Standard Linear Programming Problem Research Articles

A fuzzy linear programming model with fuzzy parameters and decision variables

Linear programming (LP) is an optimisation technique most widely used for optimal allocation of limited resources amongst competing activities. Precise data are fundamentally indispensable in standard LP problems. However, the observed values of the data in real-world problems are often imprecise or vague. Fuzzy set theory has been extensively used to represent ambiguous, uncertain or imprecise data in LP by formalising the inaccuracies inherent in human decision-making. We propose a new method for solving fuzzy LP (FLP) problems in which the right-hand side parameters and the decision variables are represented by fuzzy numbers. A new fuzzy ranking model and a new supplementary variable are utilised in the proposed FLP method to obtain the fuzzy and crisp optimal solutions by solving one LP model. Moreover, we introduce an alternative model with deterministic variables and parameters derived from the proposed FLP model. Interestingly, the result of the alternative model is identical to the crisp solution of the proposed FLP model. We use a numerical example from the FLP literature for comparison purposes and to demonstrate the applicability of the proposed method and exhibit the efficacy of the procedure.

Formula omitted]-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach

This paper is concerned with the l1-gain performance analysis and positive filter design for a positive discrete-time Markov jump linear system (MJLS) by using a linear programming (LP) approach. First, by constructing a linear stochastic Lyapunov function and introducing an “equivalent” deterministic positive discrete-time linear system, necessary and sufficient conditions in the form of LP are derived for stochastic stability and l1-gain performance of the positive discrete-time MJLS. Then a sufficient condition on the existence of the desired positive l1-gain filter is provided. The desired positive l1-gain filter can be designed by solving a standard LP problem. The pest’s structured population dynamic model is employed to illustrate the effectiveness of the proposed method.

Computational investigation of the feasibility of factor price equalization

We propose an empirical procedure for verifying the factor price equalization (FPE) condition. The FPE condition defined by Deardorff (J Int Econ 36:167–175, 1994) is converted to a standard linear programming problem. Although the procedure does not provide a graphical representation, it can be applied for any number of primary factors of production. In addition, based on the solution vector of the linear programming problem, we can find what quantities of the primary factors of production should be reallocated between which regions to satisfy the FPE condition when the condition does not hold.

Multi-choice linear programming for matrix game

The aim of this paper is to develop a multi-choice model for matrix game problem where a payoff matrix is multi-choice in nature. Then it transfer to a standard mathematical programming problem such a way of alternatives, out of which one is to be selected. The selection of alternatives should be the combination of choices provides an optimal solution to the matrix game. There may be more than one combination which will provide an optimal solution. However, the problem cannot be solved by standard linear programming problem (LPP) techniques. This paper proposed a technique to formulate mixed-integer programming model. Using the standard soft ware, the proposed model can be solved. Finally numerical example is presented to illustrate the proposed model and solution procedure.

Multiple object tracking using the shortest path faster association algorithm.

To solve the persistently multiple object tracking in cluttered environments, this paper presents a novel tracking association approach based on the shortest path faster algorithm. First, the multiple object tracking is formulated as an integer programming problem of the flow network. Then we relax the integer programming to a standard linear programming problem. Therefore, the global optimum can be quickly obtained using the shortest path faster algorithm. The proposed method avoids the difficulties of integer programming, and it has a lower worst-case complexity than competing methods but better robustness and tracking accuracy in complex environments. Simulation results show that the proposed algorithm takes less time than other state-of-the-art methods and can operate in real time.

Reformulation and Solution Algorithms for Absolute and Percentile Robust Shortest Path Problems

To model a driver's route choice behavior under inherent traffic system stochasticity and to further provide better route guidance with travel-time reliability guarantees, this paper examines two models to evaluate the travel-time robustness: absolute robust shortest path (ARSP) and α -percentile robust shortest path (PRSP) problems. A Lagrangian relaxation approach and a scenario-based representation scheme are integrated to reformulate the minimax and percentile criteria under day-dependent random travel times. The complex problem structure is decomposed into several subproblems that can be efficiently solved as standard shortest path problems or univariate linear programming problems. Large-scale numerical experiments with real-world data are provided to demonstrate the efficiency of the proposed algorithms.

The LP-Newton method for standard form linear programming problems

Fujishige et al. propose the LP-Newton method, a new algorithm for solving linear programming problems (LPs). They address LPs which have a lower and an upper bound for each variable. They reformulate the problem by introducing a related zonotope. Their algorithm solves the problem by repeating projections to the zonotope.In this paper, we develop the LP-Newton method for solving standard form LPs. We recast the LP by introducing a related convex cone. Our algorithm solves the problem by iterating projections to the convex cone.

Sparse Representation for Target Detection in Hyperspectral Imagery

In this paper, we propose a new sparsity-based algorithm for automatic target detection in hyperspectral imagery (HSI). This algorithm is based on the concept that a pixel in HSI lies in a low-dimensional subspace and thus can be represented as a sparse linear combination of the training samples. The sparse representation (a sparse vector corresponding to the linear combination of a few selected training samples) of a test sample can be recovered by solving an l0-norm minimization problem. With the recent development of the compressed sensing theory, such minimization problem can be recast as a standard linear programming problem or efficiently approximated by greedy pursuit algorithms. Once the sparse vector is obtained, the class of the test sample can be determined by the characteristics of the sparse vector on reconstruction. In addition to the constraints on sparsity and reconstruction accuracy, we also exploit the fact that in HSI the neighboring pixels have a similar spectral characteristic (smoothness). In our proposed algorithm, a smoothness constraint is also imposed by forcing the vector Laplacian at each reconstructed pixel to be minimum all the time within the minimization process. The proposed sparsity-based algorithm is applied to several hyperspectral imagery to detect targets of interest. Simulation results show that our algorithm outperforms the classical hyperspectral target detection algorithms, such as the popular spectral matched filters, matched subspace detectors, adaptive subspace detectors, as well as binary classifiers such as support vector machines.

Understanding linear semi-infinite programming via linear programming over cones

In this article, the standard primal and dual linear semi-infinite programming (DLSIP) problems are reformulated as linear programming (LP) problems over cones. Therefore, the dual formulation via the minimal cone approach, which results in zero duality gap for the primal–dual pair for LP problems over cones, can be applied to linear semi-infinite programming (LSIP) problems. Results on the geometry of the set of the feasible solutions for the primal LSIP problem and the optimality criteria for the DLSIP problem are also discussed.

Modification of Karmarkar’s projective scaling algorithm

This paper presents a new conversion technique of the standard linear programming problem into a homogenous form desired for the Karmarkar’s algorithm, where we employed the primal–dual method. The new converted linear programming problem provides initial basic feasible solution, simplex structure, and homogenous matrix. Apart from the transformation, Hooker’s method of projected direction is employed in the Karmarkar’s algorithm and the modified algorithm is presented. The modified algorithm has a faster convergence with a suitable choice of step size.

A New Methodology for the General Multiparametric Mixed-Integer Linear Programming (MILP) Problems

In this paper, a general algorithm is developed to address the multiparametric mixed-integer linear programming (mpMILP) problem with uncertain parameters in the left-hand side (LHS), right-hand side (RHS), and objective function coefficients simultaneously. The algorithm is based on a general multiparametric linear programming (mpLP) algorithm, which is derived using the optimality conditions of standard linear programming (LP) problem. The intent of the proposed framework is to propose a general framework to address different uncertainties in the process engineering problems. In addition, the paper also discusses the solution of the special problem where LHS uncertainty is included in the optimization model as a coefficient of continuous variable, and it notes the high computational complexity needed to retrieve the rigorous solution of large-scale problems, because of the nonlinearities of the objective function and nonconvex critical regions. Several numerical examples are presented to illustrate the effectiveness and applicability of the proposed method.

An optimal design of collateralized mortgage obligation with PAC-companion structure using dynamic cash reserve

This paper presents a model for optimally designing a collateralized mortgage obligation (CMO) with a planned amortization class (PAC)-companion structure using dynamic cash reserve. In this structure, the mortgage pool’s cash flow is allocated by rule to the two bond classes such that PAC bondholders receive substantial prepayment protection, that protection being provided by the companion bondholders. The structure we propose provides greater protection to the PAC bondholders than current structures during periods of rising interest rates when this class of bondholders faces greater extension risk. We do so by allowing a portion of the cash flow from the collateral to be reserved to meet the PAC’s scheduled cash flow in subsequent periods. The greater protection is provided by the companion bondholders exposure to interest loss. To tackle this problem, we transform the problem of designing the optimal PAC-companion structure into a standard stochastic linear programming problem which can be solved efficiently. Moreover, we present an extended model by considering the quality of the companion bond and by relaxing the PAC bondholder shortfall constraint. Based on numerical experiments through Monte Carlo simulation, we show the utility of the proposed model.

Steepest Descent Day-to-Day Dynamic Toll

Day-to-day dynamic congestion pricing schemes have been recently proposed to force the traffic system to evolve from the status quo to a stationary state of system optimum instead of user equilibrium, considering drivers’ day-to-day behavior adjustments. From the perspective of traffic management, it may be desirable to expedite the evolution process such that the total travel cost across the process can be reduced. A novel steepest descent dynamic toll scheme is proposed that minimizes the derivative of the total system cost with regard to day t or reduces the total system cost the most for each day. The problem of finding the steepest descent scheme is first formulated as a piecewise linear nonsmooth optimization problem and then transformed into a standard linear programming problem. Its mathematical properties are discussed further and a solution procedure is proposed for specifying the steepest descent pricing scheme. A numerical study of the well-known Braess network and the Sioux Falls, South Dakota, network is conducted to compare the performance of different dynamic pricing schemes.

Knowledge-based semidefinite linear programming classifiers

In this paper, we present knowledge-based support vector machine (SVM) classifiers using semidefinite linear programming. SVMs are an optimization-based solution method for large-scale data classification problems. Knowledge-based SVM classifiers, where prior knowledge is in the form of ellipsoidal constraints, result in a semidefinite linear programme with a set containment constraint. These problems are reformulated as standard semidefinite linear programming problems by the application of a dual characterization of the set containment under a mild regularity condition. The reformulated semidefinite linear programme is solved by the publicly available solvers. Computational results show that prior knowledge can often improve correctness of the classifier.

SOLVING FUZZY GOAL PROGRAMMING PROBLEMS

In standard goal programming (GP) it is assumed that the decision maker (DM) is able to accurately determine goal values. This is unrealistic. Imprecise DM aspiration can be expressed through fuzzy sets whose membership functions represent the DM´s degree of satisfaction. When membership functi-ons are nonlinear, the model becomes a nonlinear program that may be difficult to solve. Due to imprecise context normally the DM is not able to accurately determine the fuzzy goals membership functions. We show that few changes in membership functions produce small differences on the DM´s global satisfaction degree and on the efficient frontier. Based on these results, in this paper we present a procedure to approach the nonlinear fuzzy membership functions, whatever its shape might be, through piecewise linear functions which supply an standard goal linear programming problem that, obviously, is easier to solve than the nonlinear original one. An illustrative example is also provided

Minimax design of stable IIR digital filter with prescribed magnitude and phase responses

In this work, an iterative linear programming approach is presented to design stable IIR digital filters with prescribed magnitude and phase responses. At each iteration, the complex error of the frequency response is transformed into a linear form by treating the denominator polynomial obtained from the preceding iteration as a part of the weighting function, and the poles restricted inside the unit circle by using a set of linear constraints. After solving the standard linear programming problem at each iteration, the design algorithm converges to the minimax solution. Design examples demonstrate that our method provides better design results than the conventional linear programming method.

Symmetry Approach to Extension of Flutter Boundaries via Mistuning

A general framework is presented for analyzing and optimizing stability increases resulting from mistuning. The framework given is model independent and is based primarily on symmetry arguments. DifŽ cult practical issues are transformed to tractable mathematical questions. It is shown that mistuning analysis reduces to a block circular matrix eigenvalue/vector problem that can be solved efŽ ciently even for large problems. Similarly, the optimization becomes a standard linear constraint quadratic programming problem and can be solved numerically. Because the methods given are model-independent, they can be applied to various models and allow the researcher to easily conclude which models accurately capture mistuning and which do not. A simple quasisteady model for  utter in a cascade is used to illustrate and validate results in this paper.

Convex Hulls of f- and β-Vectors

In this paper we describe the convex hulls of the sets of f- and β-vectors of different classes of simplicial complexes on n vertices. These include flag complexes, order complexes of posets, matroid complexes, and general abstract simplicial complexes. As a result of this investigation, standard linear programming problems on these sets can be solved, including maximization of the Euler characteristics or of the sum of the Betti numbers.

The most-obtuse-angle row pivot rule for achieving dual feasibility: A computational study

We recently proposed several new pivot rules for achieving dual feasibility in linear programming, which are distinct from existing ones: the objective function value will no longer change necessarily monotonically in their solution process. In this paper, we report our further computational testing with one of them, the most-obtuse-angle rule. A two-phase dual simplex algorithm, in which the rule is used as a row selection rule for Phase- 1, has been implemented in FORTRAN 77 modules, and was tested on a set of standard linear programming problems from NETLIB. We show that, if full pricing is applied, our code unambiguously will outperform MINOS 5.3, one of the best implementations of the simplex algorithm at present.

An Infeasible-Interior-Point Predictor-Corrector Algorithm for Linear Programming

A predictor-corrector method is proposed for solving standard form linear programming problems starting from initial points that strictly satisfy the positivity constraints, but do not necessarily satisfy the equality constraints. The algorithm is globally convergent under the assumption that the linear program has an optimal solution. Under some additional assumptions on the starting point we prove that $\epsilon $-feasibility and $\epsilon $-complementarily can be obtained in $O( n \ln ( \frac{1}{\epsilon } )) $ iterations.