Robust Optimization of Sums of Piecewise Linear Functions with Application to Inventory Problems

Abstract

Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be "robustified" is not concave (or linear) with respect to the perturbing parameters. In this paper, we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semi-definite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multi-period inventory problems that is exact whether the uncertainty region reduces to a L1-norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem.

Robust Optimization with Ambiguous Stochastic Constraints Under Mean and Dispersion Information

“…The objective function in (50) has the sum-of-maxima form, which typically is problematic in RO because of the difficulty of maximizing a convex function; see, for example, Gorissen and den Hertog (2013) and Ardestani-Jaafari and Delage (2016). Models such as (50) in the multiperiod setting can lead to issues such as time (in)consistency, i.e., the question whether the chosen multiperiod strategy remains optimal at later stages for all possible outcomes of the uncertainty at the earlier stage.…”

Robust Multi-Product Newsvendor Model With Substitution Under Cardinality-Constrained Uncertainty Set

“…Under these circumstances, alternatively, one can choose the robust approach to formulate the model with partial information of the demand, which can be easily characterized or will stay the same at a relatively long period (i.e., mean, variance, or support). Therefore, some works applied the robust optimization to the newsvendor problems (Scarf 1957, Vairaktarakis 2000,Özler et al 2009, Lin and Ng 2011, Raza 2014, Hanasusanto et al 2015, Carrizosa et al 2016, Chen and Zhang 2009, Ardestani-Jaafari and Delage 2016. Especially, Scarf (1957) was the pioneer to introduce the robust idea to analyze single-product newsvendor problem with known mean and variance of the demand.…”

A Dynamic Programming Approach for a Class of Robust Optimization Problems

“…However, the rowand-column generation algorithm would then involve solving nonlinear (MP). For instance, allowing g i and h i to be bi-affine functions, as seen in the recent literature on RO [23,6], would lead to bilinear (MP) when considering the more general models described in the next section.…”

“…More interestingly, the problem can also be solved in polynomial time whenever Γ = n, which corresponds to Ξ Γ being a box uncertainty set; see [26]. More recently, the authors of [6] have shown that a closely related problem where each function f i involves only component ξ i of the uncertain vector is also easy.…”