Stochastic Decomposition Algorithm Research Articles

Stochastic Decomposition Method for Two-Stage Distributionally Robust Linear Optimization

In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen [Math. Oper. Res., 16 (1991), pp. 650--669] for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm include (1) it works with data-driven approximations of ambiguity sets that are constructed using samples of increasing size and (2) efficient construction of approximations of the worst-case expectation function that solves only two second-stage subproblems in every iteration. We identify conditions under which the ambiguity set approximations converge to the true ambiguity sets and show that the DRSD method asymptotically identifies an optimal solution, with probability one. We also computationally evaluate the performance of the DRSD method for solving distributionally robust versions of instances considered in stochastic programming literature. The numerical results corroborate the analytical behavior of the DRSD method and illustrate the computational advantage over an external sampling-based decomposition approach (distributionally robust L-shaped method).

Artificial Noise-Aided Secure Relay Communication With Unknown Channel Knowledge of Eavesdropper

In this article, a new relay-aided secure communication system is investigated, where a transmitter sends signals to a destination via an amplify-and-forward (AF) relay in the presence of an eavesdropper. We consider a general system configuration, where the source, relay, destination, and eavesdropper are all equipped with multiple antennas. In the practical scenarios of unknown eavesdropper's channel state information (CSI) and uncertainty of the eavesdropper's location, we aim to maximize the expected value of the system secrecy rate over the presumed distribution of the eavesdropper's channels, by exploiting the artificial noise (AN) transmitted by the source and relay nodes. The system design issue is formulated as a nonconvex stochastic optimization problem with a source transmission power constraint and a nonconvex relay transmission power constraint. A novel computational method is proposed to solve this challenging problem. The new method is developed based on an exact penalty function method together with a parallel stochastic decomposition algorithm. Numerical simulations are performed to study the effectiveness of the proposed scheme at various locations of the eavesdropper. Simulation results show that for most cases, secure communication can be achieved without the CSI knowledge of eavesdropper's channels, and the achievable secrecy rate follows the trend of a benchmark system where the eavesdropper's full CSI is available. In particular, the achievable system secrecy rate increases with the number of antennas at the legitimate users. Moreover, the optimal power allocated for the transmission of the AN increases with the system signal-to-noise ratio. The proposed computational method achieves a higher system secrecy rate than a conventional penalty function based approach.

Asymptotic Results of Stochastic Decomposition for Two-Stage Stochastic Quadratic Programming

This paper presents the stochastic decomposition (SD) algorithms for two classes of stochastic programming problems: (1) two-stage stochastic quadratic-linear programming (SQLP) in which a quadrati...

A hybrid robust stochastic programming for a bi-objective blood collection facilities problem (Case study: Iranian blood transfusion network)

ABSTRACTIn this paper, a generalized network optimization model is developed for a complex blood supply chain including a regionalized blood bank system. This system consist of collection sites, testing and processing facilities, storage facilities, distribution centers, as well as points of demand (hospitals). To keep the network in contradiction of the uncertainty, a consolidated approach based on a recently developed stochastic robust approach is extended. An accelerated stochastic Benders decomposition algorithm is proposed to solve the problem modeled in this paper. To speed up the convergence of the solution algorithm, valid inequalities are introduced to get better quality lower bounds. Numerical illustrations are given to verify the mathematical formulation and also to show the benefits of using the stochastic robust approach. At the end, the performance improvements attained by the valid inequalities and the Pareto-optimal cuts are demonstrated in a real-world application.

Hierarchical multiple-image encryption based on the cascaded interference structure and vector stochastic decomposition algorithm

A new kind of hierarchical multiple-image encryption method based on the cascaded interference structure and vector stochastic decomposition algorithm is proposed. In this method, by using the nonequal modulus decomposition and vector stochastic decomposition algorithm, the Kth-level secret image modulated by a random phase distribution is analytically encoded into a Kth-level phase-only mask (POM) key and the (K−1)th-level complex amplitude field whose real amplitude is the (K−1)th-level secret image. Then, served as the input condition, the generated complex amplitude field is further encoded into a (K−1)th-level POM key and the (K−2)th-level complex amplitude, and so on, until the 1st-level complex amplitude field is decomposed into two POMs: one is the 1st-level key, and the other is the ciphertext. When decryption occurs, only when the high-level users simultaneously obtain all the phase keys, the correct sequential orders of phase keys, and the correct geometrical parameters, all the secret images are retrieved successfully. Both theoretical analysis and numerical simulations verify the feasibility of this method.

Three-dimensional scene encryption and display based on computer-generated holograms.

An optical encryption and display method for a three-dimensional (3D) scene is proposed based on computer-generated holograms (CGHs) using a single phase-only spatial light modulator. The 3D scene is encoded as one complex Fourier CGH. The Fourier CGH is then decomposed into two phase-only CGHs with random distributions by the vector stochastic decomposition algorithm. Two CGHs are interleaved as one final phase-only CGH for optical encryption and reconstruction. The proposed method can support high-level nonlinear optical 3D scene security and complex amplitude modulation of the optical field. The exclusive phase key offers strong resistances of decryption attacks. Experimental results demonstrate the validity of the novel method.

A Two-Stage Stochastic Dynamic Economic Dispatch Model Considering Wind Uncertainty

This paper proposes a two-stage stochastic dynamic economic dispatch model aiming at better managing system variability and uncertainty influenced by wind generation. Stochastic decomposition algorithm is proposed to solve the model in order to facilitate real-time application. The proposed method is tested on PJM-5 and RTS-24 systems and the results verify that the model inherits advantages from conventional dynamic economic dispatch, which is able to make out of merit order dispatch instructions for future benefits. Furthermore, the proposed model can correct false generator pre-ramp instructions due to forecast error and evaluate potential nodal wise reserve requirement therefore improving reserve deliverability. For computational efficiency assessment, stochastic decomposition is compared with sample average approximation. A modified IEEE-118 bus case study is shown.

Multistage Stochastic Decomposition: A Bridge between Stochastic Programming and Approximate Dynamic Programming

Multistate stochastic programs pose some of the more challenging optimization problems. Because such models can become rather intractable in general, it is important to design algorithms that can provide approximations which, in the long run, yield solutions that are arbitrarily close to an optimum. In this paper, we propose such a sequential sampling method which is applicable to multistage stochastic linear programs, and we refer to it as the multistage stochastic decomposition (MSD) algorithm. This algorithm represents a dynamic extension of a regularized version of stochastic decomposition (SD). While the method allows general correlation structures, specialized streamlined versions are also possible for special cases of stagewise independent and autoregressive processes commonly incorporated in stochastic programming. As with its two-stage counterpart, the MSD algorithm is shown to provide an asymptotically optimal solution, with probability one. As a by-product of this study, we also show that SD al...

Strategic Capacity Allocation under Uncertainty by Using a Two-Stage Stochastic Decomposition Algorithm with Incumbent Solutions

This paper addresses the optimization of strategic capacity allocation problems under uncertainty. Uncertain parameters include the availabilities and the demands of chemicals. The optimization problem is formulated as a two-stage mixed-integer stochastic program (first-stage integer) that maximizes the expected net present value of the project over the time horizon. The net present value includes the discounted investment costs, operating costs, and revenues from sales in the markets. The decisions about the selection of processes and about the plant capacities are defined as the first stage decision variables; given such first stage decisions, recourse is then provided by calculating the operating costs in the second stage. The paper presents the two-stage model and the solution framework. A stochastic decomposition algorithm (SD) based on incumbent solution is used as the solution algorithm. The use of incumbent solutions as convergence criterion allows us to solve the problem in reasonable computational time. Two capacity allocation problems are used as case studies: the first one is a small size problem and is used to compare the stochastic optimal solution against other solution methods; the second one is a larger problem, with an increased number of variables and stochastic parameters. The results show that the stochastic decomposition algorithm with incumbent solutions is stable and robust and greatly reduces the number of samples required for convergence.

Stochastic scenario decomposition for multistage stochastic programs

Over the past decade, several stochastic approaches have been proposed for two-stage stochastic programs. Many of these algorithms have attractive computational as well as conceptual properties (e.g. convergence with probability one). This paper expands the realm of such approaches to multistage convex stochastic programming problems. We present a stochastic scenario decomposition (SSD) algorithm which is a statistically motivated cutting plane algorithm for the solution of multistage stochastic programs. The method is based on solving a dual problem in which the variables correspond to multipliers associated with the non-anticipativity constraints of the primal problem. Our analytical results verify conditions under which SSD identifies an optimal solution asymptotically. We also overcome some computational hurdles resulting from increases in the column dimension of the SSD master problem. We propose a variable aggregation scheme that allows us to solve much smaller master programs without sacrificing solution quality. Our computational results demonstrate the effectiveness of this aggregation scheme in solving the SSD master program.

Analysis of stochastic problem decomposition algorithms in computational grids

Stochastic programming usually represents uncertainty discretely by means of a scenario tree. This representation leads to an exponential growth of the size of stochastic mathematical problems when better accuracy is needed. Trying to solve the problem as a whole, considering all scenarios together, yields to huge memory requirements that surpass the capabilities of current computers. Thus, decomposition algorithms are employed to divide the problem into several smaller subproblems and to coordinate their solution in order to obtain the global optimum. This paper analyzes several decomposition strategies based on the classical Benders decomposition algorithm, and applies them in the emerging computational grid environments. Most decomposition algorithms are not able to take full advantage of all the computing power available in a grid system because of unavoidable dependencies inherent to the algorithms. However, a special decomposition method presented in this paper aims at reducing dependency among subproblems, to the point where all the subproblems can be sent simultaneously to the grid. All algorithms have been tested in a grid system, measuring execution times required to solve standard optimization problems and a real-size hydrothermal coordination problem. Numerical results are shown to confirm that this new method outperforms the classical ones when used in grid computing environments.

Enhancements of two-stage stochastic decomposition

This paper presents some enhancements associated with stochastic decomposition (SD). Specifically, we study two issues: (a) Are there any conditions under which the regularized version of SD generates a unique solution? (b) Is there a way to modify the SD algorithm so that a user can trade-off solution times with solution quality? The second issue addresses the scalability of SD for very large scale problems for which computational resources may be limited and the user may be willing to accept solutions that are “nearly optimal”. We show that by using bootstrapping (re-sampling) the regularized SD algorithm can be accelerated without significant loss of optimality. We report computational results that demonstrate the viability of this approach.

Two-Stage Stochastic Approach to the Optimal Location of Booster Disinfection Stations

Secondary or booster disinfection is the addition of disinfectant at distinct locations of a water network to achieve control of the microorganisms in drinking water. This work describes a simplified stochastic approach for the optimal location of booster disinfection stations in water distribution networks. The model minimizes the expected total cost involved in the installation of booster stations and the mass of disinfectant needed to satisfy the residual concentration constraints within the network. Inherent uncertainties in water quality simulations, such as water demand and the chemical reactions of the disinfectant that are occurring within the system, indirectly incorporate uncertainties on the model parameters. Hence, the problem has been formulated as a first-stage-integer two-stage stochastic mixed-integer linear program with recourse (SMILPwR). The parameters needed for the solution are obtained through the water quality simulator EPANET 2.0, and the resulting formulation has been solved through a generalized framework based on the stochastic decomposition algorithm. Our paper describes the model and the algorithmic framework, and it compares the deterministic and stochastic optimal solutions. Differences between the stochastic and deterministic objective functions as high as 112% show the significant impact of uncertainties in booster disinfection.

Water networks security: A two-stage mixed-integer stochastic program for sensor placement under uncertainty

This work describes a stochastic approach for the optimal placement of sensors in municipal water networks to detect maliciously injected contaminants. The model minimizes the expected fraction of the population at risk and the cost of the sensors. Our work explicitly includes uncertainties in the attack risk and population density, so that the resulting problem involves optimization under uncertainty. In our formulation, we include the location of a number of sensors as first stage decision variables of a two-stage mixed-integer stochastic linear problem; the second stage evaluates the population at risk for the scenario obtained in the first stage and that information is then used to modify the first stage decisions for the next iteration. Since the model is integer in the first stage, a generalized framework based on the stochastic decomposition algorithm allows us to solve the problem in a reasonable computational time. The paper describes the mixed-integer stochastic model and the algorithmic framework, and compares the deterministic and stochastic optimal solutions. The network used as our case study has been derived through the water network simulator EPANET 1.0; four acyclic water flow patterns are considered. Results show a significant effect of uncertainty in sensor placement and total cost.

Improving convergence of the stochastic decomposition algorithm by using an efficient sampling technique

This work focuses on the basic stochastic decomposition (SD) algorithm of Higle and Sen [J.L. Higle, S. Sen, Stochastic Decomposition, Kluwer Academic Publishers, 1996] for two-stage stochastic linear programming problems with complete recourse. The algorithm uses sampling when the random variables are represented by continuous distribution functions. Traditionally, this method has been applied by using Monte Carlo (MC) sampling to generate the samples of the stochastic variables. However, Monte Carlo methods can result in large error bounds and variance. Hence, some other approaches use importance sampling to reduce variance and achieving convergence faster that the method based on the Monte Carlo sampling technique. This work proposes an improvement on this respect. Hence, we propose to replace the use of the Monte Carlo sampling technique or the importance sampling in the SD algorithm by the use of the novel Hammersley sequence sampling (HSS) technique. Recently, such a technique has proved to provide better uniformity properties than other sampling techniques and, as a consequence, the variance and the number of samples required for convergence are reduced. Also, we use a fractal dimension approach to characterize the error of the estimation of the recourse function based on sampling. The computational implementation of the algorithm involves a framework that integrates the GAMS modeling environment, the HSS sampling code (FORTRAN) and a C++ program which generates appropriate LP problems for each SD iteration. The algorithm has been tested with several case studies representing chemical engineering applications and the results are discussed.

DESIGN OF ENGINEERING SYSTEMS USING STOCHASTIC DECOMPOSITION:WATER SUPPLY PLANNING APPLICATION

To effectively design engineering systems, the future operation of the system which usually involves many uncertainties must be considered. A two-stage stochastic programming formulation can aid in satisfying this requirement. The first stage of this formulation represents the design criteria at the present time when a decision must be made. The second stage represents the future operation or the system response to the design where other actions (recourse decisions) are to be made after observing the random input. To solve this type of problem, the Regularized Stochastic Decomposition (RSD) algorithm, which allows the consideration of continuous random variables, was employed and extensions to better handle real engineering problems were investigated. The algorithm is applied to a regional water supply problem that seeks the optimal design capacities of water treatment plants, secondary and tertiary wastewater treatment plants, and recharge facilities while meeting future demands. Results are generated based on different forms of uncertainties for both linear and nonlinear first-stage objective functions. The advantages of using stochastic programming in engineering decision making are evaluated.

A regularized stochastic decomposition algorithm for two-stage stochastic linear programs

In this paper a regularized stochastic decomposition algorithm with master programs of finite size is described for solving two-stage stochastic linear programming problems with recourse. In a deterministic setting cut dropping schemes in decomposition based algorithms have been used routinely. However, when only estimates of the objective function are available such schemes can only be properly justified if convergence results are not sacrificed. It is shown that almost surely every accumulation point in an identified subsequence of iterates produced by the algorithm, which includes a cut dropping scheme, is an optimal solution. The results are obtained by including a quadratic proximal term in the master program. In addition to the cut dropping scheme, other enhancements to the existing methodology are described. These include (i) a new updating rule for the retained cuts and (ii) an adaptive rule to determine when additional reestimation of the cut associated with the current solution is needed. The algorithm is tested on problems from the literature assuming both descrete and continuous random variables.

An exact penalty algorithm for recourse-constrained stochastic linear programs

This paper presents an algorithm for solving stochastic linear programs in which a recourse problem that appears in the constraint set. This algorithm, which is based on a stochastic decomposition (SD) algorithm of Higle and Sen, involves the use of an exact penalty function in the master program. Results from applications to several problems are reported.

Statistical verification of optimality conditions for stochastic programs with recourse

Statistically motivated algorithms for the solution of stochastic programming problems typically suffer from their inability to recognize optimality of a given solution algorithmically. Thus, the quality of solutions provided by such methods is difficult to ascertain. In this paper, we develop methods for verification of optimality conditions within the framework of Stochastic Decomposition (SD) algorithms for two stage linear programs with recourse. Consistent with the stochastic nature of an SD algorithm, we provide termination criteria that are based on statistical verification of traditional (deterministic) optimality conditions. We propose the use of “bootstrap methods” to confirm the satisfaction of generalized Kuhn-Tucker conditions and conditions based on Lagrange duality. These methods are illustrated in the context of a power generation planning model, and the results are encouraging.