Stochastic Programming with Cauchy Distribution

Many planning problems involve choosing a set of optimal decisions for a system in the face of uncertainty of elements that may play a central role in the way the system is analyzed and operated. During the past decade, there has been a renewed interest in the modelling, analysis, and solution of such problems due to a remarkable development of both new theoretical results and novel computational techniques in stochastic optimization. A prominent approach is to develop upper and lower bounding approximations to the problem along with procedures to sharpen bounds until an acceptable tolerance is satisfied. The contributions of this dissertation are concerned with the latter approach. The thesis first studies the stochastic linear programming problem with randomness in both the objective coefficients and the constraints. A convex concave saddle property of the value function is utilized to derive new bounding techniques which generalize previously known results. These approximations require discretizing bounded domains of the random variables in such a way that tight upper and lower bounds result. Such techniques will prove attractive with the recent advances in large scale linear programming. The above results are also extended to obtain new upper and lower bounds when the domains of random variables are unbounded. While these bounds are tight, the approximating models are large-scale deterministic linear programs. In particular, with a proposed order-cone decomposition for the domains, these linear programs are well-structured, thus enabling one to use efficient techniques for solution, such as parallel computation. The thesis next considers convex stochastic programs. Using aggregation concepts from the deterministic literature, new bounds are developed for the problem which are computable using standard convex programming algorithms. Finally, the discussion is focused on a stochastic convex program arising in a certain resource allocation problem. Exploiting the problem structure, bounds are developed via the Karush-Kuhn-Tucker conditions. Rather than discretizing domains, these approximations advocate replacing difficult multidimensional integrals by a series of simple univariate integrals. Such practice allows one to preserve differentiability properties so that smooth convex programming methods can be applied for solution.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

Although many varied techniques have been proposed for handling deterministic non-linear programming problems there apperars to have been little success in solving the more realistic problem of stochastic non-linear programming, despite the many results that have been obtained for stochastic linear programming. In this paper the stochastic non-linear problem is treated by means of an adaptation of a method used by Berkovitz [1] in obtaining an exiatence theorem for a type of inequality constrained variational problem involving one independent variable. The stochastic programming problem of course involves many independent variables. Necessary conditions are obtained for the existence of a solution of a fairly general type of non-linear problem, and these conditions are shown to be also sufficient for the convex problem. A duality theorem is given for the latter problem.

For a class of stochastic linear bilevel programming problem, we firstly transform it into a deterministic linear bilevel covariance programming problem. Then, the deterministic bilevel covariance programming problem is solved by backpropagation artificial neural network based on elite particle swam optimization algorithm (BPANN-PSO). Finally, we perform the simulation experiments and the results show that the computational efficiency of the proposed algorithm has a potential upside compared with the classical algorithm.

A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. This is the “wait and see” problem of stochastic linear programming. Explicit results for the distribution problem are extremely difficult to obtain; indeed, previous results are known only if the right hand side coefficients have an exponential distribution [1]. To date, no explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s. In this paper, we obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. A transformation is utilized that greatly reduces computational time.

We formulate and discuss a model of two-period stochastic optimization which allows for the possibility of the probability distribution over a non-empty finite set of states of nature-exactly one member of which will be realized in the future period- to depend on the control variable chosen in the initial period. In the second and terminal period, the decision maker chooses the value of a control variable vector which along with the control variable vector chosen in the initial period, solve a linear programming problem. The parameters of the linear programming are determined by an uncertain state of nature that is realized at the beginning of the terminal period. In order to obtain specific results, we assume that in the initial period, the decision maker can choose any one alternative from a finite set of alternatives, each such alternative being “a system of linear inequality constraints.” It is further assumed that solution sets of any two different systems are mutually disjoint and the probability distribution of the uncertain components of the state variables that is realized in the second period is determined by the system of linear inequalities chosen in the initial period. We refer to the overall problem as a “two-period stochastic linear programming game”. Our first result for a two-period stochastic linear programming game is that the optimal value for such a game as well as the optimal solutions for each of the finite number of stochastic linear programs can be found by solving a linear programming problem and our second result in this context says that the (optimal) solution for the dual of this linear programming contains a component that determines “at least one” stochastic linear programming problem any solution of which when substituted in its objective function, yields the optimal value of the two-period stochastic linear programming game. There is no need to solve any other subproblem or the primal problem to derive the optimal value of the game. AMS Classifications: 90C05, 90C15, 90C39, 90C46, 91A65 JEL Classifications: C61, C79, D81

Raw-materials base (hereinafter RMB) is one of the largest industries for financial investments in Russia. There are various mathematical descriptions for the development of regions with resource-based economy. Earlier in [1] the researchers considered the model based on bilevel integer stochastic programming problems with Boolean variables. This paper proposes a new approach to public-private partnership modelling, including a bilevel linear stochastic programming problem. This model assumes that budget constraints of the state and the investor can vary in a random manner with a specific probability distribution. We put forward two methods to solve this problem: problem’s reduction to the deterministic bilevel one and formulation of deterministic problems sequence with help of Monte Carlo methods. In order to solve deterministic problems of integer programming, we suggest two approaches: direct enumeration and heuristic “Game” approach. The numerical experiments for proposed algorithms validation are conducted on the basis of actual data of Zabaykalsky Krai development. Multiple input parameters of the model vary in these experiments. Finally, we present a brief analysis of the obtained solutions to the stochastic linear programming problems with Boolean variables.

A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values

As mentioned in chapter I the formulation of stochastic programming problems goes back to the mid 50s (Dantzig, Beale, Tintner). Since then a lot of research activities have resulted in developments of approaches for solving these problems. We intend next to focus on some of existing approaches and outline their main characteristica. The motivation to do this lies in the variety of stochastic two-stage programming problems entailing different and challenging features that have been exploited within the last decades. Some of the approaches discussed below have already been treated in Ermoliev and Wets 1988 [42], and even much more detailed therein than we will do here; however, the achieved progress within stochastic programming in the last five years have raised new ideas.

Multistage stochastic programming problems well correspond to many practical situations in which a random element exists and moreover it is reasonable to treat them with respect to some discrete time interval. In particular, this type of problems correspond to practical situations that can be considered with respect to some time interval and simultanously decomposed with respect to the individual time points. The aim of this paper is to investigate the objective functions corresponding to the individual problems belonging to the one multistage stochastic programming problem. A special attention is paid to the Lipschitz property.

Chapter Three - Introduction to Stochastic Programming

The self Help Group (SHG) is group of rural poor who have organized themselves into a group for eradicationof poverty. The members of the group belong to families below the poverty line. This will help the families of occupational groups like agricultural labourers, marginal farmers, designers and artisans marginally above the poverty line, or who may have been excluded from the Below Poverty Line (BPL) list to become members of the Self Help Group. A self help group consists of two categories. One named as magalier thittam and another is non- magalier thittam. The factors of Self help group categories are random in nature. These factors can be handled using stochastic linear programming problem (SLPP). Here the data is collected from Tuticorin district. The optimization technique such as two stage programming and chance constrained programming can be adopted for SLPP. In this paper chance constrained programming (CPP) is used to obtain optimal solution.

A novel microgrid support management system based on stochastic mixed-integer linear programming

Practical planning problems with deterministic forecasts of inherently uncertain parameters often yield unsatisfactory solutions. Stochastic programming formulations allow uncertain parameters to be modeled as random variables with known distributions, but the size of the resulting mathematical programs can be formidable. Decomposition-based algorithms take advantage of special structure and provide an attractive approach to such problems. We consider two classes of decomposition-based stochastic programming algorithms. The first type of algorithm addresses problems with a ``manageable`` number of scenarios. The second class incorporates Monte Carlo sampling within a decomposition algorithm. We develop and empirically study an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs within a prespecified tolerance. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of ``real-world`` multistage stochastic hydroelectric scheduling problems. Recently, there has been an increased focus on decomposition-based algorithms that use sampling within the optimization framework. These approaches hold much promise for solving stochastic programs with many scenarios. A critical component of such algorithms is a stopping criterion to ensure the quality of the solution. With this as motivation, we develop a stopping rule theory for algorithms in which bounds on the optimal objective function value are estimated by sampling. Rules are provided for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence interval statements for the quality of the proposed solution can be verified. Issues associated with the application of this theory to two sampling-based algorithms are considered, and preliminary empirical coverage results are presented.

This paper considers a two-level linear programming problem involving random variable coefficients to cope with hierarchical decision making problems under uncertainty. Two decision making models are provided to optimize the mean of the objective function value or to minimize the variance. It is shown that the original problem is transformed into a deterministic problem. The computational methods are constructed to obtain the Stackelberg solution to the two-level programming problems. An illustrative numerical example is provided to understand the geometrical properties of the solutions