Essays on Approximation Algorithms for Robust Linear Optimization Problems

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

Dynamic Pricing through Sampling Based Optimization

In this paper we develop an approach to dynamic pricing that combines ideas from data-driven and robust optimization to address the uncertain and dynamic aspects of the problem. In our setting, a firm off ers multiple products to be sold over a fixed discrete time horizon. Each product sold consumes one or more resources, possibly sharing the same resources among di fferent products. The firm is given a fixed initial inventory of these resources and cannot replenish this inventory during the selling season. We assume there is uncertainty about the demand seen by the fi rm for each product and seek to determine a robust and dynamic pricing strategy that maximizes revenue over the time horizon. While the traditional robust optimization models are tractable, they give rise to static policies and are often too conservative. The main contribution of this paper is the exploration of closed-loop pricing policies for di fferent robust objectives, such as MaxMin, MinMax Regret and MaxMin Ratio. We introduce a sampling based optimization approach that can solve this problem in a tractable way, with a con fidence level and a robustness level based on the number of samples used. We will show how this methodology can be used for data-driven pricing or adapted for a random sampling optimization approach when limited information is known about the demand uncertainty. Finally, we compare the revenue performance of the di fferent models using numerical simulations, exploring the behavior of each model under diff erent sample sizes and sampling distributions.National Science Foundation (U.S.) (Grant 0556106-CMII)National Science Foundation (U.S.) (Grant 0824674-CMII)Singapore-MIT Allianc

Two-Stage Robust Optimization for the Orienteering Problem with Stochastic Weights

In this paper, the two-stage orienteering problem with stochastic weights is studied, where the first-stage problem is to plan a path under the uncertain environment and the second-stage problem is a recourse action to make sure that the length constraint is satisfied after the uncertainty is realized. First, we explain the recourse model proposed by Evers et al. (2014) and point out that this model is very complex. Then, we introduce a new recourse model which is much simpler with less variables and less constraints. Based on these two recourse models, we introduce two different two-stage robust models for the orienteering problem with stochastic weights. We theoretically prove that the two-stage robust models are equivalent to their corresponding static robust models under the box uncertainty set, which indicates that the two-stage robust models can be solved by using common mathematical programming solvers (e.g., IBM CPLEX optimizer). Furthermore, we prove that the two two-stage robust models are equivalent to each other even though they are based on different recourse models, which indicates that we can use a much simpler model instead of a complex model for practical use. A case study is presented by comparing the two-stage robust models with a one-stage robust model for the orienteering problem with stochastic weights. The numerical results of the comparative studies show the effectiveness and superiority of the proposed two-stage robust models for dealing with the two-stage orienteering problem with stochastic weights

Tractable Policies in Dynamic Robust Optimization

In many sequential decision problems, uncertainty is revealed over time and we need to make decisions in the face of uncertainty. This is a fundamental problem arising in many applications such as facility location, resource allocation and capacity planning under demand uncertainty. Robust optimization is an approach to model uncertainty where we optimize over the worst-case realization of parameters within an uncertainty set. While computing an optimal solution in dynamic robust optimization is usually intractable, affine policies (or linear decision rules) are widely used as an approximate solution approach. However, there is a stark contrast between the observed good empirical performance and the bad worst-case theoretical performance bounds. In the first part of this thesis, we address this stark contrast between theory and practice. In particular, we introduce a probabilistic approach in Chapter 2 to analyze the performance of affine policies on randomly generated instances and show they are near-optimal with high probability under reasonable assumptions. In Chapter 3, we study these policies under important models of uncertainty such as budget of uncertainty sets and intersection of budgeted sets and show that affine policies give an optimal approximation matching the hardness of approximation. In the second part of the thesis and based on our analysis of affine policies, we design new tractable policies for dynamic robust optimization. In particular, in Chapter 4, we present a tractable framework to design piecewise affine policies that can be computed efficiently and improve over affine policies for many instances. In Chapter 5, we introduce extended affine policies and threshold policies and show that their performance guarantees are significantly better than previous policies. Finally, in Chapter 6, we study piecewise static policies and their limitations for solving some classes of dynamic robust optimization problems