Showing papers on "Stochastic programming published in 2012"

An optimal method for stochastic composite optimization

Abstract: This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems.

A Chance-Constrained Two-Stage Stochastic Program for Unit Commitment With Uncertain Wind Power Output

Abstract: In this paper, we present a unit commitment problem with uncertain wind power output. The problem is formulated as a chance-constrained two-stage (CCTS) stochastic program. Our model ensures that, with high probability, a large portion of the wind power output at each operating hour will be utilized. The proposed model includes both the two-stage stochastic program and the chance-constrained stochastic program features. These types of problems are challenging and have never been studied together before, even though the algorithms for the two-stage stochastic program and the chance-constrained stochastic program have been recently developed separately. In this paper, a combined sample average approximation (SAA) algorithm is developed to solve the model effectively. The convergence property and the solution validation process of our proposed combined SAA algorithm is discussed and presented in the paper. Finally, computational results indicate that increasing the utilization of wind power output might increase the total power generation cost, and our experiments also verify that the proposed algorithm can solve large-scale power grid optimization problems.

Optimal Stochastic Approximation Algorithms for Strongly Convex Stochastic Composite Optimization I: A Generic Algorithmic Framework

Abstract: In this paper we present a generic algorithmic framework, namely, the accelerated stochastic approximation (AC-SA) algorithm, for solving strongly convex stochastic composite optimization (SCO) problems. While the classical stochastic approximation algorithms are asymptotically optimal for solving differentiable and strongly convex problems, the AC-SA algorithm, when employed with proper stepsize policies, can achieve optimal or nearly optimal rates of convergence for solving different classes of SCO problems during a given number of iterations. Moreover, we investigate these AC-SA algorithms in more detail, such as by establishing the large-deviation results associated with the convergence rates and introducing an efficient validation procedure to check the accuracy of the generated solutions.

Tracking the articulated motion of two strongly interacting hands

Abstract: We propose a method that relies on markerless visual observations to track the full articulation of two hands that interact with each-other in a complex, unconstrained manner. We formulate this as an optimization problem whose 54-dimensional parameter space represents all possible configurations of two hands, each represented as a kinematic structure with 26 Degrees of Freedom (DoFs). To solve this problem, we employ Particle Swarm Optimization (PSO), an evolutionary, stochastic optimization method with the objective of finding the two-hands configuration that best explains observations provided by an RGB-D sensor. To the best of our knowledge, the proposed method is the first to attempt and achieve the articulated motion tracking of two strongly interacting hands. Extensive quantitative and qualitative experiments with simulated and real world image sequences demonstrate that an accurate and efficient solution of this problem is indeed feasible.

Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

Abstract: Preface.- 1. Conditional Expectation and Linear Parabolic PDEs.- 2. Stochastic Control and Dynamic Programming.- 3. Optimal Stopping and Dynamic Programming.- 4. Solving Control Problems by Verification.- 5. Introduction to Viscosity Solutions.- 6. Dynamic Programming Equation in the Viscosity Sense.- 7. Stochastic Target Problems.- 8. Second Order Stochastic Target Problems.- 9. Backward SDEs and Stochastic Control.- 10. Quadratic Backward SDEs.- 11. Probabilistic Numerical Methods for Nonlinear PDEs.- 12. Introduction to Finite Differences Methods.- References.

Data centers power reduction: A two time scale approach for delay tolerant workloads

Abstract: In this work we focus on a stochastic optimization based approach to make distributed routing and server management decisions in the context of large-scale, geographically distributed data centers, which offers significant potential for exploring power cost reductions. Our approach considers such decisions at different time scales and offers provable power cost and delay characteristics. The utility of our approach and its robustness are also illustrated through simulation-based experiments under delay tolerant workloads.

Robot Motion Planning in Dynamic, Uncertain Environments

Abstract: This paper presents a strategy for planning robot motions in dynamic, uncertain environments (DUEs). Successful and efficient robot operation in such environments requires reasoning about the future evolution and uncertainties of the states of the moving agents and obstacles. A novel procedure to account for future information gathering (and the quality of that information) in the planning process is presented. To approximately solve the stochastic dynamic programming problem that is associated with DUE planning, we present a partially closed-loop receding horizon control algorithm whose solution integrates prediction, estimation, and planning while also accounting for chance constraints that arise from the uncertain locations of the robot and obstacles. Simulation results in simple static and dynamic scenarios illustrate the benefit of the algorithm over classical approaches. The approach is also applied to more complicated scenarios, including agents with complex, multimodal behaviors, basic robot-agent interaction, and agent information gathering.

Kullback-Leibler divergence constrained distributionally robust optimization

Abstract: In this paper we study distributionally robust optimization (DRO) problems where the ambiguity set of the probability distribution is defined by the Kullback-Leibler (KL) divergence. We consider DRO problems where the ambiguity is in the objective function, which takes a form of an expectation, and show that the resulted minimax DRO problems can be formulated as a one-layer convex minimization problem. We also consider DRO problems where the ambiguity is in the constraint. We show that ambiguous expectation-constrained programs may be reformulated as a one-layer convex optimization problem that takes the form of the Benstein approximation of Nemirovski and Shapiro (2006). We further consider distributionally robust probabilistic programs. We show that the optimal solution of a probability minimization problem is also optimal for the distributionally robust version of the same problem, and also show that the ambiguous chance-constrained programs (CCPs) may be reformulated as the original CCP with an adjusted confidence level. A number of examples and special cases are also discussed in the paper to show that the reformulated problems may take simple forms that can be solved easily. The main contribution of the paper is to show that the KL divergence constrained DRO problems are often of the same complexity as their original stochastic programming problems and, thus, KL divergence appears a good candidate in modeling distribution ambiguities in mathematical programming.

Design under uncertainty of hydrocarbon biorefinery supply chains: Multiobjective stochastic programming models, decomposition algorithm, and a Comparison between CVaR and downside risk

Abstract: A bicriterion, multiperiod, stochastic mixed-integer linear programming model to address the optimal design of hydrocarbon biorefinery supply chains under supply and demand uncertainties is presented. The model accounts for multiple conversion technologies, feedstock seasonality and fluctuation, geographical diversity, biomass degradation, demand variation, government incentives, and risk management. The objective is simultaneous minimization of the expected annualized cost and the financial risk. The latter criterion is measured by conditional value-at-risk and downside risk. The model simultaneously determines the optimal network design, technology selection, capital investment, production planning, and logistics management decisions. Multicut L-shaped method is implemented to circumvent the computational burden of solving large scale problems. The proposed modeling framework and algorithm are illustrated through four case studies of hydrocarbon biorefinery supply chain for the State of Illinois. Comparisons between the deterministic and stochastic solutions, the different risk metrics, and two decomposition methods are discussed. The computational results show the effectiveness of the proposed strategy for optimal design of hydrocarbon biorefinery supply chain under the presence of uncertainties. © 2012 American Institute of Chemical Engineers AIChE J, 2012

Bioethanol supply chain system planning under supply and demand uncertainties

Abstract: A mixed integer stochastic programming model is established to support strategic planning of bioenergy supply chain systems and optimal feedstock resource allocation in an uncertain decision environment. The two-stage stochastic programming model, together with a Lagrange relaxation based decomposition solution algorithm, was implemented in a real-world case study in California to explore the potential of waste-based bioethanol production. The model results show that biowaste-based ethanol can be a viable part of sustainable energy solution for the future.

Automatic robust convex programming

Abstract: This paper presents the robust optimization framework in the modelling language YALMIP, which carries out robust modelling and uncertainty elimination automatically and allows the user to concentrate on the high-level model. While introducing the software package, a brief summary of robust optimization is given, as well as some comments on modelling and tractability of complex convex uncertain optimization problems.

Multilevel Optimization: Algorithms and Applications

Abstract: In many decision processes there is an hierarchy of decision-makers and decisions are taken at different levels in this hierarchy. Multilevel programming focuses on the whole hierarchy structure. In terms of modeling, the constraint domain associated with a multilevel programming problem is implicitly determined by a series of optimization problems which must be solved in a predetermined sequence. The field of multilevel optimization has become a well-known and important research field. Hierarchical structures can be found in scientific disciplines such as environment, ecology, biology, chemical engineering, mechanics, classification theory, databases, network design, transportation, game theory and economics. Moreover, new applications are constantly being introduced. This has stimulated the development of new theory and efficient algorithms. This volume contains 16 chapters written by various leading researchers and presents a cohesive authoritative overview of developments and applications in their emerging field of optimization. Audience: Researchers whose work involves the application of mathematical programming and optimization to hierarchical structures.

Modeling with Stochastic Programming

Abstract: Uncertainty in Optimization.-Modeling Feasibility and Dynamics.-Modeling the Objective Function.- Scenario tree generation, With Michal Kaut.-Service network design, With Arnt-Gunnar Lium and Teodor Gabriel Crainic.- A multi-dimensional newsboy problem with substitution, With Hajnalka Vaagen.- Stochastic Discount Factors.- Long Lead Time Production, With Aliza Heching.- References.- Index"/p>

Hub location under uncertainty

Abstract: Hub location problems are network design problems which are solved as part of a strategic decision making process. In strategic planning, decisions may have a long lasting effect and the implementation may take considerable time. Moreover, input data is not precisely known in advance. Hence, decisions have to be made anticipating uncertainty. In this paper, we address several aspects concerning hub location problems under uncertainty. Two sources of uncertainty are considered: the set-up costs for the hubs and the demands to be transported between the nodes. Generic models are presented for single and multiple allocation versions of the problems. Firstly, the two sources of uncertainty are analyzed separately and afterwards a more comprehensive model is proposed considering all sources of uncertainty. Using a set of computational tests performed, we analyze the changes in the solutions driven by the different sources of uncertainty considered isolated and combined.

An Energy Management Controller to Optimally Trade Off Fuel Economy and Drivability for Hybrid Vehicles

Abstract: Hybrid vehicle fuel economy performance is highly sensitive to the energy management strategy used to regulate power flow among the various energy sources and sinks. Optimal non-causal solutions are easy to determine if the drive cycle is known a priori. It is very challenging to design causal controllers that yield good fuel economy for a range of possible driver behavior. Additional challenges come in the form of constraints on powertrain activity, such as shifting and starting the engine, which are commonly called “drivability” metrics and can adversely affect fuel economy. In this paper, drivability restrictions are included in a shortest path stochastic dynamic programming (SP-SDP) formulation of the real-time energy management problem for a prototype vehicle, where the drive cycle is modeled as a stationary, finite-state Markov chain. When the SP-SDP controllers are evaluated with a high-fidelity vehicle simulator over standard government drive cycles, and compared to a baseline industrial controller, they are shown to improve fuel economy more than 11% for equivalent levels of drivability. In addition, the explicit tradeoff between fuel economy and drivability is quantified for the SP-SDP controllers.

A multi-stage stochastic supply network design problem with financial decisions and risk management

Abstract: In this paper, a multi-period supply chain network design problem is addressed. Several aspects of practical relevance are considered such as those related with the financial decisions that must be accounted for by a company managing a supply chain. The decisions to be made comprise the location of the facilities, the flow of commodities and the investments to make in alternative activities to those directly related with the supply chain design. Uncertainty is assumed for demand and interest rates, which is described by a set of scenarios. Therefore, for the entire planning horizon, a tree of scenarios is built. A target is set for the return on investment and the risk of falling below it is measured and accounted for. The service level is also measured and included in the objective function. The problem is formulated as a multi-stage stochastic mixed-integer linear programming problem. The goal is to maximize the total financial benefit. An alternative formulation which is based upon the paths in the scenario tree is also proposed. A methodology for measuring the value of the stochastic solution in this problem is discussed. Computational tests using randomly generated data are presented showing that the stochastic approach is worth considering in these types of problems.

A two-echelon stochastic facility location model for humanitarian relief logistics

Abstract: We develop a two-stage stochastic programming model for a humanitarian relief logistics problem where decisions are made for pre- and post-disaster rescue centers, the amount of relief items to be stocked at the pre-disaster rescue centers, the amount of relief item flows at each echelon, and the amount of relief item shortage. The objective is to minimize the total cost of facility location, inventory holding, transportation and shortage. The deterministic equivalent of the model is formulated as a mixed-integer linear programming model and solved by a heuristic method based on Lagrangean relaxation. Results on randomly generated test instances show that the proposed solution method exhibits good performance up to 25 scenarios. We also validate our model by calculating the value of the stochastic solution and the expected value of perfect information.

Tractable stochastic analysis in high dimensions via robust optimization

Abstract: Modern probability theory, whose foundation is based on the axioms set forth by Kolmogorov, is currently the major tool for performance analysis in stochastic systems. While it offers insights in understanding such systems, probability theory, in contrast to optimization, has not been developed with computational tractability as an objective when the dimension increases. Correspondingly, some of its major areas of application remain unsolved when the underlying systems become multidimensional: Queueing networks, auction design in multi-item, multi-bidder auctions, network information theory, pricing multi-dimensional options, among others. We propose a new approach to analyze stochastic systems based on robust optimization. The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory, with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem. In addition, we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization. In this way, the performance analysis questions become highly structured optimization problems (linear, semidefinite, mixed integer) for which there exist efficient, practical algorithms that are capable of solving problems in high dimensions. We demonstrate that the proposed approach achieves computationally tractable methods for (a) analyzing queueing networks, (b) designing multi-item, multi-bidder auctions with budget constraints, and (c) pricing multi-dimensional options.

Validation analysis of mirror descent stochastic approximation method

Abstract: The main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation (SA) type algorithms. To this end we show that while running a Mirror Descent Stochastic Approximation procedure one can compute, with a small additional effort, lower and upper statistical bounds for the optimal objective value. We demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method, while their computational cost is considerably smaller.

Dynamic Resource Allocation for Heterogeneous Services in Cognitive Radio Networks With Imperfect Channel Sensing

Abstract: Resources in cognitive radio networks (CRNs) should dynamically be allocated according to the sensed radio environment Although some work has been done for dynamic resource allocation in CRNs, many works assume that the radio environment can perfectly be sensed However, in practice, it is difficult for the secondary network to have the perfect knowledge of a dynamic radio environment in CRNs In this paper, we study the dynamic resource allocation problem for heterogeneous services in CRNs with imperfect channel sensing We formulate the power and channel allocation problem as a mixed-integer programming problem under constraints The computational complexity is enormous to solve the problem To reduce the computational complexity, we tackle this problem in two steps First, we solve the optimal power allocation problem using the Lagrangian dual method under the assumption of known channel allocation Next, we solve the joint power and channel allocation problem using the discrete stochastic optimization method, which has low computational complexity and fast convergence to approximate to the optimal solution Another advantage of this method is that it can track the changing radio environment to dynamically allocate the resources Simulation results are presented to demonstrate the effectiveness of the proposed scheme