Theses Doctoral

# Essays on Approximation Algorithms for Robust Linear Optimization Problems

Lu, Brian Yin

Solving optimization problems under uncertainty has been an important topic since the appearance of mathematical optimization in the mid 19th century. George Dantzig’s 1955 paper, “Linear Programming under Uncertainty” is considered one of the ten most inﬂuential papers in Management Science [26]. The methodology introduced in Dantzig’s paper is named stochastic programming, since it assumes an underlying probability distribution of the uncertain input parameters. However, stochastic programming suffers from the “curse of dimensionality”, and knowing the exact distribution of the input parameter may not be realistic. On the other hand, robust optimization models the uncertainty using a deterministic uncertainty set. The goal is to optimize the worst-case scenario from the uncertainty set. In recent years, many studies in robust optimization have been conducted and we refer the reader to Ben-Tal and Nemirovski [4–6], El Ghaoui and Lebret [19], Bertsimas and
Sim [15, 16], Goldfarb and Iyengar [23], Bertsimas et al. [8] for a review of robust optimization. Computing an optimal adjustable (or dynamic) solution to a robust optimization problem is generally hard. This motivates us to study the hardness of approximation of the problem and provide efﬁcient approximation algorithms. In this dissertation, we consider adjustable robust linear optimization problems with packing and covering formulations and their approximation algorithms. In particular, we study the performances of static solution and afﬁne solution as approximations for the adjustable robust problem.
Chapter 2 and 3 consider two-stage adjustable robust linear packing problem with uncertain second-stage constraint coefﬁcients. For general convex, compact and down-monotone uncertainty sets, the problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters [22]. In particular, for a fairly general class of uncertainty sets, we show that the two-stage adjustable robust problem is NP-hard to approximate within a factor that is better than Ω(logn), where n is the number of columns of the uncertain coefﬁcient matrix. On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efﬁciently. We study the performance of static solutions an approximation for the adjustable robust problem and relate its optimality to a transformation of the uncertain set. With this transformation, we show that for a fairly general class of uncertainty sets, static solution is optimal for the adjustable robust problem. This is surprising since the static solution is widely perceived as highly conservative. Moreover, when the static solution is not optimal, we provide an instance-based tight approximation bound that is related to a measure of non-convexity of the transformation of the uncertain set. We also show that for two-stage problems, our bound is at least as good (and in many case signiﬁcantly better) as the bound given by the symmetry of the uncertainty set [11, 12]. Moreover, our results can be generalized to the case where the objective coefﬁcients and right-hand-side are also uncertainty.
In Chapter 3, we focus on the two-stage problems with a family of column-wise and constraint-wise uncertainty sets where any constraint describing the set involves entries of only a single column or a single row. This is a fairly general class of uncertainty sets to model constraint coefﬁcient uncertainty. Moreover, it is the family of uncertainty sets that gives the previous hardness result. On the positive side, we show that a static solution is an
O(\log n · min(\log \Gamma, \log(m+n))-approximation for the two-stage adjustable robust problem where m and n denote the numbers of rows and columns of the constraint matrix and \Gamma is the maximum possible ratio of upper bounds of the uncertain constraint coefﬁcients. Therefore, for constant \Gamma, surprisingly the performance bound for static solutions matches
the hardness of approximation for the adjustable problem. Furthermore, in general the static solution provides nearly the best efﬁcient approximation for the two-stage adjustable robust problem.
In Chapter 4, we extend our result in Chapter 2 to a multi-stage adjustable robust linear optimization problem. In particular, we consider the case where the choice of the uncertain constraint coefﬁcient matrix for each stage is independent of the others. In real world applications, decision problems are often of multiple stages and a iterative implementation of two-stage solution may result in a suboptimal solution for multi-stage problem. We consider the static solution for the adjustable robust problem and the transformation of the uncertainty sets introduced in Chapter 2. We show that the static solution is optimal for the adjustable robust problem when the transformation of the uncertainty set for each stage is convex.
Chapters 5 considers a two-stage adjustable robust linear covering problem with uncertain right-hand-side parameter. As mentioned earlier, such problems are often intractable due to astronomically many extreme points of the uncertainty set. We introduce a new approximation framework where we consider a “simple” set that is “close” to the original uncertainty set. Moreover, the adjustable robust problem can be solved efﬁciently over the extended set. We show that the approximation bound is related to a geometric factor that represents the Banach-Mazur distance between the two sets. Using this framework, we provide approximation bounds that are better than the bounds given by an afﬁne policy in [7] for a large class of interesting uncertainty sets. For instance, we provide an approximation solution that gives a m^{1/4}-approximation for the two-stage adjustable robust problem with hypersphere uncertainty set, while the afﬁne policy has an approximation ratio of O(\sqrt{m}).
Moreover, our bound for general p-norm ball is m^{\frac{p-1}{p^2}} as opposed to m^{1/p} as given by an affine policy.