The Probabilistic Minimum Spanning Tree, Part II: Probabilistic Analysis and Asymptotic Results

In this paper, which is a sequel to [3], we perform probabilistic analysis under the random Euclidean and the random length models of the probabilistic minimum spanning tree (PMST) problem and the two re-optimization strategies, in which we find the MST or the Steiner tree respectively among the points that are present at a particular instance. Under the random Euclidean model we prove that with probability 1, as the number of points goes to infinity, the expected length of the PMST is the same with the expectation of the MST re-optimization strategy and within a constant of the Steiner re-optimization strategy. In the random length model, using a result of Frieze [6], we prove that with probability 1 the expected length of the PMST is asymptotically smaller than the expectation of the MST re-optimization strategy. These results add evidence that a priori strategies may offer a useful and practical method for resolving combinatorial optimization problems on modified instances. Key words: Probabilistic analysis, combinatorial optimization, minimum spanning tree, Steiner tree

On the approximability of adjustable robust convex optimization under uncertainty

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.National Science Foundation (U.S.) (NSF Grant CMMI-1201116

The Minimum Spanning Tree Constant in Geometrical Probability and Under the Independent Model; A Unified Approach

Given n uniformly and independently points in the d dimensional cube of unit volume, it is well established that the length of the minimum spanning tree on these n points is asymptotic to /3MsT(d)n(d-l)/d,where the constant PMST(d) depends only on the dimension d. It has been a major open problem to determine the constant 3MST(d). In this paper we obtain an exact expression of the constant MST(d) as a series expansion. Truncating the expansion after a finite number of terms yields a sequence of lower bounds; the first 3 terms give a lower bound which is already very close to the empirically estimated value of the constant. Our proof technique unifies the derivation for the MST asymptotic behavior for the Euclidean and the independent model

Dynamic Pricing: A learning Approach

We present an optimization approach for jointly learning the demand as a functionof price, and dynamically setting prices of products in an oligopoly environment in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, but rather assume parametric families of demand functions that are learned over time. We first consider the noncompetitive case and present dynamic programming algorithms of increasing computational intensity with incomplete state information for jointly estimating the demand and setting prices as time evolves. Our computational results suggest that dynamic programming based methods outperform myopic policies often significantly. We then extend our analysis in a competitive environment with two firms. We introduce a more sophisticated model of demand learning, in which the price elasticities are slowly varying functions of time, and allows for increased flexibility in the modeling of the demand. We propose methods based on optimization for jointly estimating the Firm's own demand, its competitor's demand, and setting prices. In preliminary computational work, we found that optimization based pricing methods offer increased expected revenue for a firm independently of the policy the competitor firm is following

Multistage Air Traffic Flow Management under Capacity Uncertainty: A Robust and Adaptive Optimization Approach

In this paper, we study the first application of robust and adaptive optimization in the Air Traffic Flow Management (ATFM) problem. The existing models for network-wide ATFM assume deterministic capacity estimates across airports and sectors without taking into account the uncertainty in capacities induced by weather. We introduce a weather-front based approach to model the uncertainty inherent in airspace capacity estimates resulting from the impact of a small number of weather fronts moving across the National Airspace (NAS). The key advantage of our uncertainty set construction is the low-dimensionality (uncertainty in only two parameters govern the overall uncertainty set for each airspace element). We formulate the consequent ATFM problem under capacity uncertainty within the robust and adaptive optimization framework and propose tractable solution methodologies. Our theoretical contributions are as follows: i) we propose a polyhedral description of the convex hull of the discrete uncertainty set; ii) we prove the equivalence of the robust problem to a modified instance of the deterministic problem; and iii) we solve optimally the LP relaxation of the adaptive problem using piece-wise affine policies where the number of pieces in an optimal policy are governed by the number of extreme points in the uncertainty set. A particularly attractive feature is that for most practically encountered instances, an affine policy suffices to solve the adaptive problem optimally. Finally, we report empirical results from the proposed models on real world flight schedules augmented with simulated weather fronts that illuminate the merits of our proposal. The key insights from our computational results are: i) the robust problem inherits all the attractive properties of the deterministic problem (e.g., superior integrality properties and fast computational times); and ii) the price of robustness and adaptability is typically small.National Science Foundation (U.S.) (NSF Grant EFRI-0735905

Optimization of Polling Systems and Dynamic Vehicle Routing Problems on Networks

We consider the problem of optimizing a polling system, i.e., of optimally sequencing a server in a multi-class queueing system with switch-over times in order to minimize a linear objective function of the waiting times. The problem has important applications in computer, communication, production and transportation networks. We propose nonlinear programming relaxations that provide strong lower bounds to the optimal cost for all static policies. We also obtain lower bounds for dynamic policies as well, which are primarily useful under light traffic conditions and/or small switch-over times. We conjecture that the lower bounds developed in this paper for the class of static policies are also valid for dynamic policies under heavy traffic conditions. We use the information from the lower bound and integer programming techniques to construct static policies that are very close (0-3%) to the lower bounds. We compare numerically our proposed policies with static policies proposed in the literature as well as with dynamic policies and find that the policies we propose outperform all static policies proposed in the literature and at least in heavier traffic outperform dynamic policies as well

Design of Near Optimal Decision Rules in Multistage Adaptive Mixed-Integer Optimization

In recent years, decision rules have been established as the preferred solution method for addressing computationally demanding, multistage adaptive optimization problems. Despite their success, existing decision rules (a) are typically constrained by their a priori design and (b) do not incorporate in their modeling adaptive binary decisions. To address these problems, we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions, respectively. We then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed-integer optimization. We demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems. We provide global lower bounds and show that our approach is (i) practically tractable and (ii) provides high quality solutions that outperform alternative methods

Multiclass queueing systems in heavy traffic: an asymptotic approach based on distributional and conservation laws

We propose a new approach to analyze multiclass queueing systems in heavy traffic based on what we consider as fundamental laws in queueing systems, namely distributional and conservation laws. Methodologically, we extend the distributional laws from single class queueing systems to multiple classes and combine them with conservation laws to find the heavy traffic behavior of the following systems: a)EGI/G/1 queue under FIFO, b) EGI/G/1 queue with priorities, c) Polling systems with general arrival distributions. Compared with traditional heavy traffic analysis via Brownian processes, our approach gives more insight to the asymptotics used, solves systems that traditional heavy traffic theory has not fully addressed, and more importantly leads to closed form answers, which compared to simulation are very accurate even for moderate traffic

Survivable Networks, Linear Programming Relaxations and the Parsimonious Property

We consider the survivable network design problem - the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem

Multistage Robust Mixed-Integer Optimization with Adaptive Partitions

We present a new partition-and-bound method for multistage adaptive mixed-integer optimization (AMIO) problems that extends previous work on finite adaptability. The approach analyzes the optimal solution to a static (nonadaptive) version of an AMIO problem to gain insight into which regions of the uncertainty set are restricting the objective function value. We use this information to construct partitions in the uncertainty set, leading to a finitely adaptable formulation of the problem. We use the same information to determine a lower bound on the fully adaptive solution. The method repeats this process iteratively to further improve the objective until a desired gap is reached. We provide theoretical motivation for this method, and characterize its convergence properties and the growth in the number of partitions. Using these insights, we propose and evaluate enhancements to the method such as warm starts and smarter partition creation. We describe in detail how to apply finite adaptability to multistage AMIO problems to appropriately address nonanticipativity restrictions. Finally, we demonstrate in computational experiments that the method can provide substantial improvements over a nonadaptive solution and existing methods for problems described in the literature. In particular, we find that our method produces high-quality solutions versus the amount of computational effort, even as the problem scales in the number of time stages and the number of decision variables