Abstract
Linear optimization problems are investigated that have random parameters in their \(m\ge 1\) constraints. In constructing a robust solution \({\mathbf {x}}\in \mathbb {R}^d\), we control the risk arising from violations of the constraints. This risk is measured by set-valued risk measures, which extend the usual univariate coherent distortion (=spectral) risk measures to the multivariate case. To obtain a robust solution in \(d\) variables, the linear goal function is optimized under the restrictions holding uniformly for all parameters in a \(d\)-variate uncertainty set. This set is built from uncertainty sets of the single constraints, each of which is a weighted-mean trimmed region in \(\mathbb {R}^d\) and can be efficiently calculated. Furthermore, a possible substitution of violations between different constraints is investigated by means of the admissable set of the multivariate risk measure. In the case of no substitution, we give an exact geometric algorithm, which possesses a worst-case polynomial complexity. We extend the algorithm to the general substitutability case, that is, to robust polyhedral optimization. The consistency of the approach is shown for generally distributed parameters. Finally, an application of the model to supervised machine learning is discussed.
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Notes
An \(\alpha \)-central region is a set of possible outcome vectors that are taken into reasonable account by the decision maker (dependent on his or her risk posture \(\alpha \)).
Rüschendorf (2013) proposes a different notion of a multivariate distortion risk measure, which is scalar-valued: Given a \(d\)-variate distribution having p.d.f. \(F\), he considers the level set \(Q(t)\) of \(F\) at level \(t\) and defines some scalar measure of \(Q(t)\) as the \(t\)-quantile. Then, based on these scalar-valued quantiles, he introduces multivariate risk measures in the same way as univariate ones.
Here, \(\mathcal{F}\) coincides with the admissable set of the vector-valued multivariate risk measure \(\nu \) described in Bazovkin (2014). This measure consists in the smallest vector which, when being added to the random returns vector \({\mathbf {Y}}\), puts \(-\mu ^m\left( {\mathbf {Y}}+\nu ({\mathbf {Y}})\right) \) into the admissable set. The latter is the set of returns which appear to be acceptable to the risk taker (or the regulator). In this model, we obtain (16) with only \(\mu ^m\) and \(\mathcal{F}\) being involved:
$$\begin{aligned} \nu (\tilde{\mathbf {A}}{\mathbf {x}}-{\mathbf {b}}) \le {\mathbf {0}}\quad \Longleftrightarrow \quad -\mu ^m(\tilde{\mathbf {A}}{\mathbf {x}}-{\mathbf {b}})\subset \mathcal{F}. \end{aligned}$$The support function of a set \(S\) in \(\mathbb {R}^m\) is defined as \(h^m_S({\mathbf {p}})=\sup \{ {\mathbf {x}}'{\mathbf {p}}: {\mathbf {x}}\in S\}\), \({\mathbf {p}}\in \mathbb {R}^m\).
Consequently we do not need to calculate WM regions (see Fig. 2).
Thus, the above approach can be also employed as an alternative to the regular simplex method.
References
Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. (1996). The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software, 22, 469–483.
Bazovkin, P. (2014). Geometrical framework for robust portfolio optimization. Discussion Papers in Econometrics and Statistics, Institute of Econometrics and Statistics, University of Cologne (01/14).
Bazovkin, P., & Mosler, K. (2012). An exact algorithm for weighted-mean trimmed regions in any dimension. Journal of Statistical Software, 47(13).
Ben-Tal, A., Ghaoui, E., & Nemirovski, A. (2009). Robust Optimization. Princeton: Princeton University Press.
Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., & Rennen, G. (2013). Robust solutions of optimization problems affected by uncertain probabilities. Management Science, 59, 341–357.
Bertsimas, D., & Brown, D. B. (2009). Constructing uncertainty sets for robust linear optimization. Operations Research, 57, 1483–1495.
Bertsimas, D., Brown, D. B., & Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53, 464–501.
Borgwardt, K. (2001). Optimierung, Operations Research, Spieltheorie: Mathematische Grundlagen. Basel: Birkhäuser.
Buckley, C. E. (1988). A divide-and-conquer algorithm for computing 4-dimensional convex hulls. In Proceedings International Workshop on Computational Geometry on Computational Geometry and its Applications (pp. 113–135). New York: Springer.
Cascos, I., & Molchanov, I. (2007). Multivariate risks and depth-trimmed regions. Finance and Stochastics, 11, 373–397.
Chazelle, B. (1993). An optimal convex hull algorithm in any fixed dimension. Discrete and Computational Geometry, 10, 377–409.
Chen, W., & Sim, M. (2009). Goal-driven optimization. Operations Research, 57, 342–357.
Chen, W., Sim, M., Sun, J., & Teo, C. P. (2010). From cvar to uncertainty set: Implications in joint chance-constrained optimization. Operations Research, 58, 470–485.
Delage, E., & Ye, Y. (2010). Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 58, 595–612.
Delbaen, F. (2002). Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics (pp. 1–37). Berlin: Springer-Verlag.
Dyckerhoff, R., & Mosler, K. (2011). Weighted-mean trimming of multivariate data. Journal of Multivariate Analysis, 102, 405–421.
Dyckerhoff, R., & Mosler, K. (2012). Weighted-mean trimming of a probability distribution. Statistics and Probability Letters, 82, 318–325.
Fabozzi, F. J., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176, 191–220.
Föllmer, H., & Schied, A. (2004). Stochastic finance: An introduction in discrete time. Berlin: Walter de Gruyter.
Gabrel, V., Murat, C., & Thiele, A. (2014). Recent advances in robust optimization: An overview. European Journal of Operational Research, 235, 471–483.
Grünbaum, B. (2003). Convex polytopes (2nd ed.). New York: Springer.
Kall, P., & Mayer, J. (2010). Stochastic linear programming. Models, theory, and computation (2nd ed.). New York: Springer.
Mosler, K., & Bazovkin, P. (2014). Stochastic linear programming with a distortion risk constraint. OR Spectrum, 36(4), 949–969.
Natarajan, K., Pachamanova, D., & Sim, M. (2009). Constructing risk measures from uncertainty sets. Operations Research, 57, 1129–1141.
Rüschendorf, L. (2013). Mathematical risk analysis. Berlin, Heidelberg: Springer.
Vapnik, V. N. (1998). Statistical learning theory. New York: Wiley.
Acknowledgments
The detailed suggestions of two referees have much improved the clarity and readability of the paper; they are gratefully acknowledged. Pavel Bazovkin was partly supported by a grant of the German Research Foundation (DFG).
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Bazovkin, P., Mosler, K. A general solution for robust linear programs with distortion risk constraints. Ann Oper Res 229, 103–120 (2015). https://doi.org/10.1007/s10479-015-1786-8
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DOI: https://doi.org/10.1007/s10479-015-1786-8
Keywords
- Robust optimization
- Data depth
- Weighted-mean trimmed regions
- Coherent risk measure
- Multivariate risk measure
- Robust classification
- Algorithm