On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive. If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain.National Science Foundation (U.S.) (NSF Grant DMI-0556106)National Science Foundation (U.S.) (NSF Grant EFRI-0735905

A First Derivative Potts Model for Segmentation and Denoising Using ILP

Unsupervised image segmentation and denoising are two fundamental tasks in image processing. Usually, graph based models such as multicut are used for segmentation and variational models are employed for denoising. Our approach addresses both problems at the same time. We propose a novel ILP formulation of the first derivative Potts model with the $\ell_1$ data term, where binary variables are introduced to deal with the $\ell_0$ norm of the regularization term. The ILP is then solved by a standard off-the-shelf MIP solver. Numerical experiments are compared with the multicut problem.Comment: 6 pages, 2 figures. To appear at Proceedings of International Conference on Operations Research 2017, Berli

On the Power and Limitations of Affine Policies in Two-Stage Adaptive Optimization

We consider a two-stage adaptive linear optimization problem under right hand side uncertainty with a min–max objective and give a sharp characterization of the power and limitations of affine policies (where the second stage solution is an affine function of the right hand side uncertainty). In particular, we show that the worst-case cost of an optimal affine policy can be Omega(m12−) times the worst-case cost of an optimal fully-adaptable solution for any delta > 0, where m is the number of linear constraints. We also show that the worst-case cost of the best affine policy is O(m) times the optimal cost when the first-stage constraint matrix has non-negative coefficients. Moreover, if there are only k ≤ m uncertain parameters, we generalize the performance bound for affine policies to O(k) , which is particularly useful if only a few parameters are uncertain. We also provide an O(k) -approximation algorithm for the general case without any restriction on the constraint matrix but the solution is not an affine function of the uncertain parameters. We also give a tight characterization of the conditions under which an affine policy is optimal for the above model. In particular, we show that if the uncertainty set, R+m is a simplex, then an affine policy is optimal. However, an affine policy is suboptimal even if is a convex combination of only (m + 3) extreme points (only two more extreme points than a simplex) and the worst-case cost of an optimal affine policy can be a factor (2 − delta) worse than the worst-case cost of an optimal fully-adaptable solution for any delta > 0.National Science Foundation (U.S.) (NSF Grants DMI-0556106)National Science Foundation (U.S.) (EFRI-0735905

Delay, memory, and messaging tradeoffs in distributed service systems

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate $\lambda n$, with $0<\lambda<1$, and are immediately dispatched by a centralized dispatcher to one of $n$ First-In-First-Out queues associated with $n$ identical servers. The dispatcher is endowed with a finite memory, and with the ability to exchange messages with the servers. We propose and study a resource-constrained "pull-based" dispatching policy that involves two parameters: (i) the number of memory bits available at the dispatcher, and (ii) the average rate at which servers communicate with the dispatcher. We establish (using a fluid limit approach) that the asymptotic, as $n\to\infty$, expected queueing delay is zero when either (i) the number of memory bits grows logarithmically with $n$ and the message rate grows superlinearly with $n$, or (ii) the number of memory bits grows superlogarithmically with $n$ and the message rate is at least $\lambda n$. Furthermore, when the number of memory bits grows only logarithmically with $n$ and the message rate is proportional to $n$, we obtain a closed-form expression for the (now positive) asymptotic delay. Finally, we demonstrate an interesting phase transition in the resource-constrained regime where the asymptotic delay is non-zero. In particular, we show that for any given $\alpha>0$ (no matter how small), if our policy only uses a linear message rate $\alpha n$, the resulting asymptotic delay is upper bounded, uniformly over all $\lambda<1$; this is in sharp contrast to the delay obtained when no messages are used ($\alpha = 0$), which grows as $1/(1-\lambda)$ when $\lambda\uparrow 1$, or when the popular power-of-$d$-choices is used, in which the delay grows as $\log(1/(1-\lambda))$

Probabilistic combinatorial optimization: Moments, semidefinite programming, and asymptotic bounds

10.1137/S1052623403430610SIAM Journal on Optimization151185-20

Decomposition Algorithms for Analyzing Transient Phenomena in Multi-class Queueing Networks in Air Transportation

In a previous paper (Peterson, Bertsimas, and Odoni 1992), we studied the phenomenon of transient congestion in landings at a hub airport and developed a recursive approach for computing moments of queue lengths and waiting times. In this paper we extend our approach to a network, developing two approximations based on the method used for the single hub. We present computational results for a simple 2-hub network and indicate the usefulness of the approach in analyzing the interaction between hubs. Although our motivation is drawn from air transportation, our method is applicable to all multi-class queuing networks where service capacity at a station may be modeled as a Markov or semi-Markov process. Our method represents a new approach for analyzing transient congestion phenomena in such networks. Airport congestion and delay have grown significantly over the last decade. By 1986 ground delays at domestic airports averaged 2000 hours per day, the equivalent of grounding the entire fleet of Delta Airlines at that tillie (250 aircraft) for one day (Donoghue 1986). In 1990, 21 airports in the U.S. exceeded 20, 000 hours of delay, with 12 more projected to exceed this total by 1997 (National Transportation Research Board 1991). This amounts to *School of Public and Environmental Affairs, Indiana University, Bloomington, Indiana tSloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts ;Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusett

On the probabilistic min spanning tree Problem

We study a probabilistic optimization model for min spanning tree, where any vertex vi of the input-graph G(V,E) has some presence probability pi in the final instance G′ ⊂ G that will effectively be optimized. Suppose that when this “real” instance G′ becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G ′. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G ′ ⊆ G is optimal for G ′. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

A Geometric Characterization of the Power of Finite Adaptability in Multistage Stochastic and Adaptive Optimization

In this paper, we show a significant role that geometric properties of uncertainty sets, such as symmetry, play in determining the power of robust and finitely adaptable solutions in multistage stochastic and adaptive optimization problems. We consider a fairly general class of multistage mixed integer stochastic and adaptive optimization problems and propose a good approximate solution policy with performance guarantees that depend on the geometric properties of the uncertainty sets. In particular, we show that a class of finitely adaptable solutions is a good approximation for both the multistage stochastic and the adaptive optimization problem. A finitely adaptable solution generalizes the notion of a static robust solution and specifies a small set of solutions for each stage; the solution policy implements the best solution from the given set, depending on the realization of the uncertain parameters in past stages. Therefore, it is a tractable approximation to a fully adaptable solution for the multistage problems. To the best of our knowledge, these are the first approximation results for the multistage problem in such generality. Moreover, the results and the proof techniques are quite general and also extend to include important constraints such as integrality and linear conic constraints.National Science Foundation (U.S.) (Grant EFRI-0735905