Distributionally robust optimization under endogenous uncertainty with an application in retrofitting planning

Highlights

• Distributionally robust optimization model under endogenous uncertainty with Fréchet class of distributions.

• Efficient constraint generation framework with dominant scenarios in stochastic optimization.

• Solutions for the retrofitting planning application under decision-dependent distributional ambiguity.

Abstract

Endogenous uncertainty concerns uncertainty which is dependent of decisions such as link failure in the retrofitting planning application. We propose a marginal-based distributionally robust optimization framework for integer stochastic optimization with decision-dependent discrete distributions that can be applied for the retrofitting planning application. We show that the resulting model can be formulated as a mixed-integer linear optimization problem. In order to solve the problem, we develop a constraint generation algorithm given the exponentially large number of constraints. Numerical results for the retrofitting planning application show that the proposed algorithm once tailored can solve the problem efficiently.

Introduction

Optimization under uncertainty concerns how to make (optimal) decisions when there is uncertainty in problem parameters and data (Diwekar, 2020). Uncertainty is traditionally represented with distributional information of random parameters in stochastic optimization (see, e.g., Birge & Louveaux, 2011). They could be stock prices in the application of portfolio management, customer demands in revenue management, or wind speed in energy management. In general, these random parameters are exogenous, i.e., they are not affected by decisions. However, there are cases when decisions can influence (distributional) information of these random parameters, which implies endogenous or decision-dependent uncertainty. For example, if a bridge is retrofitted (planning decisions), its survivability, i.e., the probability that it has no damage after a natural disaster, increases. This is the first type of endogenous uncertainty where decisions affect the probability distribution of random parameters such as the survival probabilities in the above example. The second type of endogenous uncertainty happens when decisions affect the resolution of uncertainty, i.e, when random parameters are realized. This type of endogenous uncertainty usually occurs in the multi-stage setting in which different decisions result in the realization of different parameters at different stages. For example, complete information of an oil field is only obtained when a facility is installed at that field as an investment decision in one particular period (stage) while the information of other unexplored oil fields remains unknown (see, e.g., Goel & Grossmann, 2006).

Stochastic optimization models with endogenous uncertainty are more difficult to handle in general due to various technical difficulties introduced by the dependence of distributions of random parameters on decisions such as the potential loss of convexity in these models (see, e.g., Dupačová, 2006). Jonsbråten, Wets, and Woodruff (1998) were among the first to investigate stochastic optimization with endogenous uncertainty, which initially focuses mainly on the second type of endogenous uncertainty. They developed implicit enumeration algorithms for the models in which the realization of random parameters only depends on first-stage decisions. More recently, Goel and Grossmann (2004) and Gupta, Grossmann, 2011, Gupta, Grossmann, 2014 have investigated further this type of endogenous uncertainty and proposed different solution approaches including non-anticipativity constraint relaxation and decomposition-based approximation algorithms. Vayanos, Kuhn, and Rustem (2011) studied decision rules for multi-stage stochastic optimization problems with endogenous uncertainty. As for the first type of endogenous uncertainty, Peeta, Salman, Gunnec, and Viswanath (2010) handled the resulting highly non-linear models with linear approximation while (Flach & Poggi, 2010) applied other convexification techniques to approximate them. Prestwich, Laumanns, and Kawas (2014) used the idea of distribution shaping and scenario bundling to handle potentially large scenario sets in these models and they were able to solve them efficiently without any approximation. More recently, Hellemo (2016) considered a combined type of decision-dependent uncertainty and applied it in the context of oil field exploration.

In addition to stochastic optimization, robust optimization is another research area in optimization under uncertainty which follows the principle of “immunized against worst case” (see, e.g., Ben-Tal, Ghaoui, & Nemirovski, 2009) instead of expected performance. Robust optimization assumes uncertain parameters belong to uncertainty sets without any distributional information. Distributionally robust optimization, on the other hand, makes the assumption that random parameters follow unknown probability distributions which belong to ambiguity sets. Distributionally robust optimization or robust/minimax stochastic optimization was first investigated by Žáčková (1966), and has been studied extensively more recently within the research communities of both stochastic and robust optimization. With respect to endogenous uncertainty, there are only few very recent research publications discussing decision-dependent uncertainty sets for robust optimization models (see, e.g., Nohadani, Sharma, 2018, Lappas, Gounaris, 2018 and references therein). Similarly, there has not much research focussing on distributionally robust optimization with endogenous uncertainty. Royset & Wets (2017) investigated the approximation of general optimization problems under stochastic ambiguity (which includes both endogenous and exogenous uncertainty) using cummulative distribution functions and their hypo distance as a metric to establish convergence results. Zhang, Xu, and Zhang (2016) analyzed the stability of a general distributionally robust optimization problem under endogenous uncertainty with parametric ambiguity sets. Noyan, Rudolf, and Lejeune (2018) studied the earth mover’s distance-based ambiguity sets for decision-dependent distributions while (Luo & Mehrotra, 2020) extended the framework of distributionally robust optimization with decision-dependent parametric ambiguity sets. In this paper, motivated by the stochastic optimization problem under the first type of endogenous uncertainty considered by Peeta et al. (2010), we focus on a different distributionally robust optimization model under endogenous uncertainty which focuses on probability dependence of decision-dependent distributions.

Specifically, our contributions and the structure of the paper are as follows.

• We propose a new model of distributionally robust optimization under the first type of endogenous uncertainty for integer stochastic optimization problems in Section 2. More specifically, we are going to use Fréchet classes of distributions, i.e., classes of distributions defined by known marginal distributions, as ambiguity sets in the proposed model. We show that the resulting model can be reformulated as a mixed-integer linear optimization problem.

• We provide a general constraint generation algorithm to solve the proposed distributionally robust optimization model in Section 3 to handle exponentially large number of constraints. Numerical results are reported for the retrofitting planning application in Section 4 with a tailored algorithm which can generate constraints efficiently by exploiting some structural properties of the retrofitting problem.