Using agents for solving a multi-commodity-flow problem

We investigate a commodity trading problem in a flow network with arbitrary topology where sinks combine commodities into bundles in order to generate profits. Our focus is the profit maximization problem for the trading network under both central and distributed control. We compute solutions for the central control problem using an integer linear program while we compute solutions for the distributed case by implementing the nodes in the network as software-agents that exchange messages in order to establish profitable trades. We report on computational results using both methods and demonstrate that there is a connection between agent profits and a centrality measure developed for the problem. We also demonstrate that with our current agent strategy, there is a trade-off between the agents acting too quickly before enough information is available and waiting too long and thus giving each agent too much information and thus too much power over the outcome.     

Numerical procedure for solving a minimization eigenvalue problem

A numerical procedure for solving the left eigenvalue problem

A method for solving a general multi-valued complementarity problem

We propose an extended version of Chandrasekaran’s method for general complementarity problems with multi-valued weakly off-diagonally antitone costmappings. It allows one either to construct a sequence converging to a solution or to recognize that the problem has no solutions. We also propose versions of Jacobi’s methods for multi-valued inclusions subject to one- and two-side constraints.     

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.     

Cutting plane algorithms for solving a stochastic edge-partition problem

We consider the edge-partition problem, which is a graph theoretic problem arising in the design of Synchronous Optical Networks. The deterministic edge-partition problem considers an undirected graph with weighted edges, and simultaneously assigns nodes and edges to subgraphs such that each edge appears in exactly one subgraph, and such that no edge is assigned to a subgraph unless both of its incident nodes are also assigned to that subgraph. Additionally, there are limitations on the number of nodes and on the sum of edge weights that can be assigned to each subgraph. In this paper, we consider a stochastic version of the edge-partition problem in which we assign nodes to subgraphs in a first stage, realize a set of edge weights from a finite set of alternatives, and then assign edges to subgraphs. We first prescribe a two-stage cutting plane approach with integer variables in both stages, and examine computational difficulties associated with the proposed cutting planes. As an alternative, we prescribe a hybrid integer programming/constraint programming algorithm capable of solving a suite of test instances within practical computational limits.     

A method for solving a quadratic optimal control problem

Hestenes' method of multipliers is used to approximate a quadratic optimal control problem. The global existence of a family of unconstrained problems is established. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original differentially constrained problem.The preparation of this paper was sponsored in part by the U.S. Army Research Office under Grant No. DA-31-124-ARO(D)-355.     

An exact algorithm for solving a capacitated location-routing problem

In location-routing problems, the objective is to locate one or many depots within a set of sites (representing customer locations or cities) and to construct delivery routes from the selected depot or depots to the remaining sites at least system cost. The objective function is the sum of depot operating costs, vehicle acquisition costs and routing costs. This paper considers one such problem in which a weight is assigned to each site and where sites are to be visited by vehicles having a given capacity. The solution must be such that the sum of the weights of sites visited on any given route does not exceed the capacity of the visiting vehicle. The formulation of an integer linear program for this problem involves degree constraints, generalized subtour elimination constraints, and chain barring constraints. An exact algorithm, using initial relaxation of most of the problem constraints, is presented which is capable of solving problems with up to twenty sites within a reasonable number of iterations.     

An approach for solving a fuzzy multiobjective programming problem

In this paper we present a new approach, based on the Nearest Interval Approximation Operator, for dealing with a multiobjective programming problem with fuzzy-valued objective functions.     

Discontinuous Galerkin methods for solving a quasistatic contact problem

We consider the numerical solution of a nonlinear evolutionary variational inequality, arising in the study of quasistatic contact problems. We study spatially semi-discrete and fully discrete schemes for the problem with several discontinuous Galerkin discretizations in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for the schemes, reaching the optimal convergence order for linear elements. Numerical results are presented on a two dimensional test problem to illustrate numerical convergence orders.     

Iterative method for solving a nonlinear boundary value problem

In this paper, a boundary value problem for a nonlinear second-order ordinary differential equation is studied. By means of the maximum principle we established the existence and the uniqueness of a solution of the problem. Then for finding the solution an iterative method is proposed. It is proved that this method converges much faster than the Picar successive approximations and in a particular case it gives two-sided monotone approximations to the exact solution of the problem. Finally, some illustrative examples are considered to confirm the efficiency of the method.     

Numerical method for solving a boundary-value problem of thermoelasticity

We consider the problem of determining the stress-strain state of an elastoplastic layer under impulse heating. The theory of small elastoplastic strains with linear hardening is used. A boundary-value problem is obtained for the equations of thermoelasticity whose coefficients at any time are functionals of strain history. A method is developed for solving this problem, based on discretization by space and time variables and application of an appropriate difference scheme. This scheme constructs a recursive evolution process for the state column at the nodes of the space grid. Numerical implementation of the method has demonstrated its reliability and efficiency.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 66–71, 1986.     

A certain spline method for solving a boundary-value problem

Translated from Ukrainskii Matematichskii Zhurnal, Vol. 41, No. 5, pp. 703–707, May, 1989.     

A method for solving a boundary problem for a nonlinear controlled system

An algorithm is proposed for constructing a control function that transfers a wide class of nonlinear systems of ordinary differential equations from an initial state to an arbitrarily small neighborhood of a given terminal state. The algorithm is convenient for numerical implementation. Taking into account the restrictions on the control and phase coordinates, a constructive criterion is obtained for choosing terminal states for which this transfer is possible. The problem of an interorbital flight is considered and modeled numerically.     

Stochastic DEA with ordinal data applied to a multi-attribute pricing problem

Multiple attribute pricing problems are highly challenging due to the dynamic and uncertain features in the associated market. In this paper, we address the condominium multiple attribute pricing problem using data envelopment analysis (DEA). In this study, we simultaneously consider stochastic variables, non-discretionary variables, and ordinal data, and present a new type of DEA model. Based on our proposed DEA, an effective performance measurement tool is developed to provide a basis for understanding the condominium pricing problem, to direct and monitor the implementation of pricing strategy, and to provide information regarding the results of pricing efforts for units sold as well as insights for future building design. A case study is executed on a leading Canadian condominium developer.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.