Random Linear Programming Problems Research Articles

О генерации случайных задач линейного программирования на кластерных вычислительных системах

The article presents and evaluates a scalable FRaGenLP algorithm for generating random linear programming problems of large dimension n on cluster computing systems. To ensure the consistency of the problem and the boundedness of the feasible region, the constraint system includes 2 n +1 standard inequalities, called support inequalities. New random inequalities are generated and added to the system in a manner that ensures the consistency of the constraints. Furthermore, the algorithm uses two likeness metrics to prevent the addition of a new random inequality that is similar to one already present in the constraint system. The algorithm also rejects random inequalities that cannot affect the solution of the linear programming problem bounded by the support inequalities. The parallel implementation of the FRaGenLP algorithm is performed in C++ through the parallel BSF-skeleton, which encapsulates all aspects related to the MPI-based parallelization of the program. We provide the results of large-scale computational experiments on a cluster computing system to study the scalability of the FRaGenLP algorithm.

The simplest model of jamming

We study a well known neural network model—the perceptron—as a simple statistical physics model of jamming of hard objects. We exhibit two regimes: (1) a convex optimization regime where jamming is hypostatic and non-critical; (2) a non-convex optimization regime where jamming is isostatic and critical. We characterize the critical jamming phase through exponents describing the distribution laws of forces and gaps. Surprisingly we find that these exponents coincide with the corresponding ones recently computed in high dimensional hard spheres. In addition, modifying the perceptron to a random linear programming problem, we show that isostaticity is not a sufficient condition for singular force and gap distributions. For that, fragmentation of the space of solutions (replica symmetry breaking) appears to be a crucial ingredient. We hypothesize universality for a large class of non-convex constrained satisfaction problems with continuous variables.

Interactive fuzzy programming for multiobjective fuzzy random linear programming problems through possibility-based probability maximization

This paper presents interactive fuzzy programming for multiobjective fuzzy random linear programming problems through possibility-based probability maximization. In our proposed approach, it is assumed that the decision maker has fuzzy goals for not only original objective functions but also the corresponding distribution functions in a probability maximization model, and such fuzzy goals are quantified by eliciting the corresponding membership functions. Using the fuzzy decision, such two kinds of membership functions are integrated. The satisfactory solution is derived from among an extended Pareto optimal solution set through the interaction with the decision maker. An illustrative numerical example is provided to demonstrate the efficiency of the proposed method.

Interactive Multiobjective Fuzzy Random Linear Programming through Fractile Criteria

We propose an interactive fuzzy decision making method for multiobjective fuzzy random linear programming problems through fractile criteria optimization. In the proposed method, it is assumed that the decision maker has fuzzy goals for not only objective functions but also permissible probability levels in a fractile optimization model, and such fuzzy goals are quantified by eliciting the corresponding membership functions. Using the fuzzy decision, such two kinds of membership functions are integrated. In the integrated membership space, the satisfactory solution is obtained from among an extended Pareto optimal solution set through the interaction with the decision maker. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

Average case complexity results for a centering algorithm for linear programming problems under Gaussian distributions

Abstract To solve linear programming problems by interior point methods an approximately centered interior point has to be known. Such a point can be found by an algorithmic approach – a so-called phase 1 algorithm or centering algorithm. For random linear programming problems distributed according to the rotation symmetry model, especially with normal distribution, we present probabilistic results on the quality of the origin as starting point and the average number of steps of a centering algorithm.

Fuzzy random programming with equilibrium chance constraints

To model fuzzy random decision systems, this paper first defines three kinds of equilibrium chances via fuzzy integrals in the sense of Sugeno. Then a new class of fuzzy random programming problems is presented based on equilibrium chances. Also, some convex theorems about fuzzy random linear programming problems are proved, the results provide us methods to convert primal fuzzy random programming problems to their equivalent stochastic convex programming ones so that both the primal problems and their equivalent problems have the same optimal solutions and the techniques developed for stochastic convex programming can apply. After that, a solution approach, which integrates simulations, neural network and genetic algorithm, is suggested to solve general fuzzy random programming problems. At the end of this paper, three numerical examples are provided. Since the equivalent stochastic programming problems of the three examples are very complex and nonconvex, the techniques of stochastic programming cannot apply. In this paper, we solve them by the proposed hybrid intelligent algorithm. The results show that the algorithm is feasible and effectiveness.

Automatic generation of linear programming problems for computer aided instruction

The paper presents a method for generating random linear programming problems with a previously selected type of solution. The user can choose a problem whose solution is unbounded, bounded for minima, maxima or both, unique or multiple, with given structure, at wish. Initially, the feasible solution of the LPP is generated as the sum of a linear space, a cone, and a polytope, depending on the desired properties of the solution. With the aim of obtaining a simple set of constraints, the generators of these three structures are selected as random vectors with integer simple components, the range of which can be given. Next, an objective function that satisfies the required conditions, i.e. leads to a solution of the desired type, is obtained. The generating algorithms have been implemented in Mathematica and some illustrative examples are given to clarify the generation process. With this tool, a LPP can be generated, according to the instructor requirements, where this is a human or an expert system. They can control student progress and generate a sequence of problems covering all possible cases, in steps of increasing difficulty. Combining this tool with another (also produced by the same authors) that solves the problems and explains the whole process, step by step, a computer aided module for learning LPP, which is completely autonomous, can be easily obtained.

Average number of iterations of some polynomial interior-point

We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the average number of iterations of these algorithms, coupled with a finite termination technique, is bounded above by O ( n 1,5 ) . The random LP problem is Todd's probabilistic model with the standard Gauss distribution.

Erratum: Probabilistic Models for Linear Programming

In the paper “Probabilistic models for linear programming,” published in Mathematics of Operations Research, Vol. 16, No. 4, pp. 671–693 , the author proposed various models for generating random linear programming problems and investigated several properties of these models, including the probability that the feasible region is bounded, the distribution of the distance from a particular interior point to each constraint hyperplane, and some properties of the vertices of the feasible region. Unfortunately, one of the results about these vertices is incorrect, and this affects some of the corollaries. Here it is described what is invalid, and explained where the difficulty arises in the proof.

Probabilistic Models for Linear Programming

We propose and investigate new probabilistic models for linear programming. In contrast to previous models, ours guarantee the existence of optimal solutions and are symmetric under duality. While in some respects our distributions are very special, there is sufficient flexibility to permit an arbitrary degree of primal and/or dual degeneracy, either just at the optimal solution or throughout the feasible region using null variables. Moreover, the precision of the distributions allows us to compute the probability that the feasible region is bounded as well as the distribution of the distance to a constraint hyperplane and that of the components of a vertex. Interest in these measures stems from Karmarkar's algorithm, and we also introduce a model for generating random linear programming problems on a simplex.

Stochastic linear programming method for right-hand sides random vector

A method of solving a random linear programming problem, worked out for the case with non-deterministic right-hand sides, most often occurring in practice, is presented. Definitions of a solution feasibility measure are introduced. They are based on the probability value of a constraints satisfaction for a given basis solution. A method of calculating the probability value of a feasibility conditions satisfaction and rules for replacing a basis solution for another one with a greater probability of the feasibility conditions satisfaction are included. The method does not lead to a dimension extension of a linear problem being solved in comparison with the deterministic case. This is a post-optimization method, applied after using the Simplex method, commonly applied in deterministic linear programming.

線形計画問題に対する乗法的罰金関数法の拡張

We shall extend Iri's multiplicative penalty function method for linear programming [4] so that it can handle the problem of unknown optimum value of the objective function, without solving both primal and dual problems simultaneously, and generate convergent dual solutions. By making use of these dual variables, lower bounds of the optimum objective function value are updated efficiently, which makes the total number of iterations required in the extended algorithm small. In doing so, a new duality on the multiplicative penalty function is discussed. A sufficient condition for a constraint to be inactive at all optimum solutions is given, which can be checked in the extended algorithm. Several computational techniques for enhancing the efficiency of the algorithm are also discussed. Some connection of the proposed algorithm with Sonnevend's and Renegar's methods [ 10, 11] is touched upon. Furthermore a method of estimating the optimum objective function value is given. Preliminary computational results on the random linear programming problem are finally shown.