Showing papers on "Stochastic programming published in 2008"

Robust Stochastic Approximation Approach to Stochastic Programming

Abstract: In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte Carlo sampling techniques, namely, the stochastic approximation (SA) and the sample average approximation (SAA) methods. Both approaches, the SA and SAA methods, have a long history. Current opinion is that the SAA method can efficiently use a specific (say, linear) structure of the considered problem, while the SA approach is a crude subgradient method, which often performs poorly in practice. We intend to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems. We extend the analysis to the case of convex-concave stochastic saddle point problems and present (in our opinion highly encouraging) results of numerical experiments.

Stochastic Joint Optimization of Wind Generation and Pumped-Storage Units in an Electricity Market

Abstract: One of the main characteristics of wind power is the inherent variability and unpredictability of the generation source, even in the short-term. To cope with this drawback, hydro pumped-storage units have been proposed in the literature as a good complement to wind generation due to their ability to manage positive and negative energy imbalances over time. This paper investigates the combined optimization of a wind farm and a pumped-storage facility from the point of view of a generation company in a market environment. The optimization model is formulated as a two-stage stochastic programming problem with two random parameters: market prices and wind generation. The optimal bids for the day-ahead spot market are the ldquohere and nowrdquo decisions while the optimal operation of the facilities are the recourse variables. A joint configuration is modeled and compared with an uncoordinated operation. A realistic example case is presented where the developed models are tested with satisfactory results.

Modeling and Control of a Power-Split Hybrid Vehicle

Abstract: Toyota hybrid system (THS) is used in the current best selling hybrid vehicle on the market-the Toyota Prius. This hybrid system contains a power-split planetary gear system which combines the benefits of series and parallel hybrid vehicles. In this paper, we developed a dynamic model of the THS powertrain and then apply it for model-based control development. Two control algorithms are introduced: one based on the stochastic dynamic programming method, and the other based on the equivalent consumption minimization strategy. Both approaches determine the engine power based on the overall vehicle efficiency and apply the electrical machines to optimize the engine operation. The performance of these two algorithms is assessed by comparing against the dynamic programming results, which are non-causal but provide theoretical benchmarks for other implementable control algorithms.

A Sample Approximation Approach for Optimization with Probabilistic Constraints

Abstract: We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.

Pareto-Based Multiobjective Machine Learning: An Overview and Case Studies

Abstract: Machine learning is inherently a multiobjective task. Traditionally, however, either only one of the objectives is adopted as the cost function or multiple objectives are aggregated to a scalar cost function. This can be mainly attributed to the fact that most conventional learning algorithms can only deal with a scalar cost function. Over the last decade, efforts on solving machine learning problems using the Pareto-based multiobjective optimization methodology have gained increasing impetus, particularly due to the great success of multiobjective optimization using evolutionary algorithms and other population-based stochastic search methods. It has been shown that Pareto-based multiobjective learning approaches are more powerful compared to learning algorithms with a scalar cost function in addressing various topics of machine learning, such as clustering, feature selection, improvement of generalization ability, knowledge extraction, and ensemble generation. One common benefit of the different multiobjective learning approaches is that a deeper insight into the learning problem can be gained by analyzing the Pareto front composed of multiple Pareto-optimal solutions. This paper provides an overview of the existing research on multiobjective machine learning, focusing on supervised learning. In addition, a number of case studies are provided to illustrate the major benefits of the Pareto-based approach to machine learning, e.g., how to identify interpretable models and models that can generalize on unseen data from the obtained Pareto-optimal solutions. Three approaches to Pareto-based multiobjective ensemble generation are compared and discussed in detail. Finally, potentially interesting topics in multiobjective machine learning are suggested.

A Linear Decision-Based Approximation Approach to Stochastic Programming

Abstract: Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse.

Cost of Reliability Analysis Based on Stochastic Unit Commitment

Abstract: This paper presents a model for calculating the cost of power system reliability based on the stochastic optimization of long-term security-constrained unit commitment. Random outages of generating units and transmission lines as well as load forecasting inaccuracy are modeled as scenario trees in the Monte Carlo simulation. Unlike previous reliability analyses methods in the literature which considered the solution of an economic dispatch problem, this model solves an hourly unit commitment problem, which incorporates spatial constraints of generating units and transmission lines, random component outages, and load forecast uncertainty into the reliability problem. The classical methods considered predefined reserve constraints in the deterministic solution of unit commitment. However, this study considers possible uncertainties when calculating the optimal reserve in the unit commitment solution as a tradeoff between minimizing operating costs and satisfying power system reliability requirements. Loss-of-load-expectation (LOLE) is included as a constraint in the stochastic unit commitment for calculating the cost of supplying the reserve. The proposed model can be used by a vertically integrated utility or an ISO. In the first case, the utility considers the impact of long-term fuel and emission scheduling on power system reliability studies. In the second case, fuel and emission constraints of individual generating companies are submitted as energy constraints when solving the ISO's reliability problem. Numerical simulations indicate the effectiveness of the proposed approach for minimizing the cost of reliability in stochastic power systems.

Learning by mirror averaging

Abstract: Given a finite collection of estimators or classifiers, we study the problem of model selection type aggregation, that is, we construct a new estimator or classifier, called aggregate, which is nearly as good as the best among them with respect to a given risk criterion. We define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original nonlinear one and constitutes a special case of the mirror averaging algorithm. We show that the aggregate satisfies sharp oracle inequalities under some general assumptions. The results are applied to several problems including regression, classification and density estimation.

Relaxations of Weakly Coupled Stochastic Dynamic Programs

Abstract: We consider a broad class of stochastic dynamic programming problems that are amenable to relaxation via decomposition. These problems comprise multiple subproblems that are independent of each other except for a collection of coupling constraints on the action space. We fit an additively separable value function approximation using two techniques, namely, Lagrangian relaxation and the linear programming (LP) approach to approximate dynamic programming. We prove various results comparing the relaxations to each other and to the optimal problem value. We also provide a column generation algorithm for solving the LP-based relaxation to any desired optimality tolerance, and we report on numerical experiments on bandit-like problems. Our results provide insight into the complexity versus quality trade-off when choosing which of these relaxations to implement.

Stochastic Approximation Approaches to the Stochastic Variational Inequality Problem

Abstract: Stochastic approximation methods have been extensively studied in the literature for solving systems of stochastic equations and stochastic optimization problems where function values and first order derivatives are not observable but can be approximated through simulation. In this paper, we investigate stochastic approximation methods for solving stochastic variational inequality problems (SVIP) where the underlying functions are the expected value of stochastic functions. Two types of methods are proposed: stochastic approximation methods based on projections and stochastic approximation methods based on reformulations of SVIP. Global convergence results of the proposed methods are obtained under appropriate conditions.

Block-diagonal preconditioning for spectral stochastic finite-element systems

Abstract: Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

Optimal Involvement in Futures Markets of a Power Producer

Abstract: This paper addresses the optimal involvement in a futures electricity market of a power producer to hedge against the risk of pool price volatility. The considered trading horizon spans one whole year. Recognizing the highly uncertain nature of future pool prices, a stochastic programming framework with recourse is used to model this decision-making problem. The resulting problem is a large scale mixed-integer linear programming problem. Scenario reduction techniques are used to make this problem tractable. Risk is properly modeled using the CVaR methodology. Results from a realistic case study are provided and analyzed. Some conclusions are finally drawn.

Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation

Abstract: In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both–the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush–Kuhn–Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results.

Stability of Solutions for Stochastic Impulsive Systems via Comparison Approach

Abstract: This note studies stability problem of solutions for stochastic impulsive systems. By employing Lyapunov-like function method and It's formula, comparison principles of existence and uniqueness and stability of solutions for stochastic impulsive systems are established. Based on these comparison principles, the stability properties of stochastic impulsive systems are derived by the corresponding stability properties of a deterministic impulsive system. As the application, the stability results are used to design impulsive control for the stabilization of unstable stochastic systems. Finally, one example is given to illustrate the obtained results.

Solving Chance-Constrained Stochastic Programs via Sampling and Integer Programming

Abstract: Various applications in reliability and risk management give rise to optimization prob- lems with constraints involving random parameters, which are required to be satisfied with a prespecified probability threshold. There are two main difficulties with such chance-constrained problems. First, checking feasibility of a given candidate solution exactly is, in general, impossible because this requires evaluating quantiles of random functions. Second, the feasible region induced by chance constraints is, in general, nonconvex, leading to severe optimization challenges. In this tutorial, we discuss an approach based on solving approximating problems using Monte Carlo samples of the random data. This scheme can be used to yield both feasible solutions and statistical optimality bounds with high confidence using modest sample sizes. The approximat- ing problem is itself a chance-constrained problem, albeit with a finite distribution of modest support, and is an NP-hard combinatorial optimization problem. We adopt integer-programming-based methods for its solution. In particular, we discuss a fam- ily valid inequalities for a integer programming formulations for a special but large class of chance-constrained problems that have demonstrated significant computa- tional advantages.

The Innovest Austrian Pension Fund Financial Planning Model InnoALM

Abstract: This paper describes the financial planning model InnoALM we developed at Innovest for the Austrian pension fund of the electronics firm Siemens. The model uses a multiperiod stochastic linear programming framework with a flexible number of time periods of varying length. Uncertainty is modeled using multiperiod discrete probability scenarios for random return and other model parameters. The correlations across asset classes, of bonds, stocks, cash, and other financial instruments, are state dependent using multiple correlation matrices that correspond to differing market conditions. This feature allows InnoALM to anticipate and react to severe as well as normal market conditions. Austrian pension law and policy considerations can be modeled as constraints in the optimization. The concave risk-averse preference function is to maximize the expected present value of terminal wealth at the specified horizon net of expected discounted convex (piecewise-linear) penalty costs for wealth and benchmark targets in each decision period. InnoALM has a user interface that provides visualization of key model outputs, the effect of input changes, growing pension benefits from increased deterministic wealth target violations, stochastic benchmark targets, security reserves, policy changes, etc. The solution process using the IBM OSL stochastic programming code is fast enough to generate virtually online decisions and results and allows for easy interaction of the user with the model to improve pension fund performance. The model has been used since 2000 for Siemens Austria, Siemens worldwide, and to evaluate possible pension fund regulation changes in Austria.

Assessing marginal water values in multipurpose multireservoir systems via stochastic programming

Abstract: The International Conference on Water and the Environment held in Dublin in 1992 emphasized the need to consider water as an economic good. Since water markets are usually absent or ineffective, the value of water cannot be directly derived from market activities but must rather be assessed through shadow prices. Economists have developed various valuation techniques to determine the economic value of water, especially to handle allocation issues involving environmental water uses. Most of the nonmarket valuation studies reported in the literature focus on long-run policy problems, such as permanent (re) allocations of water, and assume that the water availability is given. When dealing with short-run allocation problems, water managers are facing complex spatial and temporal trade-offs and must therefore be able to track site and time changes in water values across different hydrologic conditions, especially in arid and semiarid areas where the availability of water is a limiting and stochastic factor. This paper presents a stochastic programming approach for assessing the statistical distribution of marginal water values in multipurpose multireservoir systems where hydropower generation and irrigation crop production are the main economic activities depending on water. In the absence of a water market, the Lagrange multipliers correspond to shadow prices, and the marginal water values are the Lagrange multipliers associated with the mass balance equations of the reservoirs. The methodology is illustrated with a cascade of hydroelectric-irrigation reservoirs in the Euphrates river basin in Turkey and Syria.

Combining (Integer) Linear Programming Techniques and Metaheuristics for Combinatorial Optimization

Abstract: Summary. Several different ways exist for approaching hard optimization problems. Mathematical programming techniques, including (integer) linear programming based methods, and metaheuristic approaches are two highly successful streams for combinatorial problems. These two have been established by different communities more or less in isolation from each other. Only over the last years a larger number of researchers recognized the advantages and huge potentials of building hybrids of mathematical programming methods and metaheuristics. In fact, many problems can be practically solved much better by exploiting synergies between these different approaches than by “pure” traditional algorithms. The crucial issue is how mathematical programming methods and metaheuristics should be combined for achieving those benefits. Many approaches have been proposed in the last few years. After giving a brief introduction to the basics of integer linear programming, this chapter surveys existing techniques for such combinations and classifies them into ten methodological categories.