Robust combinatorial optimization with variable cost uncertainty

Abstract

a b s t r a c tWe present in this paper a new model for robust combinatorial optimization with cost uncertainty that generalizes the classical budgeted uncertainty set. We suppose here that the budget of uncertainty is given by a function of the problem variables, yielding an uncertainty multifunction. The new model is less conservative than the classical model and approximates better Value-at-Risk objective functions, especially for vectors with few non-zero components. An example of budget function is constructed from the probabilistic bounds computed by Bertsimas and Sim. We provide an asymptotically tight bound for the cost reduction obtained with the new model. We turn then to the tractability of the resulting optimization problems. We show that when the budget function is affine, the resulting optimization problems can be solved by solving n þ 1 deterministic problems. We propose combinatorial algorithms to handle problems with more general budget functions. We also adapt existing dynamic programming algorithms to solve faster the robust counterparts of optimization problems, which can be applied both to the traditional budgeted uncertainty model and to our new model. We evaluate numerically the reduction in the price of robustness obtained with the new model on the shortest path problem and on a survivable network design problem.

Robust constrained shortest path problems under budgeted uncertainty

“…The results presented in what follows are obtained by applying tools similar to those described in [22,Section 3]. However, their importance should not be undervalued since they show that U γ -CSP is pseudo-polynomial, and therefore, belongs to the same complexity class as its deterministic counterpart.…”

“…, |A|}, U γ (x) reduces to U Γ . However, it has been shown in [21,22] that U γ is in general less conservative than U Γ . A measure of conservatism of a robust model is its price of robustness, that is, the increase in solution cost when imposing a probabilistic protection with a specified guarantee > 0.…”

“…For instance, [4] and [15] study how to generalize the complexity results of [6] to combinatorial optimization problems with uncertainty in the constraints and integer variables, respectively, while [21,22] discuss how the probabilistic guarantee can be improved by allowing Γ to depend on the decision variables, yielding variable budgeted uncertainty, denoted U γ in what follows. While the robust shortest path problem has been investigated in several works (see [3,6,9], among others), we are not aware of previous works on robust approaches for the constrained shortest path problem.…”

“…In particular, we define an algorithm based on the solution of |A| + 1 deterministic CSP by employing similar techniques to those developed in [6,22,4,15]. We show then in Section 4 that U Γ -T W SP is N P-hard in the strong sense even when G is a directed and acyclic graph, the mean weight vector r = 0, and b = 0, which highlights a fundamental difference between U Γ -CSP and U Γ -T W SP .…”

A Dynamic Programming Approach for a Class of Robust Optimization Problems

“…The approach is compared numerically to other solution algorithms in [29], where the authors also study the space complexity of the algorithm. The seminal idea of [28] is extended by [33] to any robust combinatorial optimization problems with cost uncertainty whose deterministic counterpart can be solved by a DPA, including its variant with variable uncertainty [32]. Differently from [28,29,33], which solve the full RO problem by a DPA, the authors of [3] use a DPA to solve the AP that arises in robust vehicle routing problems with time windows.…”