Abstract
Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations.
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Notes
The ACC representation of Example 3.4.4 has typos and, for completeness, we present the corrected representation in Appendix C. We test the corrected ACC representation in this case study.
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Acknowledgements
This research has been supported in part by the National Science Foundation (NSF) under Grants CMMI-1555983 and CCF-1442495.
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Appendices
Appendix A: proof of Observation 2
Proof
As \(f^k_+(z) = \bigl (\frac{\alpha }{\alpha + 1}\bigr )(1 - k^{-\alpha -1})z - (1 - k^{-\alpha }) \beta \), we have
for all \(k \ge 1\). As \([f^{k+1}_+(z)]_+ \ge 0\) for all \(z \in \mathbb {R}\), to show that \([f^{k+1}_+(z)]_+ \ge [f^k_+(z)]_+\), it suffices to prove that \([f^{k+1}_+(z)]_+ \ge f^k_+(z)\) for all \(z \in \mathbb {R}\). First, as \(\beta \le 0\) and \(\frac{1 - k^{-\alpha }}{1 - k^{-\alpha -1}}\) increases in k, we have \(\bigl (\frac{\alpha +1}{\alpha }\bigr ) \bigl (\frac{1 - k^{-\alpha }}{1 - k^{-\alpha -1}}\bigr ) \beta \ge \bigl (\frac{\alpha +1}{\alpha }\bigr ) \bigl (\frac{1 - (k+1)^{-\alpha }}{1 - (k+1)^{-\alpha -1}}\bigr ) \beta \), i.e., \(z_0(k) \ge z_0(k+1)\). It follows that, when \(z < z_0(k)\), \(f^k_+(z) \le 0\) and hence \([f^{k+1}_+(z)]_+ \ge f^k_+(z)\). Second, when \(z \ge z_0(k)\), \(f^{k+1}_+(z) \ge 0\) because \(z \ge z_0(k) \ge z_0(k+1)\) and \(f^{k+1}_+(z)\) increases in z. As both \(f^k_+(z)\) and \(f^{k+1}_+(z)\) are affine functions of z, we have \(f^{k+1}_+(z) = f^{k+1}_+(z_0(k)) + (\frac{\alpha }{\alpha + 1})(1 - (k+1)^{-\alpha -1})(z - z_0(k))\) and \(f^k_+(z) = (\frac{\alpha }{\alpha + 1})(1 - k^{-\alpha -1})(z - z_0(k))\) for \(z \ge z_0(k)\). It follows that \(f^{k+1}_+(z) - f^k_+(z) = f^{k+1}_+(z_0(k)) + (\frac{\alpha }{\alpha + 1})[k^{-\alpha -1} - (k+1)^{-\alpha -1}](z - z_0(k)) \ge 0\). Hence, \([f^{k+1}_+(z)]_+ \ge f^k_+(z)\) when \(z \ge z_0(k)\) and the proof is complete. \(\square \)
Appendix B
For random variable Z and constant \(\beta \in \mathbb {R}\), we make the following observation on the worst-case expectation \(\sup _{{\mathbb P}_Z \in \mathcal {D}(\mu _0, \Sigma _0)} \mathbb {E}_{{\mathbb P}_Z}[Z-\beta ]_+\). Note that this observation can be made following the derivations in [30], and we present a proof below for completeness.
Observation 3
Given \(\beta \in {\mathbb R}\), we have
Proof
We represent \(\sup _{{\mathbb P}_Z \in \mathcal {D}(\mu _0, \Sigma _0)} \mathbb {E}_{{\mathbb P}_Z}[Z-\beta ]_+\) as the following optimization problem
The weak duality between (P) and (D), i.e., \(v_D \le v_P\), holds because \(\mu _0 p + \Sigma _0 q + r = \mathbb {E}_{{\mathbb P}_Z}[qZ^2 + pZ + r] \le \mathbb {E}_{{\mathbb P}_Z}[Z-\beta ]_+\) for any feasible solution (q, p, r) to (D) and feasible solution \({\mathbb P}_Z\) to (P). Now we prove the strong duality by constructing two feasible solutions to (P) and (D), respectively, that have the same objective value. On the one hand, the primal solution \(\hat{{\mathbb P}}_Z\) is supported on two points \(z_1\) and \(z_2\) with probability masses \(p_1\) and \(p_2\), respectively, where \(\Delta = \sqrt{(\beta - \mu _0)^2 + (\Sigma _0 - \mu _0^2)}\) and
We have \(p_1, p_2 \ge 0\) because \(\Delta \ge |\beta - \mu _0|\). Meanwhile, we have
and
Hence, \(\hat{{\mathbb P}}_Z\) is feasible to (P). On the other hand, the dual solution \((\hat{q}, \hat{p}, \hat{r})\) is such that
Hence, \(\hat{q}z^2 + \hat{p}z + \hat{r} = \frac{1}{4\Delta }(z + \Delta - \beta )^2\). It follows that \(\hat{q}z^2 + \hat{p}z + \hat{r} \ge 0\) for all \(z \in {\mathbb R}\). Meanwhile, \((\hat{q}z^2 + \hat{p}z + \hat{r}) - (z - \beta ) = \frac{1}{4\Delta }(z - \beta - \Delta )^2 \ge 0\), i.e., \(\hat{q}z^2 + \hat{p}z + \hat{r} \ge z - \beta \). Thus, \(\hat{q}z^2 + \hat{p}z + \hat{r} \ge [z-\beta ]_+\) and so \((\hat{q}, \hat{p}, \hat{r})\) is feasible to (D).
Finally, the primal objective value associated with \(\hat{{\mathbb P}}_Z\) is \(p_2(z_2 - \beta ) = \frac{(\mu _0 - \beta + \Delta )\Delta }{2\Delta } = \frac{1}{2}(\Delta - \beta + \mu _0)\). Meanwhile, the dual objective value associated with \((\hat{q}, \hat{p}, \hat{r})\) is
which coincides with the primal objective value associated with \(\hat{{\mathbb P}}_Z\). \(\square \)
Appendix C: corrected ACC representation of Example 3.4.4 in [17]
The ACC representation (3.50) in Example 3.4.4 in [17] has typos and is corrected as follows:
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Li, B., Jiang, R. & Mathieu, J.L. Ambiguous risk constraints with moment and unimodality information. Math. Program. 173, 151–192 (2019). https://doi.org/10.1007/s10107-017-1212-x
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DOI: https://doi.org/10.1007/s10107-017-1212-x
Keywords
- Ambiguity
- Chance constraints
- Conditional Value-at-Risk
- Second-order cone representation
- Separation
- Golden section search