Robust Chance-Constrained Geometric Programming with Application to Demand Risk Mitigation

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.