Update readme

README.md

@@ -28,18 +28,24 @@ pip install ".[examples]"
 ## Numerical Experiments
 
 1) Toy Problem (abbreviated as tp). 
-We compare ROBIST with the methods of Calafiore & Campi (2005) and Yanıkoglu & den Hertog (2013) in [`examples/tp_experiments_cal2005_yan2013.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp_experiments_cal2005_yan2013.py). Furthermore we analyze the performance of ROBIST in more detail in [`examples/tp_analysis.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp_analysis.py).
+We compare ROBIST with the methods of Calafiore & Campi (2005) and Yanıkoglu & den Hertog (2013) in [`examples/tp_experiments.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp_experiments.py). 
 
-2) Portfolio Management Problem (abbreviated as pm). 
-We compare ROBIST with the data-driven robust optimization approach proposed by Bertsimas et al. (2018) and the scenario optimization approach of Calafiore (2013) in [`examples/pm_experiments.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/pm_experiments.py).
+2) Altered Toy Problem (abbreviated as tp2).
+We analyze the performance of ROBIST in more detail in [`examples/tp2_analysis.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp2_analysis.py), [`examples/tp2_extra_experiments_1.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp2_extra_experiments_1.py) and [`examples/tp2_extra_experiments_2.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp2_extra_experiments_2.py).
+
+3) Linear CCP Problem (abbreviated as jiang2022).
+We compare ROBIST with the SAA-based methods of Ahmed et al. (2017) and Jiang and Xie (2022) in [`examples/jiang2022_experiments.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/jiang2022_experiments.py).
 
-3) Weighted Distribution Problem (abbreviated as wdp). 
+4) Weighted Distribution Problem (abbreviated as wdp). 
 We compare ROBIST with the scenario optimization methods of Calafiore & Campi (2005), Caré et al. (2014), Calafiore (2016) and Garatti et al. (2022) in [`examples/wdp_experiments.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/wdp_experiments.py).
 
-4) Two-Stage Lot-Sizing Problem (abbreviated as ls). 
+5) Portfolio Management Problem (abbreviated as pm). 
+We compare ROBIST with the data-driven robust optimization approach proposed by Bertsimas et al. (2018) and the scenario optimization approach of Calafiore (2013) in [`examples/pm_experiments.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/pm_experiments.py).
+
+6) Two-Stage Lot-Sizing Problem (abbreviated as ls). 
 We compare ROBIST with the method of Vayanos et al. (2012) in [`examples/ls_experiments.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/ls_experiments.py).
 
-For more information about these problems we refer to [preprint-paper].
+For more information on these experiments and the results we refer to [preprint-paper].
 
 ## Illustrative Example
 
@@ -59,9 +65,9 @@ where $z_1$ and $z_2$ are uncertain parameters, both uniformly distributed with
 Suppose we have access to a data set of $N=200$ realizations of $(\tilde{z}_1, \tilde{z}_2)$ and would like the solution to be feasible with probability of at least 90%. 
 We illustrate the application of ROBIST for this toy problem using the following figures. 
 
-First, we randomly split the data set into two equal-sized sets $\mathcal{D}^{train}\_{N_1}$ and $\mathcal{D}^{\text{test}}\_{N_2}$, each containing $100$ scenarios.
+First, we randomly split the data set into three equal-sized sets $\mathcal{D}^{train}\_{N_1}$, $\mathcal{D}^{\text{valid}}\_{N_2}$ and $\mathcal{D}^{\text{test}}\_{N_3}$, each containing $100$ scenarios.
 
-![Data](https://github.com/JustinStarreveld/ROBIST/raw/main/docs/illustrative_figures/Illustrate_data_split_N=200.png)
+![Data](https://github.com/JustinStarreveld/ROBIST/raw/main/docs/illustrative_figures/Illustrate_data_split_N=300.png)
 
 We initialize the algorithm by optimizing for the expected/nominal scenario, i.e., $\bar{\mathbf{z}} = (z_1, z_2) = (0,0)$. This provides an initial solution: $\mathbf{x}\_{0} = (x_1, x_2) = (1,1)$ with an objective value of 2.
 The next step is to use the training data $\mathcal{D}^{\text{train}}\_{N_1}$ to evaluate the robustness of $\mathbf{x}\_{0}$. This evaluation is illustrated in the following figure.
@@ -77,19 +83,21 @@ Again, we can evaluate the robustness of our newly generated solution $\mathbf{x
 We find that $\mathbf{x}\_{1}$ exceeds our desired level of robustness, thus the algorithm will remove a randomly picked scenario from our set of sampled scenarios in the following iteration. 
 The algorithm continues adding or removing scenarios and evaluating the resulting solutions on $\mathcal{D}^{\text{train}}\_{N_1}$ in this manner until either the time limit or iteration limit is reached. 
 
-Once the stopping criteria is reached, we use the "out-of-sample" test data $\mathcal{D}^{\text{test}}\_{N_2}$ to properly evaluate each solution $\mathbf{x}\_{i}$ and obtain valid "feasibility certificates". 
-These evaluations can then be used to construct a trade-off curve and aid in choosing a solution. The orange line in the figure below depicts such a trade-off curve. 
+Once the stopping criteria is reached, we use the validation data $\mathcal{D}^{\text{valid}}\_{N_2}$ to properly evaluate each solution $\mathbf{x}\_{i}$ and obtain valid "feasibility certificates". 
+These evaluations can then be used to construct a trade-off curve and aid in choosing a final solution (the orange line in the figure below depicts such a trade-off curve, where each orange square
+represents a non-dominated solution with respect to the validation data). Recall that we would like our solution to be feasible with probability $\geq 90\%$, thus we select the solution with the highest 
+objective value while requiring that the proxy certificate derived from the validation data is greater than or equal to 0.90. Finally, we utilize $\mathcal{D}^{\text{test}}\_{N_3}$ to provide
+a valid feasibility certificate for this solution. In this example, the algorithm returns a solution with objective value 1.37 and feasibility certificate 0.93. 
 
- ![Trade-off curve](https://github.com/JustinStarreveld/ROBIST/raw/main/docs/illustrative_figures/TradeOffCurves_N=100_alpha=0.01_epsilon=0.1_iMax=1000.png)
+ ![Trade-off curve](https://github.com/JustinStarreveld/ROBIST/raw/main/docs/illustrative_figures/TradeOffCurves_N=300_alpha=0.01_epsilon=0.1_iMax=100.png)
 
 The script used to create the figures in this illustrative example is [`examples/tp_illustrative_plots.py`](https://github.com/JustinStarreveld/ROBIST/blob/main/examples/tp_illustrative_plots.py).
 
 ## Contact Information
 Our code is not flawless. In case you have any questions or suggestions, please reach us at [email protected]. 
 
 ## Citation
-
-Was our software useful to you? Great! You can cite us using:
+Was our software useful to you? You can cite us using:
 
 ```
 @misc{ROBIST,