in\mathbb{R}^{n_{d}\times n_{x}},C\in\mathbb{R}^{n_{d}\times
n_{y}},E\in\mathbb{R}^{n_{f}\times n_{p}},G\in\mathbb{R}^{n_{h}\times n_{x}}$
for dimensions $n_{x},n_{p},n_{y},n_{d},n_{f},n_{h}\in\mathbb{N}$.

This problem can be reformulated as

$\displaystyle\min_{x}\max_{p,x^{*},y^{*}}\min_{y}\;$ $\displaystyle
c^{T}x+p^{T}Ay-c^{T}x^{*}-p^{T}Ay^{*}$ (2a) $\displaystyle s.t.\;$
$\displaystyle Bx+Cy\geq d$ (2b) $\displaystyle Bx^{*}+Cy^{*}\geq d$ (2c)
$\displaystyle Ep\geq f$ (2d) $\displaystyle Gx\geq h$ (2e) $\displaystyle
x,p,y,x^{*},y^{*}\geq 0.$ (2f)

By dualizing the second stage problem, we obtain with dual variable vector
$\pi\in\mathbb{R}^{n_{d}}$

$\displaystyle\min_{x}\max_{p,x^{*},y^{*},\mathbf{\pi}}\;$ $\displaystyle
c^{T}x+(d-Bx)^{T}\mathbf{\pi}-c^{T}x^{*}-p^{T}Ay^{*}$ (3a) $\displaystyle
s.t.\;$ $\displaystyle C^{T}\mathbf{\pi}\leq A^{T}p$ (3b) $\displaystyle
Bx^{*}+Cy^{*}\geq d$ (3c) $\displaystyle Ep\geq f$ (3d) $\displaystyle Gx\geq
h$ (3e) $\displaystyle x,p,\mathbf{\pi},x^{*},y^{*}\geq 0.$ (3f)

We can observe that the feasible set of the second stage problem is
independent of $x$ and has bilinear terms. We will exploit this structure in
the following solution algorithms.

###  B.2 Algorithm Formulation

The algorithms we propose are based on the algorithms described by Zeng and
Zhao (2013), with the difference that in our dualized problem $p$ appears not
only in the objective function but also in the constraints. Therefore, an
argument is necessary to observe that the optimal value in this case is still
in a vertex of a certain polyhedron. To state the algorithms, we first have to
split the problem in main- and subproblem. There are two possible algorithms,
a constraint generation (CG) and a column and constraint generation (C&CG)
one, with slightly different main problem, which we look at in the following.

Let us start with the subproblem. In the following, we assume relatively
complete recourse. This means that for every first and middle stage decision,
there is a feasible second stage decision. We obtain the subproblem

$\displaystyle SP(x):\;\;sp(x)=\max_{p,x^{*},y^{*},\mathbf{\pi}}\;$
$\displaystyle(d-Bx)^{T}\mathbf{\pi}-c^{T}x^{*}-p^{T}Ay^{*}$ $\displaystyle
s.t.\;$ $\displaystyle C^{T}\mathbf{\pi}\leq A^{T}p$ $\displaystyle
Bx^{*}+Cy^{*}\geq d$ $\displaystyle Ep\geq f$ $\displaystyle
p,\mathbf{\pi},x^{*},y^{*}\geq 0,$

which is a bilinear problem (BLP). We will observe later that the solution to
this problem can be assumed to be in a vertex of the polyhedron that defines
the feasible region of the problem. For the rest of the article, we assume
that whenever we talk about a solution to the problem, it is chosen in this
way. Similar BLPs can be linearized, for example in (Yokoyama et al., 2014),
but as the constraints involving $p$ are not independent of other variables
but also contain $\mathbf{\pi}$, we cannot apply that to the structure of our
problem and therefore have to keep the bilinear form.

For the main problem, we consider with additional variable $\eta\in\mathbb{R}$
the epigraph reformulation of (3)

$\displaystyle\min_{x,\eta}\;$ $\displaystyle c^{T}x+\eta$ $\displaystyle
s.t.\;$ $\displaystyle\eta\geq(d-Bx)^{T}\mathbf{\pi}-c^{T}x^{*}-p^{T}Ay^{*}$
$\displaystyle\forall(p,x^{*},y^{*},\mathbf{\pi})\in S$ $\displaystyle Gx\geq
h$ $\displaystyle x\geq 0,$

where

$\displaystyle S:=\left\\{(p,x^{*},y^{*},\mathbf{\pi})\in\mathbb{R}_{\geq
0}^{n_{p}+n_{x}+n_{y}+n_{d}}\mid C^{T}\mathbf{\pi}\leq
A^{T}p,\;Bx^{*}+Cy^{*}\geq d,\;Ep\geq f\right\\}.$

Problems like this with an infinite number of constraints are call semi-
infinite. By relaxing the semi-infinite constraint to a still to be
concretized, finite subset

$\displaystyle
S_{k}:=\left\\{(p_{l},x_{l}^{*},y_{l}^{*},\mathbf{\pi}_{l})\mid(p_{l},x_{l}^{*},y_{l}^{*},\mathbf{\pi}_{l})\in
S,\;l\in\\{0,...,k\\}\right\\},$

we obtain the main problem

$\displaystyle MP^{CG}(k):\;\;\min_{x,\eta}\;$ $\displaystyle c^{T}x+\eta$
$\displaystyle s.t.\;$
$\displaystyle\eta\geq(d-Bx)^{T}\mathbf{\pi}_{l}-c^{T}x_{l}^{*}-p_{l}^{T}Ay_{l}^{*}$
$\displaystyle\forall l\leq k$ $\displaystyle Gx\geq h$ $\displaystyle x\geq
0,\eta\in\mathbb{R}.$

The algorithm that is based on this problem formulation is the CG algorithm.
It is also possible to consider an epigraph reformulation of (2), which leads
to the C&CG algorithm with main problem

$\displaystyle MP^{C\&CG}(k):\;\;\min_{x,\eta,y_{1},...,y_{k}}\;$
$\displaystyle c^{T}x+\eta$ $\displaystyle s.t.\;$ $\displaystyle\eta\geq
p_{l}^{T}Ay_{l}-c^{T}x_{l}^{*}-p_{l}^{T}Ay_{l}^{*}$ $\displaystyle\forall
l\leq k$ $\displaystyle Bx+Cy_{l}\geq d$ $\displaystyle\forall l\leq k$
$\displaystyle Gx\geq h$ $\displaystyle x\geq 0,\eta\in\mathbb{R}.$

Both problems are LPs, so without loss of generality we assume in the
following that solutions are obtained at vertices of the polyhedron defining
the feasible region. Now that we have stated all partial problems, we can
state Algorithm 1. Depending on whether we use $MP^{CG}$ or $MP^{C\&CG}$
therein as $MP$, we are in the case of the CG or the C&CG algorithm.

Algorithm 1 (Column and) Constraint Generation

$\varepsilon\geq 0,lb=-\infty,ub=+\infty,k=0$

$x_{0},\eta_{0}\leftarrow MP(0)$

$p_{1},x_{1}^{*},y_{1}^{*},\mathbf{\pi}_{1}\leftarrow SP(x_{0})$

$lb\leftarrow\max\\{lb,c^{T}x_{0}+\eta_{0}\\}$

$ub\leftarrow\min\\{ub,c^{T}x_{0}+sp(x_{0})\\}$

$k=1$

while $ub-lb>\varepsilon$ do

$x_{k},\eta_{k}\leftarrow MP(k)$

$p_{k+1},x_{k+1}^{*},y_{k+1}^{*},\mathbf{\pi}_{k+1}\leftarrow SP(x_{k})$

$lb\leftarrow\max\\{lb,c^{T}x_{k}+\eta_{k}\\}$

$ub\leftarrow\min\\{ub,c^{T}x_{k}+sp(x_{k})\\}$

$k\leftarrow k+1$

end while

return $x_{k}$

By iteratively solving main- and subproblem, we compute lower bounds ($lb$)
and upper bounds ($ub$) for the problem. This is done until a predefined
solution quality $\varepsilon\geq 0$ is reached. As the solution to the
bilinear subproblem can be very time-consuming, it can be beneficial in
practice not to solve it to full optimality but to use upper bounds instead of
solutions in the algorithm.

###  B.3 Proof of Convergence

###### Theorem 1

The CG algorithm terminates in a finite number of steps. The difference
between the returned solution and the global optimal solution is at most
$\varepsilon$.

###### Proof 1

The proof is similar to the proof in A 2.2. by Yokoyama et al. (2014). It
follows from the definition of the main- and the subproblem that the
respective solutions are lower and upper bounds for the problem. With the
definitions

$\displaystyle h_{1}=\begin{pmatrix}\mathbf{\pi}\\\
p\end{pmatrix},h_{2}=\begin{pmatrix}x^{*}\\\ y^{*}\end{pmatrix}$

the subproblem is a bilinear problem

$\displaystyle SP(x):\;\;\max_{h_{1},h_{2}}\;$
$\displaystyle\begin{pmatrix}d-Bx&0\end{pmatrix}h_{1}-\begin{pmatrix}c&0\end{pmatrix}h_{2}-h_{1}^{T}\begin{pmatrix}0&0\\\
0&A\end{pmatrix}h_{2}$ $\displaystyle s.t.\;$
$\displaystyle\begin{pmatrix}-C^{T}&A^{T}\\\
0&E\end{pmatrix}h_{1}\geq\begin{pmatrix}0\\\ f\end{pmatrix}$
$\displaystyle\begin{pmatrix}B&C\end{pmatrix}h_{2}\geq d$ $\displaystyle
h_{1},h_{2}\geq 0$

to which Theorem 2.1 from Konno (1976) can be applied, and therefore, the
solution has to be obtained in a vertex of the polyhedron defining the
feasible region, which is independent of $x$. When generating constraints in
the main problem corresponding to all vertices of the polyhedron that defines
the feasible region of the subproblem, the solution is obtained and the
algorithm terminates as main- and subproblem solution coincide.

It remains to show that the algorithm cannot run into an infinite loop where
the same constraints are added to the mainproblem repeatedly, while others are
never considered. In this case, there would be an iteration, where all
considered constraints are added and all further iterations of the loop only
add the same constraints. From this iteration on, the feasible region of the
main problem will not change anymore. As we assume mainproblem solutions to be
in vertices of the feasible region, there will be $x_{k-n}$ and $x_{k}$ such
that $x_{k-n}=x_{k}$ for $k,n\in\mathbb{N},k\geq n$ in a finite number of
steps. But then, $x_{k}$ has to be the solution to the problem and the
algorithm terminates.

###### Remark 2

The C&CG algorithm converges in a finite number of steps, where the maximum
possible number of steps is at most as large as the maximum possible number of
steps for the dual algorithm. The convergence proof is analogous to that of
Theorem 1, with the only difference that the constraints only correspond to
$p$-values of the vertices of the polyhedron that defines the feasible region.
As multiple vertices might share the same $p$-value, this number can be
significantly smaller than the number of constraints that are required in the
dual algorithm.

###### Remark 3

Both variants of Algorithm 1 also work with discrete or mixed integer first
stage decision.

## CRediT authorship contribution statement

E. Halser: Conceptualization, Data curation, Formal analysis, Investigation,
Methodology, Visualization, Writing - original draft. E. Finhold: Supervision,
Writing - review & editing. N. Leithäuser: Supervision, Writing - review &
editing. T. Seidel: Supervision, Writing - review & editing. K.-H. Küfer:
Supervision.

## Declaration of Generative AI and AI-assisted technologies in the writing
process

During the preparation of this work, the authors used FhGenie and DeepL Write
in order to improve the language. After using these services, the authors
reviewed and edited the content as needed and take full responsibility for the
content of the publication.

## Data Availability

Data will be made available on request.

## Funding Sources

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.

## Acknowledgements

We thank our colleagues Jean Bertrand Gauthier for helpful discussions
concerning column and constraint generation and Kerstin Schneider for
formulation help while writing the article.

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