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31
Convex relaxations of chance constrained optimization problems
, 2011
"... In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed ..."
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In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance constraint function is affine with respect to the underlying random vector and the random vector has independent components. We also propose an iterative improvement scheme for refining the bounds. 1
Resource allocation with stochastic demands
 in Proceedings of the 2012 IEEE 8th International Conference on Distributed Computing in Sensor Systems, DCOSS ’12
, 2012
"... AbstractResources in modern computer systems include not only CPU, but also memory, hard disk, bandwidth, etc. To serve multiple users simultaneously, we need to satisfy their requirements in all resource dimensions. Meanwhile, their demands follow a certain distribution and may change over time. ..."
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AbstractResources in modern computer systems include not only CPU, but also memory, hard disk, bandwidth, etc. To serve multiple users simultaneously, we need to satisfy their requirements in all resource dimensions. Meanwhile, their demands follow a certain distribution and may change over time. Our goal is then to admit as many users as possible to the system without violating the resource capacity more often than a predefined overflow probability. In this paper, we study the problem of allocating multiple resources among a group of users/tasks with stochastic demands. We model it as a stochastic multidimensional knapsack problem. We extend and apply the concept of effective bandwidth in order to solve this problem efficiently. Via numerical experiments, we show that our algorithms achieve nearoptimal performance with specified overflow probability.
Risk based Optimization for Improving Emergency Medical Systems
"... In emergency medical systems, arriving at the incident location a few seconds early can save a human life. Thus, this paper is motivated by the need to reduce the response time – time taken to arrive at the incident location after receiving the emergency call – of Emergency Response Vehicles, ERVs ..."
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In emergency medical systems, arriving at the incident location a few seconds early can save a human life. Thus, this paper is motivated by the need to reduce the response time – time taken to arrive at the incident location after receiving the emergency call – of Emergency Response Vehicles, ERVs (ex: ambulances, fire rescue vehicles) for as many requests as possible. We expect to achieve this primarily by positioning the ”right ” number of ERVs at the ”right ” places and at the ”right ” times. Given the exponentially large action space (with respect to number of ERVs and their placement) and the stochasticity in location and timing of emergency incidents, this problem is computationally challenging. To that end, our contributions building on existing datadriven approaches are three fold: 1. Based on real world evaluation metrics, we provide a risk based optimization criterion to learn from past incident data. Instead of minimizing expected response time, we minimize the largest value of response time such that the risk of finding requests that have a higher value is bounded (ex: Only 10 % of requests should have a response time greater than 8 minutes). 2. We develop a mixed integer linear optimization formulation to learn and compute an allocation from a set of input requests while considering the risk criterion. 3. To allow for ”live ” reallocation of ambulances, we provide a decomposition method based on Lagrangian Relaxation to significantly reduce the runtime of the optimization formulation. Finally, we provide an exhaustive evaluation on realworld datasets from two asian cities that demonstrates the improvement provided by our approach over current practice and the best known approach from literature.
TESLA: an extended study of an energysaving agent that leverages schedule flexibility. Autonomous Agents and MultiAgent Systems
, 2013
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A constraint sampling approach for multistage robust optimization
, 2010
"... We propose a tractable approximation scheme for convex (not necessarily linear) multistage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computatio ..."
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We propose a tractable approximation scheme for convex (not necessarily linear) multistage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization. We obtain apriori probabilistic guarantees on the feasibility properties of the optimal decision rule by extending existing constraint sampling techniques from the single to the multistage case. We demonstrate that for a suitable choice of basis functions, the approximation converges as the size of the basis and the number of sampled constraints tend to infinity. The approach yields an algorithm parameterized in the basis size, the probability of constraint violation and the confidence that this probability will not be exceeded. These three parameters serve to tune the tradeoff between optimality and feasibility of the decision rules and the computational cost of the algorithm. We assess the convergence and scalability properties of our approach in the context of two inventory management problems.
Optimization approaches for solving chance constrained stochastic orienteering problems
 In Proceedings of the International Conference on Algorithmic Decision Theory (ADT
"... Abstract. Orienteering problems (OPs) are typically used to model routing and trip planning problems. OP is a variant of the well known traveling salesman problem where the goal is to compute the highest reward path that includes a subset of nodes and has an overall travel time less than the specif ..."
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Abstract. Orienteering problems (OPs) are typically used to model routing and trip planning problems. OP is a variant of the well known traveling salesman problem where the goal is to compute the highest reward path that includes a subset of nodes and has an overall travel time less than the specified deadline. Stochastic orienteering problems (SOPs) extend OPs to account for uncertain travel times and are significantly harder to solve than deterministic OPs. In this paper, we contribute a scalable mixed integer LP formulation for solving risk aware SOPs, which is a principled approximation of the underlying stochastic optimization problem. Empirically, our approach provides significantly better solution quality than the previous best approach over a range of synthetic benchmarks and on a realworld theme park trip planning problem. 1
Resource Allocation With NonDeterministic Demands and Profits
"... Abstract—Support for intelligent and autonomous resource management is one key factor to the success of modern sensor network systems. The limited resources, such as exhaustible battery life, moderate processing ability and finite bandwidth, restrict the system’s ability to serve multiple users sim ..."
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Abstract—Support for intelligent and autonomous resource management is one key factor to the success of modern sensor network systems. The limited resources, such as exhaustible battery life, moderate processing ability and finite bandwidth, restrict the system’s ability to serve multiple users simultaneously. It always happens that only a subset of tasks is selected with the goal of maximizing total profit. Besides, because of uncertain factors like unreliable wireless medium or variable quality of sensor outputs, it is not practical to assume that both demands and profits of tasks are deterministic and known a priori, both of which may be stochastic following certain distributions. In this paper, we model this resource allocation challenge as a stochastic knapsack problem. We study a specific case in which both demands and profits follow normal distributions, which are then extended to Poisson and Binomial variables. A couple of tunable parameters are introduced to configure two probabilities: one limits the capacity overflow rate with which the combined demand is allowed to exceed the available supply, and the other sets the minimum chance at which expected profit is required to be achieved. We define relative values for random variables in given conditions, and utilize them to search for the best resource allocation solutions. We propose heuristics with different optimality/efficiency tradeoffs, and find that our algorithms run relatively fast and provide results considerably close to the optimum. I.
Optimization: A Journal of Mathematical Programming and Operations Research Stochastic programming problems with generalized integrated chance constraints
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Discrete Optimization A linear programming approach for linear programs with probabilistic constraints
"... a b s t r a c t We study a class of mixedinteger programs for solving linear programs with joint probabilistic constraints from random righthand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provi ..."
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a b s t r a c t We study a class of mixedinteger programs for solving linear programs with joint probabilistic constraints from random righthand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixedinteger formulation of [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worstcase solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixedinteger programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.
CHANCE CONSTRAINED PROBLEMS: PENALTY REFORMULATION AND PERFORMANCE OF SAMPLE APPROXIMATION TECHNIQUE
"... Chance constrained problems: penalty reformulation and performance of sample approximation technique ..."
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Chance constrained problems: penalty reformulation and performance of sample approximation technique