Dual Linear Programming Problems Research Articles

Descent along nodal straight lines and simplex algorithm: two variants of regression analysis based on the least absolute deviation method

A comparative analysis of the computational complexity of exact algorithms for estimating linear regression equations was conducted using the least absolute deviation method. The goal of the study is to compare the computational efficiency of exact algorithms for descent along nodal lines and algorithms based on solving linear programming problems. For this purpose, the algorithm of gradient descent along nodal lines and algorithms for solving the equivalent primal and dual linear programming problems using the simplex method were considered. The computational complexity of algorithms for implementing the method of least modules in solving direct and dual linear programming problems was estimated. A comparison between the average time for determining the regression coefficients using the primal and dual linear programming problems and the average time for gradient descent along nodal lines was conducted using the Monte Carlo method of statistical experiments. It is shown that both options are significantly inferior behind gradient descent along nodal lines, both in terms of the computational complexity of the algorithms and in terms of computation time, and this advantage increases with the sample size, reaching hundred times or more.

Improvement of the enterprise’s production program as a way to adapt to market changes

Purpose. Proving the possibility of improving the procedure to adapt the enterprise to market trends based on the self adjustment of its production program using the solutions of direct and dual linear programming problems with the help of the Microsoft Excel package “Search for solutions” add-on. Methodology. The methodological basis of the study is the provisions of modern economic theory, the fundamental works by foreign and domestic scientists on the formation of management systems and the peculiarities of enterprises’ adaptation to changes. The methodological apparatus of economic and mathematical modeling, operations research, in particular, the theory of duality of linear programming problems is widely used. Findings. The enterprise’s adaptation to changes in the external and internal environment with the help of the proposed iterative procedure of unused reserves redistribution between scarce resources made it possible to increase the products’ sale by reducing the manufacturing of one type of products and increasing the output of another one under the conditions of a stable range of products manufacturing at the enterprise. Originality. The article substantiates the procedure of the enterprise adaptation to changes in internal and external environments on the basis of self adjustment of its production program using the solutions of direct and dual problems of linear programming. Practical value. The article features the applied aspects to self adjust the enterprise production program, aimed to increase the efficiency of economic activity by means of step-by-step improvement of the optimal production plan (works, services).

Randomized Strategies for Robust Combinatorial Optimization with Approximate Separation

In this paper, we study the following robust optimization problem. Given a set family representing feasibility and candidate objective functions, we choose a feasible set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the feasible sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory, and optimization. To solve the problem, we provide a general framework based on the dual linear programming problem. In the framework, we utilize the ellipsoid algorithm with the approximate separation algorithm. We prove that there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for our robust optimization problem if there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for finding a (deterministic) feasible set that maximizes a nonnegative linear combination of the candidate objective functions. Using our result, we provide approximation algorithms for the max–min fair randomized allocation problem and the maximum cardinality robustness problem with a knapsack constraint.

A novel Lower Bound Limit Analysis model with hexahedron elements for the failure analysis of laboratory and thin infill masonry walls in two-way bending

A novel Lower Bound (LB) Limit Analysis (LA) Finite Element (FE) model for the study at failure of laboratory and thin infill masonry walls in two-way bending is presented. In the model, a masonry plate is discretized into infinitely resistant hexahedrons and quadrilateral interfaces where all plastic dissipation occurs. Three internal static variables act on interfaces, namely bending moment, out-of-plane Kirchhoff shear and torque. Equilibrium is imposed on hexahedrons, whereas admissibility is enforced exclusively on interfaces between adjoining elements. The admissibility of the internal actions on masonry interfaces is imposed with a strength domain obtained by means of an already existing LB homogenization technique where joints are reduced to interfaces. The resultant Linear Programming (LP) problem - which allows to estimate collapse loads and distribution of internal actions at collapse - is characterized by a number of variables and constraints relatively limited, which requires little computational burden to be solved through standard interior point LP software. Failure mechanisms are obtained solving the dual LP problem. The procedure is validated against two existing experimental datasets of walls in two-way bending, namely three series of panels with and without perforations tested at collapse at the University of Adelaide Australia and four series of solid and perforated panels tested at the University of Plymouth UK. Comparing the results in terms of collapse loads, active failure mechanisms and distribution of internal actions with both experimental evidence and previously presented numerical models, excellent results are found, an outcome showing the reliability of the procedure proposed.

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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.

Synthesis of a Path-Planning Algorithm for Autonomous Robots Moving in a Game Environment during Collision Avoidance

This paper describes and illustrates the optimization of a safe mobile robot control process in collision situations using the model of a multistep matrix game of many participants in the form of a dual linear programming problem. The synthesis of non-cooperative and cooperative game control software was performed in Matlab/Simulink software to determine the safe path of the robot when passing a greater number of other robots and obstacles. The operation of the game motion control algorithm of a mobile robot is illustrated by computer simulations made in the Matlab/Simulink program of two real previously recorded navigation situations while passing dozens of other autonomous mobile robots.

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.

Piecewise rigid displacement (PRD) method: a limit analysis-based approach to detect mechanisms and internal forces through two dual energy criteria

This paper proposes an extension of the Piecewise Rigid Displacement (PRD) method based on a new dual linear programming problem that minimises the complementary energy. Before, the PRD method had been applied to solve the kinematical problem for masonry-like structures composed of normal, rigid, no-tension (NRNT) material minimising the total potential energy. Specifically, the PRD method frames this minimum-energy search as a linear programming problem whose solutions are displacements and singular strain fields (cracks).Here, we show that the corresponding dual linear programming problem discretises the minimum of the complementary energy and returns, as solutions, admissible internal stress states compatible with the crack pattern obtained by solving the primal problem. Thus, these two minimum-energy criteria are dually connected, and their combined use allows coupling mechanisms and internal forces with settlements or homogeneous boundary displacements. This allows addressing different mechanical problems: equilibrium and stability of the reference configuration, effects of settlements, and mechanisms due to overloading (e.g. horizontal forces). Since the NRNT material represents the extension to continuum media of Heyman's material model, the PRD method offers an extremely fast, limit analysis-based, displacement approach that allows simultaneously finding mechanisms and compatible internal forces for any boundary condition, loads and geometry.

Masonry structures in the presence of foundation settlements and unilateral contact problems

This paper aims at presenting a new perspective on the modeling of the influence of foundation settlements in masonry structures. We show that it is possible to model in a consistent way the crack pattern and the associated mechanism induced by applied settlements starting from a proper treatment of unilateral contact constraints between contiguous blocks, in which the structure is partitioned after the onset of settlements. We extend two classic variational formulations for the Signorini problem, namely the minimum of the total potential energy and a complementary boundary formulation, to the case of contact between multiple no-tension rigid bodies satisfying Heyman's hypotheses for the limit analysis of masonry structures. The discretization of the proposed variational formulations directly entails two dual linear programming problems. The proposed formulations allow to correctly evaluate the effects of foundation settlements and their influence on the ultimate load bearing capacity of masonry structures.

Difference Schemes in the Undetermined Coefficient Space and Dual Problems of Linear Programming

Using properties of the difference schemes approximating a one-dimensional transport equation as an example, it is shown that optimization of the properties of difference schemes based on the analysis in the space of undetermined coefficients and optimization of these properties based on the method of parametric correction are dual problems. Hybrid difference schemes for the linear transport equation are built as solutions to dual linear programming problems. It is shown that Godunov’s theorem follows from the linear program optimality criterion as one of the complementary slackness conditions. A family of hybrid difference schemes is considered. It is shown that Fedorenko’s hybrid difference scheme is obtained by solving the dual linear programming problem.

Identification of efficient equilibria in multiproduct trading with indivisibilities and non-monotonicity

This paper focuses on multiproduct trading with indivisibilities and where a representative agent may have non-monotonic preferences. In this framework, the set of firms’ profits (which comes from efficient subgame perfect Nash equilibria) is the Pareto frontier of some projection of the core of the game. We show that under monotonicity efficient subgame perfect Nash equilibria are achieved by single offers and the equilibrium characterization is easy to obtain. When dealing with non-monotonic preferences the problem becomes more challenging. Then, we define a pair of primal–dual linear programming problems that fully identifies the core of the game. A set of modified versions of the dual programming problem characterizes the Pareto-optimal frontier of the core projection on firms’ coordinates. Although this approach gives us the payoff-equivalence class (Strong Nash equilibria) of all the efficient subgame perfect Nash equilibria, the number of problems to be solved may be huge.

Minimum-Euclidean-norm matrix correction for a pair of dual linear programming problems

For a pair of dual (possibly improper) linear programming problems, a family of matrix corrections is studied that ensure the existence of given solutions to these problems. The case of correcting the coefficient matrix and three cases of correcting an augmented coefficient matrix (obtained by adding the right-hand side vector of the primal problem, the right-hand-side vector of the dual problem, or both vectors) are considered. Necessary and sufficient conditions for the existence of a solution to the indicated problems, its uniqueness is proved, and the form of matrices for the solution with a minimum Euclidean norm is presented. Numerical examples are given.

Optimal Models used to Provide Urban Transport Systems Efficiency and Safety

Abstract The article under consideration presents methodology of construction of mathematical model enabling proper distribution of transport demand within the city territory. New territory assessment parameter is introduced – “city transport dependency”. System of city transport infrastructure territorial and ecological development restraints is presented. Mathematical model of direct and dual problem of linear programming is presented. Assessment of practical importance of direct and dual problem modeling results is given.

Anti-jamming Strategy Versus a Low-Power Jamming Attack When Intelligence of Adversary’s Attack Type is Unknown

In this paper, we suggest a new class of anti-jamming problems where the type of intelligence associated with ajamming attack is unknown. Specifically, we consider a problem where the nodes of a peer-to-peer network do not know whether the network is under attack by a random jammer (which might be considered as a natural background noise), or an intelligent one (i.e., the jammer who can adapt his strategy based on knowledge gained during attacks). The goal of the nodes is to identify the type of the attack based on knowledge obtained from the attack in previous time slots, and thereby to reduce the efficiency of the jamming attack. First, we model the problem as a Bayesian game for a single time slot attack, and reduce it to the solution of dual linear programming (LP) problems. Additionally, the convergence of the fictitious play algorithm for finding the equilibrium is established. Then, we develop the problem for a repeated jamming attack, where the nodes adapt their beliefs based on history of the previous attacks. In particular, we have shown that it is possible for the nodes in the network to always be able to identify the jammer's type within a finite number of time slots.

正と反の経営学

In this paper, the author discusses the theme “Thetical and Antithetical Business Management”. “Thetical Business Management” signifies “a management style in which an objective function, expressing consumer expenditure of service goods, is minimized, with a constraint condition expressing a guaranteed level of expenditure.” While “Antithetical Business Management” denotes “a management style in which an objective function expressing consumer satisfaction concerning service goods, is maximized, with a constraint condition denoting the limits of their tolerance of payment for service goods.” In other words, “Thetical Business Management” can be formulated as Primal Problem of Linear Programming, whereas “Antithetical Business Management” can be formulated as Dual Problem of Linear Programming.

Polynomially Computable Bounds for the Probability of the Union of Events

We consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the intersections of up to m events are given. It is known that the best possible bounds can be obtained by solving linear programming problems with a number of variables that is exponential in the number of events. Because of their size and structure, these large linear programs are known to be very hard to solve. In the literature simpler, polynomially sized aggregations are considered and numerous closed form or polynomially computable bounds are derived from those. We present here a new approach that introduces additional constraints to the dual linear programming problems in such a way that those become polynomially solvable. By using different sets of additional constraints, we introduce three new classes of polynomially computable upper and lower bounds. We show that they dominate almost all efficiently computable bounds known in the literature. Furthermore, by characterizing the vertices of two new classes of polyhedra, we can show that in two cases our bounds coincide with classical bounds, proving new extremal properties for those well-known bounds. Finally, we provide extensive numerical results comparing the average tightness of the various bounds on a large number of instances.

Generalized Newton method for linear optimization problems with inequality constraints

A dual problem of linear programming is reduced to the unconstrained maximization of a concave piecewise quadratic function for sufficiently large values of a certain parameter. An estimate is given for the threshold value of the parameter starting from which the projection of a given point to the set of solutions of the dual linear programming problem in dual and auxiliary variables is easily found by means of a single solution of the unconstrained maximization problem. The unconstrained maximization is carried out by the generalized Newton method, which is globally convergent in an a finite number of steps. The results of numerical experiments are presented for randomly generated large-scale linear programming problems.

Game Strategies of Ship in the Collision Situations

The paper introduced the basic model of process of safe ship control in a collision situation using a game model with j objects, which includes non-linear state equations and non-linear, time varying constraints of the state variables as well as the quality game control index in the forms of the game integral payment and the final payment. Approximated model of the process control as the model of multi-step matrix game in the form of dual linear programming problem has been adopted here. The Game Ship Control GSC computer program has been designed in the Matlab/Simulink software in order to determine the own ship’s safe trajectory. These considerations have been illustrated with examples of a computer simulation using an GSC program for determining the safe ship's trajectory in real navigational situation. Simulation research were passed for five sets of strategies of the own ship and met ships.