Showing papers on "Stochastic programming published in 2005"

Uncertain convex programs: randomized solutions and confidence levels

Abstract: Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance’ parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations. In this paper, we consider an alternative ‘randomized’ or ‘scenario’ approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems

Abstract: There is a need for a methodology to fairly compare and present evaluation study results of stochastic global optimization algorithms. This need raises two important questions of (i) an appropriate set of benchmark test problems that the algorithms may be tested upon and (ii) a methodology to compactly and completely present the results. To address the first question, we compiled a collection of test problems, some are better known than others. Although the compilation is not exhaustive, it provides an easily accessible collection of standard test problems for continuous global optimization. Five different stochastic global optimization algorithms have been tested on these problems and a performance profile plot based on the improvement of objective function values is constructed to investigate the macroscopic behavior of the algorithms. The paper also investigates the microscopic behavior of the algorithms through quartile sequential plots, and contrasts the information gained from these two kinds of plots. The effect of the length of run is explored by using three maximum numbers of function evaluations and it is shown to significantly impact the behavior of the algorithms.

Market-clearing with stochastic security-part I: formulation

Abstract: The first of this two-paper series formulates a stochastic security-constrained multi-period electricity market-clearing problem with unit commitment. The stochastic security criterion accounts for a pre-selected set of random generator and line outages with known historical failure rates and involuntary load shedding as optimization variables. Unlike the classical deterministic reserve-constrained unit commitment, here the reserve services are determined by economically penalizing the operation of the market by the expected load not served. The proposed formulation is a stochastic programming problem that optimizes, concurrently with the pre-contingency social welfare, the expected operating costs associated with the deployment of the reserves following the contingencies. This stochastic programming formulation is solved in the second companion paper using mixed-integer linear programming methods. Two cases are presented: a small transmission-constrained three-bus network scheduled over a horizon of four hours and the IEEE Reliability Test System scheduled over 24 h. The impact on the resulting generation and reserve schedules of transmission constraints and generation ramp limits, of demand-side reserve, of the value of load not served, and of the constitution of the pre-selected set of contingencies are assessed.

The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems

Abstract: This paper introduces a method for solving numerical dynamic stochastic optimization problems that avoids rootfinding operations. The idea is applicable to many microeconomic and macroeconomic problems, including life cycle, buffer-stock, and stochastic growth problems. Software is provided.

On Complexity of Stochastic Programming Problems

Abstract: The main focus of this paper is in a discussion of complexity of stochastic programming problems. We argue that two-stage (linear) stochastic programming problems with recourse can be solved with a reasonable accuracy by using Monte Carlo sampling techniques, while multistage stochastic programs, in general, are intractable. We also discuss complexity of chance constrained problems and multistage stochastic programs with linear decision rules.

Constructing bidding curves for a price-taking retailer in the norwegian electricity market

Abstract: We propose a stochastic linear programming model for constructing piecewise-linear bidding curves to be submitted to Nord Pool, which is the Nordic power exchange. We consider the case of a price-taking power marketer who supplies electricity to price-sensitive end users. The objective is to minimize the expected cost of purchasing power from the day-ahead energy market and the short-term balancing market. The model is illustrated using a case study with data from Norway.

DynDE: a differential evolution for dynamic optimization problems

Abstract: This paper presents an approach of using differential evolution (DE) to solve dynamic optimization problems. Careful setting of parameters is necessary for DE algorithms to successfully solve optimization problems. This paper describes DynDE, a multipopulation DE algorithm developed specifically to solve dynamic optimization problems that doesn't need any parameter control strategy for the F or CR parameters. Experimental evidence has been gathered to show that this new algorithm is capable of efficiently solving the moving peaks benchmark.

Optimal investments in power generation under centralized and decentralized decision making

Abstract: This work presents a novel model for optimization of investments in new power generation under uncertainty. The model can calculate optimal investment strategies under both centralized social welfare and decentralized profit objectives. The power market is represented with linear supply and demand curves. A stochastic dynamic programming algorithm is used to solve the investment problem, where uncertainty in demand is represented as a discrete Markov chain. The stochastic dynamic model allows us to evaluate investment projects in new base and peak load power generation as real options, and determine optimal timing of the investments. In a case study, we use the model to compare optimal investment strategies under centralized and decentralized decision making. A number of interesting results follow by varying the assumptions about market structure and price response on the demand side.

Genetic Algorithm for Optimal Operating Policy of a Multipurpose Reservoir

Abstract: This paper presents a Genetic Algorithm (GA) model for finding the optimal operating policy of a multi-purpose reservoir, located on the river Pagladia, a major tributary of the river Brahmaputra. A synthetic monthly streamflow series of 100 years is used for deriving the operating policy. The policies derived by the GA model are compared with that of the stochastic dynamic programming (SDP) model on the basis of their performance in reservoir simulation for 20 years of historic monthly streamflow. The simulated result shows that GA-derived policies are promising and competitive and can be effectively used for reservoir operation.

The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification

Abstract: This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C3 (Common Cut Coefficients) Theorem. Based on the C3 Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D2). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D2 algorithm.

A Method for Staffing Large Call Centers Based on Stochastic Fluid Models

Abstract: We consider a call center model withm input flows andr pools of agents; them-vector ? of instantaneous arrival rates is allowed to be time dependent and to vary stochastically. Seeking to optimize the trade-off between personnel costs and abandonment penalties, we develop and illustrate a practical method for sizing ther agent pools. Using stochastic fluid models, this method reduces the staffing problem to a multidimensional newsvendor problem, which can be solved numerically by a combination of linear programming and Monte Carlo simulation. Numerical examples are presented, and in all cases the pool sizes derived by means of the proposed method are very close to optimal.

Multimarket optimal bidding for a power producer

Abstract: This paper considers a profit-maximizing thermal producer that participates in a sequence of spot markets, namely, day-ahead, automatic generation control (AGC), and balancing markets. The producer behaves as a price-taker in both the day-ahead market and the AGC market but as a potential price-maker in the volatile balancing market. The paper provides a stochastic programming methodology to determine the optimal bidding strategies for the day-ahead market. Uncertainty sources include prices for the day-ahead and AGC markets and balancing market linear price variations with the production of the thermal producer. Results from a realistic case study are reported and analyzed. Conclusions are duly drawn.

Managing Demand Risk in Tactical Supply Chain Planning for a Global Consumer Electronics Company

Abstract: We consider the problem of managing demand risk in tactical supply chain planning for a particular global consumer electronics company. The company follows a deterministic replenishment-and-planning process despite considerable demand uncertainty. As a possible way to formally address uncertainty, we provide two risk measures, “demand-at-risk” (DaR) and “inventory-at-risk” (IaR) and two linear programming models to help manage demand uncertainty. The first model is deterministic and can be used to allocate the replenishment schedule from the plants among the customers as per the existing process. The other model is stochastic and can be used to determine the “ideal” replenishment request from the plants under demand uncertainty. The gap between the output of the two models as regards requested replenishment and the values of the risk measures can be used by the company to reallocate capacity among different products and to thus manage demand/inventory risk.

Polyhedral Risk Measures in Stochastic Programming

Abstract: We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right-hand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the Conditional-Value-at-Risk.

Stochastic Programming Applied to Model Predictive Control

Abstract: Many robust model predictive control (MPC) schemes are based on min-max optimization, that is, the future control input trajectory is chosen as the one which minimizes the performance due to the worst disturbance realization In this paper we take a different route to solve MPC problems under uncertainty Disturbances are modelled as random variables and the expected value of the performance index is minimized The MPC scheme that can be solved using Stochastic Programming (SP), for which several efficient solution techniques are available We show that this formulation guarantees robust constraint fulfillment and that the expected value of the optimum cost function of the closed loop system decreases at each time step

Stochastic Programming: Optimization When Uncertainty Matters

Abstract: Stochastic Programming (SP) was first introduced by George Dantzig in the 1950's. Since that time, tremendous progress toward an understanding of properties of SP models and the design of algorithmic approaches for solving them has been made. As a result, SP is gaining recognition as a viable approach for large scale models of decisions under uncertainty. In this paper, we present an introduction to stochastic programming models and methodology at a level that is intended to be accessible to the breadth of members within the INFORMS community.

Multiobjective Path Finding in Stochastic Dynamic Networks, with Application to Routing Hazardous Materials Shipments

Abstract: We describe a method for finding nondominated paths for multiple routing objectives in networks where the routing attributes are uncertain, and the probability distributions that describe those attributes vary by time of day. This problem is particularly important in routing and scheduling of shipments of very hazardous materials. Our method extends and integrates the work of several previous authors, resulting in a new algorithm that propagates means and variances of the uncertain attributes along paths and compares partial paths that arrive at a given node within a user-specified time window. The comparison uses an approximate stochastic dominance criterion. We illustrate the effects of changing primary parameters of the algorithm using a small test network, and we show how the nondominated solution set achieved is larger than the set that would be identified if the uncertainty in routing attributes were ignored. We then demonstrate how the algorithm creates an effective solution set in a case study using a large network.

Robust Capacity Planning in Semiconductor Manufacturing

Abstract: We present a stochastic programming approach to capacity planning under demand uncertainty in semiconductor manufacturing. Given multiple demand scenarios together with associated probabilities, our aim is to identify a set of tools that is a good compromise for all these scenarios. More precisely, we formulate a mixed-integer program in which expected value of the unmet demand is minimized subject to capacity and budget constraints. This is a dicult two-stage stochastic mixed-integer program which can not be solved to optimality in a reasonable amount of time. We instead propose a heuristic that can produce near-optimal solutions. Our heuristic strengthens the linear programming relaxation of the formulation with cutting planes and performs limited enumeration. Analyses of the results in some real-life situations are also presented.

A Scenario Aggregation-Based Approach for Determining a Robust Airline Fleet Composition for Dynamic Capacity Allocation

Abstract: Recently, airlines and aircraft manufacturers have realized the benefits of the emerging concept of dynamic capacity allocation, and have initiated advanced decision support systems to assist them in this respect. Strategic airline fleet planning is one of the major issues addressed through such systems. We present background research connected with the dynamic allocation concept, which accounts explicitly for the stochastic nature of passenger demand in the fleet composition problem. We address this problem through a scenario aggregation-based approach and present results on representative case studies based on realistic data. Our investigations establish clear benefits of a stochastic approach as compared with deterministic formulations, as well as its implementation feasibility using state-of-the-art optimization software.

Adaptive MIMO antenna selection via discrete stochastic optimization

Abstract: Recently it has been shown that it is possible to improve the performance of multiple-input multiple-output (MIMO) systems by employing a larger number of antennas than actually used and selecting the optimal subset based on the channel state information. Existing antenna selection algorithms assume perfect channel knowledge and optimize criteria such as Shannon capacity or various bounds on error rate. This paper examines MIMO antenna selection algorithms where the set of possible solutions is large and only a noisy estimate of the channel is available. In the same spirit as traditional adaptive filtering algorithms, we propose simulation based discrete stochastic optimization algorithms to adaptively select a better antenna subset using criteria such as maximum mutual information, bounds on error rate, etc. These discrete stochastic approximation algorithms are ideally suited to minimize the error rate since computing a closed form expression for the error rate is intractable. We also consider scenarios of time-varying channels for which the antenna selection algorithms can track the time-varying optimal antenna configuration. We present several numerical examples to show the fast convergence of these algorithms under various performance criteria, and also demonstrate their tracking capabilities.