Time-varying linear programming via LVI-PDNN with numerical examples

Cloud computing plays a vital role in today's IT industry, as it offers tremendous computing and storage facilities for the tasks and outsourced data respectively. The major security concern during the computations is to protect customer's confidential data. The mechanism of this paper decomposes widely applicable linear programming (LP) computation outsourcing into public LP solvers running on a cloud. Efficient privacy preserving problem transformation techniques allows customers to transform original LP into some random LP by keeping private LP parameters and protect input output information. This paper gives detailed mathematical analysis for original LP transformations and LP problem solving. It also proposes a method of adding a fake variable into linear program which will be transformed to random one so that it will be more difficult for an attacker to guess the original problem. The results obtained from the experimentation prove to be efficient and incurs close to zero additional cost.

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. Kheirfam and Hasani (Sensitivity analysis for fuzzy linear programming problems with fuzzy variables, Advanced Modeling and Optimization, 12 (2010) 257-272), proposed a method for the sensitivity analysis for fuzzy linear programming (FLP) problems with fuzzy variables. There are some important cases that are not considered by Kheirfam and Hasani. In this paper, those cases are considered and numerical examples are solved.

In this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

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.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

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

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

Fully intuitionistic fuzzy multi-level linear fractional programming problem

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The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

A projection and division method for problems of linear dynamic programming

ABSTRACTIn this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.

As wireless networks are gaining increasing popularity, the network energy efficiency has become a critical issue. In this paper, we focus on energy-efficient networking in a generic multi-radio multi-channel (MR-MC) wireless network where transmission scheduling, transmit power control, radio and channel assignment are coupled together in a multi-dimensional resource space, thus requiring joint optimization and low complexity algorithms. We propose a novel Decomposed Approach For energy-efficient (DAFEE) networking in MR-MC networks, with the objective to minimize network energy consumption while guaranteeing a certain level of performance. In particular, we leverage a multi-dimensional tuple-link based model and a concept of resource allocation pattern to transform the complex optimization problem into a linear programming (LP) problem. The LP problem however has a very large solution space due to the exponentially many possible resource allocation patterns. We then exploit delay column generation and distributed learning techniques to decompose the problem and solve it with an iterative process. Furthermore, we propose a sub-optimal algorithm to speed up the iteration with constant-bounded performance. Simulation results are presented to demonstrate the effectiveness of the proposed algorithm.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.