Robust Optimization: Using Bertsimas and Sim approach for linear model formulation (maximization problem)

Question

I tried to use Bertsimas and Sim approach for an uncertain linear model, but the thing is the answer I got for the Bertsimas and Sim linear model when Γ = 2 is different from Soyster model's result, I tried other uncertain models and I got the same result for Bertsimas' while it was completely protected against uncertainty and system model.

The Bertsimas & Sim model: $$max \ P_r+P_n $$ $$\begin{cases} -\alpha_1 P_r + \alpha_2 P_N \le \omega_{12} \\ -\alpha_1 P_r + \alpha_2 P_N \ge \omega_{11} \\ -\alpha_1 P_N + \alpha_2 P_r \le \omega_{22} \\ -\alpha_1 P_N + \alpha_2 P_r \ge \omega_{21} \end{cases} $$ $$\begin{cases}P_r\ge0 \\ P_N\ge 0 \end{cases}$$

Bertsimas & Sims model:

$$max \ P_r+P_n $$ $$\begin{cases} -\alpha_1 P_r + \alpha_2 P_N -\lambda_1 \eta -\mu_1 + \mu_2 \le \omega_{12} \\ -\alpha_1 P_r + \alpha_2 P_N -\lambda_1 \eta -\mu_1 + \mu_2 \ge \omega_{11} \\ -\alpha_1 P_N + \alpha_2 P_r -\lambda_2 \eta -\mu_1 + \mu_2\le \omega_{22} \\ -\alpha_1 P_N + \alpha_2 P_r -\lambda_2 \eta -\mu_1 + \mu_2\ge\omega_{21} \end{cases} $$

$$\begin{cases} \ \lambda_1 +\mu_1 \le \hat{\alpha_1} y_1 \\ \ \lambda_2 +\mu_1 \le \hat{\alpha_1} y_2 \\ \ \lambda_1 -\mu_2 \le -\hat{\alpha_2} y_2 \\ \ \lambda_2 -\mu_2 \le -\hat{\alpha_2} y_1 \\ \end{cases} $$ $$\begin{cases}P_r\ge0 \\ P_N\ge 0 \end{cases}$$ $$ \mu_1, \mu_2, y_1, y_2, \lambda_1, \lambda_ \ge 0 $$

I guess there's a problem with one the - behind $\mu \ or \ \hat{\alpha_2}$ but I don't know what did I wrong

Thanks for your help In forward.

Answer 1

So you have the constraints:

$$\begin{cases} \omega_{11}\le-\alpha_1 P_r + \alpha_2 P_N \le \omega_{12} \\ \omega_{21}\le-\alpha_1 P_N + \alpha_2 P_r \le \omega_{22} \end{cases} $$ with $\alpha_i \in [\bar{\alpha_i} - \hat{\alpha}_i, \bar{\alpha_i} + \hat{\alpha}_i]$ and at most $\Gamma$ of the $a_i$ can deviate at the same time (per constraint).

Unless I am mistaken, the robust counterpart is:

$$\begin{cases} \omega_{11}\le-\alpha_1 P_r + \alpha_2 P_N - \Gamma||y_1||_\infty - ||\begin{pmatrix}P_r \hat{a}_1 \\ P_N \hat{a}_2\end{pmatrix}-y_1||_1 \\ -\alpha_1 P_r + \alpha_2 P_N + \Gamma||y_2||_\infty + ||\begin{pmatrix}P_r \hat{a}_1 \\ P_N \hat{a}_2\end{pmatrix}-y_2||_1 \le \omega_{12} \\ \omega_{21}\le-\alpha_1 P_N + \alpha_2 P_r - \Gamma||y_3||_\infty - ||\begin{pmatrix}P_r \hat{a}_1 \\ P_N \hat{a}_2\end{pmatrix}-y_3||_1\\ -\alpha_1 P_N + \alpha_2 P_r + \Gamma||y_4||_\infty + ||\begin{pmatrix}P_r \hat{a}_1 \\ P_N \hat{a}_2\end{pmatrix}-y_4||_1 \le \omega_{22} \end{cases} $$ You can linearize the norms with standard reformulations. For example, the first constraint becomes $$\begin{cases} \omega_{11}\le-\alpha_1 P_r + \alpha_2 P_N - \Gamma u_1 - \sum_i v_{1i} \\ u_1 \geq y_{11}, u_1 \geq y_{12} \\ v_{11} \geq P_r\hat{a}_1 - y_{11}, v_{11} \geq -P_r\hat{a}_1 + y_{11} \\ v_{12} \geq P_N\hat{a}_2 - y_{12}, v_{12} \geq -P_N\hat{a}_2 + y_{12} \end{cases} $$ This becomes an absolute mess, which is why we love tools like YALMIP because they can do this for us.