Primal Linear Programming Research Articles

A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

Pointwise residual method for solving primal and dual ill-posed linear programming problems with approximate data

We propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved. doi:10.1017/S1446181120000243

POINTWISE RESIDUAL METHOD FOR SOLVING PRIMAL AND DUAL ILL-POSED LINEAR PROGRAMMING PROBLEMS WITH APPROXIMATE DATA

AbstractWe propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved.

Simultaneous Placements of TCSC and VSCDG for Congestion Management

Relieving congestion is a major technical issue in power transmission systems. In this paper, simultaneous placement of Thyristor controlled series capacitors (TCSC) and Voltage source converter based distribution generators (VSC-DG) are used for relieving congestion, minimize system cost and maintain the system in secure state. The system has been analyzed in three stages: (i) first stage, the congestions are made in different transmission network by overload and line outages, (ii) second stage, relieving congestion of transmission lines by using TCSC and VSCDG, where these devices are placed simultaneously at their optimal locations, (iii) comparative cost analysis is done between normal and congested systems. The proposed approach is tested on the 62-bus Indian power system network. The optimal solution of the test systems are obtained by genetic algorithm method and its solutions are compared with primal linear programming. All the results are validated through Power World Simulator and GA toolbox in MATLAB.

Development of a formal algorithm for the formulation of a dual linear optimization problem

The rigorous formal algorithm for formulating a dual problem for different forms (general, basic, standard, and canonical) of a primal linear programming problem is proposed. First, definitions of a pair of dual problems for standard form of primal linear programming are given. This approach is based on the fact that such a pair was noted first, since it had substantial interpretation.The economic interpretation of the standard problem is profit maximization in the production and sale of some types of products. Such an approach substantially indicates the existence of the primal problem (I) and the strictly corresponding dual (conjugate) (II). The problem of cost minimization is accompanying to the primal problem.The basic concept of the duality theory in linear programming problems is the fact that a pair of problems are mutually conjugate — obtaining dual of dual leads to a primal problem.The rigorous approach to obtaining an algorithm for formulating a dual problem is based on the statement that the dual problem of dual is a primal (original) problem. This approach is used in the paper. For different pairs of dual problems, this statement is rigorously proved.The existing schemes of primal to dual conversion are substantial. Given this, the algorithm of the general approach to formulating pairs of conjugate problems is proposed and rigorously proved.Formalization of the developed scheme makes it easy to get pairs of known dual problems. This allowed for the first time to propose and validate the algorithm for constructing a dual problem for an arbitrary form of the primal problem.

Data driven matrix uncertainty for robust linear programming

In this paper we consider robust linear programs with uncertainty sets defined by the convex hull of a finite number of m×n matrices. Embedded within the matrices are related robust linear programs defined by the rows, columns, and coefficients of the matrices. This results in a nested set of primal (and dual) linear programs with predictably different optimal objective values. The set of matrices also embed a covariance structure for the matrix coefficients and we show that when negative covariances predominate in the rows, more favorable optimal objective values for the primal can be expected.

Modal occupation measures and LMI relaxations for nonlinear switched systems control

This paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal linear programming (LP) problem. Its dual linear program of Hamilton–Jacobi–Bellman inequalities is also characterized. The LPs are then solved numerically with a converging hierarchy of primal–dual moment-sum-of-squares (SOS) linear matrix inequalities (LMI). Because of the special structure of switched systems, we obtain a much more efficient method than could be achieved by applying standard moment/SOS LMI hierarchies for general optimal control problems.

Economic-Mathematical Methods of Forming the Optimal Assortment Plan Trade Enterprises

The article provides a critical analysis of the existing economic and mathematical models, summarizes the maindeficiencies identified by various researchers and authors themselves, which allowed to recommend greater useof methods of linear programming. In a highly competitive consumer market, and most importantly, in thecontext of the dynamic expansion and deepening of product range, it is impossible to form the optimalassortment plans, focusing only on maximizing profits, it is also necessary to minimize distribution costs.According to the authors duality theory solves both of these problems. An algorithm for creating the optimalassortment plan commercial enterprise, based on the solution of the primal and dual linear programming problem.Given the economic interpretation of the possible reasons for which the task will not have a solution.

STEIN LINEAR PROGRAMS OVER SYMMETRIC CONES

In this paper, using Moore–Penrose inverse, we characterize the feasibility of primal and dual Stein linear programs over symmetric cones in a Euclidean Jordan algebra V. We give sufficient conditions for the solvability of the Stein linear programming problem. Further, we give a characterization of the globally uniquely solvable property for the Stein transformation in terms of a least element of a set in V in the context of the linear complementarity problem.

On the sufficiency of finite support duals in semi-infinite linear programming

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show that the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash (1987). This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all real-valued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.

On solving a class of linear semi-infinite programming by SDP method

In this paper, we present a new method to solve linear semi-infinite programming. This method bases on the fact that the nonnegative polynomial on could be turned into a positive semi-definite system, so we can use the nonnegative polynomials to approximate the semi-infinite constraint. Furthermore, we set up an approximate programming for the primal linear semi-infinite programming, and obtain an error bound between two programming problems. Numerical results show that our method is efficient.

Message-Passing Algorithms and Improved LP Decoding

Linear programming (LP) decoding for low-density parity-check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance-coming close to that of iterative decoding algorithms-and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05 fraction of correctable errors for rate-½ codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Our analysis exploits an explicit connection between LP decoding and message-passing algorithms and, unlike other techniques, directly works with the primal linear program. An interesting byproduct of our method is a notion of a “locally optimal” solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a “reweighted” version of the min-sum algorithm, obviating the need for LP. Our analysis implies, in particular, that this reweighted version of the min-sum decoder corrects up to a 0.05 fraction of errors.

The multiple-partners assignment game with heterogeneous sales and multi-unit demands: competitive equilibria

A multiple-partners assignment game with heterogeneous sales and multi-unit demands consists of a set of sellers that own a given number of indivisible units of potentially many different goods and a set of buyers who value those units and want to buy at most an exogenously fixed number of units. We define a competitive equilibrium for this generalized assignment game and prove its existence by using only linear programming. In particular, we show how to compute equilibrium price vectors from the solutions of the dual linear program associated to the primal linear program defined to find optimal assignments. Using only linear programming tools, we also show (i) that the set of competitive equilibria (pairs of price vectors and assignments) has a Cartesian product structure: each equilibrium price vector is part of a competitive equilibrium with all optimal assignments, and vice versa; (ii) that the set of (restricted) equilibrium price vectors has a natural lattice structure; and (iii) how this structure is translated into the set of agents’ utilities that are attainable at equilibrium.

Exactness conditions for a convex differentiable exterior penalty for linear programming

Sufficient conditions are given for a classical dual exterior penalty function of a linear program (LP) to be independent of its penalty parameter. This ensures that an exact solution to the primal LP can be obtained by minimizing the dual exterior penalty function. The sufficient conditions give a precise value to such a penalty parameter introduced in [O.L. Mangasarian, Exact 1-Norm support vector machines via unconstrained convex differentiable minimization, Tech. Rep. 05–03, Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 2005. J. Machine Learn. Res. 7, 2006, pp. 1517–1530], where no quantification of the parameter was given. Computational results on LPs with up to one million variables or constraints compare favourably to ILOG CPLEX 9.0 users manual [ILOG CPLEX 9.0 User's Manual. ILOG, Incline Village, Nevada, 2003] and validate the proposed approach.

Solving linear program as linear system in polynomial time

A physically concise polynomial-time iterative-cum-non-iterative algorithm is presented to solve the linear program (LP) Minctxsubject toAx=b,x≥0. The iterative part–a variation of Karmarkar projective transformation algorithm–is essentially due to Barnes only to the extent of detection of basic variables of the LP taking advantage of monotonic convergence. It involves much less number of iterations than those in the Karmarkar projective transformation algorithm. The shrunk linear system containing only the basic variables of the solution vector x resulting from Ax=b is then solved in the mathematically non-iterative part. The solution is then tested for optimality and is usually more accurate because of reduced computation and has less computational and storage complexity due to smaller order of the system. The computational complexity of the combination of these two parts of the algorithm is polynomial-time O(n3). The boundedness of the solution, multiple solutions, and no-solution (inconsistency) cases are discussed. The effect of degeneracy of the primal linear program and/or its dual on the uniqueness of the optimal solution is mentioned. The algorithm including optimality test is implemented in Matlab which is found to be useful for solving many real-world problems.

Application of He's homotopy perturbation method to linear programming problems

In this paper, we use He's homotopy perturbation method (HPM) for solving linear programming (LP) problems. By applying HPM for this class of problems, optimal solutions of a primal LP problem and its corresponding dual problem can be obtained at the same time. The efficiency of the method is shown by solving some examples.

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.

Two-dimensional Second-Order Cone Programming

We show that the primal and dual 2-dimensional second-order cone programs in standard form are equivalent to the standard-form primal and dual linear programs via a linear transformation. We show how variables in the Second-Order Cone (SOC) programs and the linear programs are related by the transformation. Based on the transformation, we interpret the simplex method for two-dimensional Second-Order Cone Programming (SOCP). We show how pivot steps change the states of variables among interior, boundary, and zero. Finally, we give some applications of two-dimensional SOCP.

Decomposition techniques for training linear programming support vector machines

In this paper, we propose three decomposition techniques for linear programming (LP) problems: (1) Method 1, in which we decompose the variables into the working set and the fixed set, but we do not decompose the constraints, (2) Method 2, in which we decompose only the constraints and (3) Method 3, in which we decompose both the variables and the constraints into two. By Method 1, the value of the objective function is proved to be non-decreasing (non-increasing) for the maximization (minimization) problem and by Method 2, the value is non-increasing (non-decreasing) for the maximization (minimization) problem. Thus, by Method 3, which is a combination of Methods 1 and 2, the value of the objective function is not guaranteed to be monotonic and there is a possibility of infinite loops. We prove that infinite loops are resolved if the variables in an infinite loop are not released from the working set and Method 3 converges in finite steps. We apply Methods 1 and 3 to LP support vector machines (SVMs) and discuss a more efficient method of accelerating training by detecting the increase in the number of violations and restoring variables in the working set that are released at the previous iteration step. By computer experiments for microarray data with huge input variables and a small number of constraints, we demonstrate the effectiveness of Method 1 for training the primal LP SVM with linear kernels. We also demonstrate the effectiveness of Method 3 over Method 1 for the nonlinear LP SVMs.

Augmented Lagrangian method for large-scale linear programming problems

The augmented Lagrangian and Newton methods are used to simultaneously solve the primal and dual linear programming problems. The proposed approach is applied to the primal linear programming problem with a very large number (≈106) of nonnegative variables and a moderate (≈103) number of equality-type constraints. Computation results such as the solution of a linear programme with 10 million primal variables are presented.