Abstract
We determine bounds on the optimal value for a chance-constrained program aiming to minimize the worst-case probability that a certain nonlinear function with random exponents will fail to exceed unity. Our approach is to design a chance-constrained inner approximation problem based on the Poisson probability measure. We then harness a property of that approximation so as to construct and solve a certain surrogate minimization problem based on a decomposable signomial (and therefore, nonconvex) program. Out of the minimization problem emerges a feasible solution and an upper bound for the original problem. Finally, we design and show how to solve a maximization problem whose optimal solution fixes a lower bound for the original problem. Motivating our study is a budget-constrained, risk-averse consumer goods firm experiencing product portfolio demand arrivals at an expected rate that jointly depends on uncertain demand responsivity parameters and the firm’s marketing effort allocation across diverse promotion channels. Numerical experiments on real and on synthetic data assess the bounds under a variety of promotion budgets, parameter uncertainty regimes and product purchase probabilities.
Similar content being viewed by others
Data Availability
All data generated or analyzed during this study are included in this published article.
References
Avriel, M., Wilde, D.J.: Stochastic geometric programming. In: Kuhn, H.W. (ed) Proceedings of the Princeton Symposium of Mathematical Programming. Princeton University Press (1970)
Boyd, S.P., Kim, S.J., Patil, D.D., Horowitz, M.A.: Digital circuit optimization via geometric programming. Oper. Res. 53(6), 899–932 (2005)
Boyd, S., Kim, S.J., Vandenberghe, L., Hassibi, A.: A tutorial on geometric programming. Optim. Eng. 8(1), 67–127 (2007)
Chang, Y.O., Karlof, J.K.: Large scale geometric programming: an application in coding theory. Comput. Oper. Res. 21(7), 747–755 (1994)
Chassein, A., Goerigk, M.: On the complexity of robust geometric programming with polyhedral uncertainty. Oper. Res. Lett. 47(1), 21–24 (2019)
Chiang, M.: Geometric programming for communication systems. Found. Trends Commun. Inf. Theory 2(1/2), 1–154 (2005)
Corstjens, M., Doyle, P.: The application of geometric programming to marketing problems. J. Mark. 49(1), 137–144 (1985)
Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)
Duffin, R., Peterson, E.L., Zener, C.: Geometric programming-theory and application. SIAM Rev. 10(2), 235–236 (1968)
Dupacova, J.: Stochastic geometric programming with an application. Kybernetika 46(3), 374–386 (2010)
Ecker, J.G.: Geometric programming: methods, computations and applications. SIAM Rev. 22(3), 338–362 (1980)
Fontem, B., Keskin, B., Melouk, S., Vaughn, C.: An exact decomposition algorithm for a chance-constrained new product risk model. Oper. Res. Lett. 47(4), 250–256 (2019)
Gao, R., Kleywegt, A.: Distributionally robust stochastic optimization with Wasserstein distance. arXiv:1604.02199 (2016)
Hosseini-Nodeh, Z., Khanjani-Shiraz, R., Pardalos, P.M.: Distributionally robust portfolio optimization with second order stochastic dominance based on Wasserstein metric. Inf. Sci. 613, 828–852 (2022)
Hsiung, Kan-Lin., Kim, Seung-Jean., Boyd, Stephen: Tractable approximate robust geometric programming. Optim. Eng. 9(2), 95–118 (2008). https://doi.org/10.1007/s11081-007-9025-z
Hsiung, K., Chen, H.: Deterministic approximations for a class of chance-constrained geometric programs. Int. J. Comput. Optim. 7(1), 13–21 (2020)
Iwata, K., Murotsu, Y., Iwatsubo, T., Fujii, S.: A probabilistic approach to the determination of the optimum cutting conditions. J. Eng. Ind. 94(4), 1099–1107 (1972)
Khanjani-Shiraz, R., Babapour-Azar, A., Hosseini-Nodeh, Z., Pardalos, P.M.: Distributionally robust maximum probability shortest path problem. J. Comb. Optim. 43, 140–167 (2022)
Khanjani-Shiraz, R., Babapour-Azar, A., Hosseini-Nodeh, Z., Pardalos, P.M.: Distributionally robust joint chance-constrained support vector machines. Optim. Lett. 1–34 (2022b)
Khanjani, S.R., Fukuyama, H.: Integrating geometric programming with rough set theory. Oper. Res. Int. J. 18, 1–32 (2018)
Khanjani-Shiraz, R., Khodayifar, S., Pardalos, P.M.: Copula theory approach to stochastic geometric programming. J. Global Optim. 81, 435–468 (2021)
Khanjani, S.R., Tavana, M., Fukuyama, H., Di Caprio, D.: Fuzzy chance-constrained geometric programming: The possibility, necessity and credibility approaches. Oper. Res. Int. J. 17, 67–97 (2017)
Li, Y., Chen, Y.C.: Geometric programming approach to doping profile design optimization of metal-oxide-semiconductor devices. Math. Comput. Model. 58(1–2), 344–354 (2013)
Liu, S.T.: Posynomial geometric programming with interval exponents and coefficients. Eur. J. Oper. Res. 186(1), 17–27 (2008)
Liu, J., Lisser, A., Chen, Z.: Stochastic geometric optimization with joint probabilistic constraints. Oper. Res. Lett. 44(5), 687–691 (2016)
Liu, J., Lisser, A., Chen, Z.: Distributionally robust chance constrained geometric optimization. Math. Oper. Res. 47(4), 2547–3399 (2022)
Namkoong, H., Duchi, J.C.: Stochastic gradient methods for distributionally robust optimization with \(f\)-divergences. Adv. Neural Inf. Process. Syst. 29, 2016 (2016)
Peterson, E.L.: Geometric programming. SIAM Rev. 18(1), 1–51 (1976)
Popescu, I.: Robust mean-covariance solutions for stochastic optimization. Oper. Res. 55(1), 98–112 (2007)
Rahimian, H., Mehrotra, S.: Frameworks and results in distributionally robust optimization. Open J. Math. Optim. 3 (2022)
Rao, S.S.: Engineering Optimization: Theory and Practice, 3rd edn. Wiley, New York (1996)
Saab, A., Burnell, E., Hoburg, W.W.: Robust designs via geometric programming (2018). arXiv:1808.07192v1
Sadjadi, S.J., Yazdian, S.A., Shahanaghi, K.: Optimal pricing, lot-sizing and marketing planning in a capacitated and imperfect production system. Comput. Ind. Eng. 62(1), 349–358 (2012)
Sadjadi, S.J., Hesarsorkh, H.A., Mohammadi, M., Naeini, A.B.: Joint pricing and production management: a geometric programming approach with consideration of cubic production cost function. J. Ind. Eng. Int. 11, 209–223 (2015)
Samadi, F., Mirzazadeh, A., Pedram, M.M.: Fuzzy pricing, marketing and service planning in a fuzzy inventory model: a geometric programming approach. Appl. Math. Model. 37(10/11), 6683–6694 (2013)
Shiraz, R.K., Tavana, M., Di Caprio, D., Fukuyama, H.: Solving geometric programming problems with normal, linear and zigzag uncertainty distributions. J. Optim. Theory Appl. 170, 1075–1078 (2016)
Singh, J., Luo, Z., Sapatnekar, S.S.: A geometric programming-based worst case gate sizing method incorporating spatial correlation. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27(2), 295–308 (2008)
Vanhonacker, W.R.: Estimation and testing of a dynamic sales response model with data aggregated over time: some results for the autoregressive current effects model. J. Mark. Res. 21(4), 445–455 (1984)
Vulcano, G., van Ryzin, G., Ratliff, R.: Estimating primary demand for substitutable products from sales transaction data. Oper. Res. 60(2), 313–334 (2012)
Wiebking, R.D.: Optimal engineering design under uncertainty by geometric programming. Manag. Sci. 23(6), 644–651 (1977)
Zhang, Q., Kortanek, K.O.: On a compound duality classification for geometric programming. J. Optim. Theory Appl. 180, 711–728 (2019)
Zhao, C., Guan, Y.: Data-driven risk-averse stochastic optimization with Wasserstein metric. Oper. Res. Lett. 46(2), 262–267 (2018)
Acknowledgements
The author is grateful to both anonymous reviewers for insightful comments and suggestions that led to a substantial improvement of the paper’s content and exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by René Henrion.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Fontem, B. Robust Chance-Constrained Geometric Programming with Application to Demand Risk Mitigation. J Optim Theory Appl 197, 765–797 (2023). https://doi.org/10.1007/s10957-023-02201-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-023-02201-8