Convex Simplex Method Research Articles

Global convergent algorithm for the bilevel linear fractional-linear programming based on modified convex simplex method

A global convergent algorithm is proposed to solve bilevel linear fractional-linear programming, which is a special class of bilevel programming. In our algorithm, replacing the lower level problem by its dual gap equaling to zero, the bilevel linear fractional-linear programming is transformed into a traditional single level programming problem, which can be transformed into a series of linear fractional programming problem. Thus, the modified convex simplex method is used to solve the infinite linear fractional programming to obtain the global convergent solution of the original bilevel linear fractional-linear programming. Finally, an example demonstrates the feasibility of the proposed algorithm.

On alternative optimal solutions to quasimonotonic programming with linear constraints

In this paper, the nonlinear programming problem with quasimonotonic (both quasiconvex and quasiconcave) objective function and linear constraints is considered. With the decomposition theorem of polyhedral sets, the structure of optimal solution set for the programming problem is depicted. Based on a simplified version of the convex simplex method, the uniqueness condition of optimal solution and the computational procedures to determine all optimal solutions are given, if the uniqueness condition is not satisfied. An illustrative example is also presented.

Computing dynamic user equilibria for large-scale transportation networks

In this paper, we present an approach for determining dynamic user equilibria. The method is suitable for disaggregated microscopic and mesoscopic simulation-based models. It is a modification of the convex-simplex method, which disposes with the line search step, and controls the subset of travelers to be re-routed at each step while updating the link travel times after each assignment. To guarantee finite termination, a suitable stopping criterion is adopted. The proposed method is implemented within TRANSIMS, the Transportation Analysis and Simulation System, as a two-stage process that employs a combined use of link performance functions and a microsimulator in order to design a framework suitable for application to real transportation systems. To demonstrate this capability, we apply the developed methodology to a large-scale network, Bignet, which is part of the transportation city network of Portland, Oregon; and a medium-scale network, Blacksburg, Virginia; and provide some comparative analyses. Our results exhibit that an improved distribution of travelers is obtained while consuming less than 17-33% of the effort required by the current version of TRANSIMS.

順圧不安定による擾乱の発達の上限を求める数値的手法

Two numerical methods are presented for calculating rigorous upper bounds on the finite-amplitude growth of barotropic instabilities to zonal jets on a rotating sphere. One of the methods is based on Shepherd (1988)'s analytic method, which uses the conservation law of domain-averaged pseudomomentum density. A variational minimization problem is solved numerically with a quasi-Newton method after discretization. The other is the authors' original method to solve a minimization problem under the constraints of the conservation laws of all Casimir invariants and total absolute angular momentum. The convex simplex method, which has been used in operations research, is applied to solve a quadratic programming problem. The two methods are applied to estimate the upper bounds for several profiles of the initial unstable jet and the bounds are compared with the results of non-linear time integrations from the unstable jet with a high-resolution model (Ishioka and Yoden, 1994). The two bounds are found to be almost completely identical to each other. Evidence from high-resolution numerical experiments is that the bounds overestimate the actual wave-enstrophy achieved in numerical experiments by a factor of 1.2 to 2.3.

An Algorithm for Non-Linear Network Programming: Implementation, Results and Comparisons

This paper evaluates an algorithm for solving network flow optimization problems with quadratic cost functions. Strategies for fast implementation are discussed and the results of extensive numerical tests are given. The performance of the algorithm measured by CPU time is compared with that of the convex simplex method specialized for quadratic network programming. Performance of the two methods is analysed with respect to network size and density, and other parameters of interest. The algorithm is shown to perform significantly better on the majority of problems. We also show how the algorithm may be used to solve non-linear convex network optimization problems by the use of sequential quadratic programming.

Solving production smoothing problems with a nonlinear candidate constraint method

We solve multiperiod nonlinear production smoothing problems by combining the convex simplex method due to Zangwill and the candidate constraint method of Sethi and Thompson. We show that, for the solved problems of size up to 120 constraints and 84 variables, computational savings of 50 percent to 60 percent over the convex simplex method can be realized by combining it with the candidate constraint method. The saving is mainly due to the reduction in the effective size of the problem and in the number of iterations performed. The method is also shown to be applicable to a wide range of problems besides production smoothing.

A forward convex-simplex method

Forward algorithms provide a managerially practical way to solve multiperiod problems by solving progressively longer finite horizon subproblems. Many procedures of this nature already exist for special structures. Aronson, Thompson, and Morton have developed a forward simplex method for general staircase-structured linear programs. In this paper this method is extended to the case in which the objective function is convex.

A family of gradient projection methods

In this paper, we give a family of gradient projection methods with three parameters and their convergent properties. This family includes Rosen's gradient projection method, Wolfe's reduced gradient method and Zangwill's convex simplex method as its special cases.

Computer experiments on quadratic programming algorithms

This paper compares the computational performance of five quadratic programming algorithms. These include Wolfe's simplex method, Lemke's complementary pivot method, convex simplex method and quadratic differential algorithm. Execution time and iteration count are used as the major criteria for comparison. Since Lemke's algorithm out-performed all other methods in the study, a detailed statistical analysis was performed to determine the relative importance of problem parameters on the efficiency of Lemke's algorithm. An analysis of variance showed that the number of variables, the percent of positive linear terms in the objective, the number of constraints, and their interactions were the significant factors for both iteration count and execution time. Finally, regression equations for iteration count and execution time are derived as a function of fifteen problem parameters.

A specialization of the convex simplex method to cubic programming

This paper presents a specialization of the Convex Simplex method to cubic objective functions. The algorithm selects a direction of improvement by looking at the partial derivative. An optimal step is chosen by maximizing the objective function in that direction. This involves considering quadratic derivatives and selecting the appropriate step size. The pivoting is done by either a simple Simplex pivoting or by addition of a constraint as in Beale's algorithm. The convergence and the computational efficiency of the algorithm are presented in the last section of the paper, with several examples of application of cubic programming.

Computational experience on an algorithm for the transportation problem with nonlinear objective functions

AbstractThis paper explores computational implications of allowing nonlinear objective functions in the transportation problem. Two types of nonlinearities, including polynomials, are studied. The choice of these functions resulted from our interest in models of integrated water management. Zangwill's convex simplex method and the primal method of transportation problem form the basis of our algorithm. Innovative features of our work are compact storage and efficient computation procedures. We study the effects on computation time of problem size; the density of nonlinear terms; the size of tolerances for stopping rules; and rules for choice of new variables to enter the solution. We find that problems up to 95 × 95 in size are capable of reasonably fast solution. A particularly surprising finding is that one‐dimensional search for improving solutions performs adequately, at least for the kinds of problems posed in this paper. We are encouraged by our results and believe that models involving nonlinear objective functions may be tractable even for relatively large problems, thus making possible more accurate descriptions of real situations.

An iterative convex simplex method for geometric programming with applications†

An iterative linear computational method which utilizes the theory of condensation of polynomials and varies parametrically the objective function of a linearized geometric programme is extended and applied hero to minimize a polynomial subject to polynomial constraints. Applications to dynamic planning and scheduling problems are also briefly referred to.

Decomposition of the convex simplex method

Rutenberg (Ref. 1) provided a decomposition method to solve the problem of minimizing a separable, nonlinear objective function with large-scale linear constraints by using the convex simplex method (Ref. 2). However, there seem to be some errors in his paper (Ref. 3). This paper will rebuild the method by using more convenient and consistent notation.

On Rutenberg's Decomposition Method

David P. Rutenberg [Rutenberg, David P. 1970. Generalized networks, generalized upper bounding and decomposition of the convex simplex methods. Management Sci. 16 (5) 388–401.] provided a method to solve separable nonlinear objective functions with large-scale linear constraints by using W. I. Zangwill's Convex Simplex Method [Zangwill, W. L. 1967. The convex simplex method. Management Sci. 14 (3) 221–238.] However, there seem to be several errors in §4.3 [Rutenberg, David P. 1970. Generalized networks, generalized upper bounding and decomposition of the convex simplex methods. Management Sci. 16 (5) 397]. Two counterexamples to his conclusions are given in this paper. A revised definition of subproblem, therefore, an optimal criterion of the large-scaled program, is also given.

An Algorithm for the Traffic Assignment Problem

The traffic assignment problem associated with a given transportation network is the process of distributing zone-to-zone trips on links of the network. A number of methods have been proposed to solve this problem, but none have been found to be entirely satisfactory. This paper is concerned with the nonlinear mathematical model of the problem, where the link-traveling costs are increasing functions of the link flows and no explicit capacity constraint is imposed on individual links. An efficient algorithm is developed, using a node-arc formulation of the problem. It is an adaptation of the convex-simplex method that takes advantage of the very special network structure of the traffic assignment problem formulated in this way. Numerical results obtained with a moderate size street network are presented.

Generalized Networks, Generalized Upper Bounding and Decomposition of the Convex Simplex Method

If the constraint matrix of a linear program has special structure it may be possible to speed computation. Techniques have been developed to take advantage of such special structures as generalized networks, generalized upper bounding, and decomposition. For these matrix structures, it is shown in this paper how to extend the techniques to Zangwill's mathematical programming algorithm, the convex simplex method.

The Convex Simplex Method

This paper presents a method, called the convex simplex method, for minimizing a convex objective function subject to linear inequality constraints. The method is a true generalization of Dantzig's linear simplex method both in spirit and in the fact that the same tableau and variable selection techniques are used. With a linear objective function the convex simplex method reduces to the linear simplex method. Moreover, the convex simplex method actually behaves like the linear simplex method whenever it encounters a linear portion of a convex objective function. Many of the sophisticated techniques designed to enhance the efficiency of the linear simplex method are applicable to the convex simplex method. In particular, as an example, a network transportation problem with a convex objective function is solved by using the standard transportation tableau and by only slightly modifying the usual procedure for a linear objective function.