A combined use of DEA (Data Envelopment Analysis) with Strong Complementary Slackness Condition and DEA–DA (Discriminant Analysis)

This study discusses a combined use of DEA (Data Environment Analysis) with SCSC (Strong Complementary Slackness Condition) and DEA–DA (Discriminant Analysis). Many studies use DEA to evaluate the performance of various organizations in private and public sectors. A conventional use of DEA is not perfect because it still contains zero in many multipliers. This implies that DEA does not fully utilize information on all inputs and outputs. As a result, DEA produces many efficient organizations. To overcome the methodological difficulty, this study proposes a new use of DEA/SCSC and DEA–DA to reduce the number of efficient organizations.     

Separating the solution sets of analytical and polynomial systems

A linear inequality system with infinitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coefficients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Research supported by DGES of Spain and FEDER of UE, Grant BFM2002-04114-C02-01. Research supported by CONACyT of Mexico, Grant 130036. Research partially supported by CONACyT of Mexico, Grant 44003.     

A sequential method for nonlinear filtering: Numerical implementation and comparisons

Exact equations are presented for sequentially updating the optimal solution for a discrete-time analog of the basic Sridhar nonlinear filtering problem as the process length increases and new observations are obtained. A tabular method is described for implementing numerically the sequential filtering equations. The accuracy and efficiency of the tabular method are illustrated by means of several numerical examples.Dedicated to R. SridharThe work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

Optimization of functional differential systems

A method of determining optimal control for functional differential systems is presented. This is a generalization of Krasovskii's results on the optimal control of time delay systems, which include several of the results published recently. A technique for the numerical solution of the resulting Riccati equations is developed, and an example is worked to illustrate the presented results.The research reported herein was partly supported by JSEP Contract No. F44620-71-C-0091 and AF Grant No. AFOSR-72-2371.     

Existence and stability of large scale nonlinear oscillations in suspension bridges

A nonlinear model of a suspension bridge is considered in which large-scale, stable oscillatory motions can be produced by constant loading and a small-scale, external oscillatory force. Loud's implicit-function theoretic method for determining existence and stability of periodic solutions or nonlinear differential equations is extended to a case of a non-differentiable nonlinearity.Author partially supported by NSF under Grant DMS 8318204 and AFOSR Grant 85-0330.Author partially supported by NSF under Grant DMS 9519882.Author partially supported by NSF under Grant DMS 8519776.     

A bound for the optimum relaxation factor for the successive overrelaxation method

Summary An upper bound on the optimum relaxation factor for use with the successive overrelaxation method is derived for a class of linear systems arising from the numerical solution by finite difference methods of a boundary value problem involving the self-adjoint differential equationWork on this paper was supported in part by the U. S. Army Research Office (Durham) through Grant DA-ARO(D)-31-124-G 1050 and by the National Science Foundation through Grant GP-8442 with The University of Texas at Austin.     

Solution of the linear inverse vector optimization problem by a single linear program

It is shown that finding a solution to a linear vector optimization problem which is efficient with respect to the constraints as well as to the objectives is equivalent to solving a single linear program.The research of this author was supported by NSF Grant DCR74-20584.The research of this author was partially supported by Canada Council Grant W760467.     

Finite element formulation based on proper orthogonal decomposition for parabolic equations

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method. This work was supported by National Natural Science Foundation of China (Grant Nos. 10871022, 10771065, and 60573158) and Natural Science Foundation of Hebei Province (Grant No. A2007001027)     

Control of non-self-adjoint distributed-parameter systems

Systems involving viscous damping forces, circulatory forces, and aerodynamic forces are non-self-adjoint. A method capable of controlling non-self-adjoint distributed systems is the independent modal-space control method, whereby the problem of controlling a distributed-parameter system is reduced to that of controlling an infinite set of independent, complex, second-order ordinary differential equations. In the case of optimal control, one must solve independent, complex, scalar Riccati equations. The transient solution of the Riccati equations can be found with relative ease and the steady-state solution can be found in closed form. A numerical example demonstrates the effectiveness of the method.This work was supported by AFOSR Research Grant No. 83-0017.     

Range conditions for the exponential Radon transform

A new approach to the range conditions for the exponential Radon transform is introduced. Partially supported by NSF EPSCOR Grant and Wesley Foundation Grant no. 9012019. Partially supported by NSF Grant 33-1807-231.     

On the integral equations of optimal signal detection and estimation theory

The analysis of many optimal signal detection and estimation processes involves the solution of Fredholm integral equations. A New approach to these equations is presented. It consists of reducing the Fredholm integral equation to a Cauchy system which is well suited to numerical solution.This research was supported by the National Institutes of Health, Grant No. GM-16437-03.     

Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

Primal-Dual Interior-Point Method for an Optimization Problem Related to the Modeling of Atmospheric Organic Aerosols

A mathematical model for the computation of the phase equilibrium related to atmospheric organic aerosols is presented. The phase equilibrium is given by the global minimum of the Gibbs free energy for a system that involves water and organic components. This minimization problem is equivalent to the determination of the convex hull of the corresponding molar Gibbs free energy function. A geometrical notion of phase simplex related to the convex hull is introduced to characterize mathematically the phases at equilibrium. A primal-dual interior-point algorithm for the efficient solution of the phase equilibrium problem is presented. A novel initialization of the algorithm, based on the properties of the phase simplex, is proposed to ensure the convergence to a global minimum of the Gibbs free energy. For a finite termination of the interior-point method, an active phase identification procedure is incorporated. Numerical results show the robustness and efficiency of the approach for the prediction of liquid-liquid equilibrium in multicomponent mixtures.Communicated by R. GlowinskiThis work was supported by US Environmental Protection Grant X-83234201. The second author was partially supported by Swiss National Science Foundation Grant PBEL2-103152.     

On the Chebyshev solution of inconsistent linear equations

A new formulation of the ascent algorithm for the Chebyshev solution of linear systems is given. This leads to two algorithms, which are similar to the ordinary simplex and the product form of inverse algorithms for the solution of linear programming problems.This research was supported by the National Aeronautics and Space Administration, Washington, D.C., Grant No. NGR-33-015-013.     

On Controllability of an Elastic Ring

We study the exact controllability problem for a ring under stretching tension that varies in time. We are looking for a couple of forces, which drive the state solution to rest. We show that applying two forces is necessary for controllability and the ring is controllable in the time interval greater than the optical length of the string. We also explain why one force would not be enough to control the ring. We use the method of moments to reduce the controllability problem to a moment problem for the controlling forces. The solution of that problem is based on an auxiliary basis property result. Both method of moments and proof of the basis property are developed for the model with time-dependent parameters. S.A.’s research was supported in part by the NSF, grant ARC–0724860. B.B.’s research was supported in part by University of Tennessee at Chattanooga Faculty Research Grant. S.I.’s research was supported in part by the Russian Foundation for Basic Research, grants 08-01-00595a and 08-01-00676a.     

An initial-value method for the determination of Green's functions

The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.This research was partially supported by UNLV Grant No. 001-060-4573.     

Nonscalarized multiobjective global optimization

A new approach to multiobjective optimization is presented which is made possible due to our ability to obtain full global optimal solutions. A distinctive feature of this approach is that a vector cost function is nonscalarized. The method provides a means for the solution of vector optimization problems with nonreconcilable objectives.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A3492.     

Dynamically Consistent Cooperative Solution in a Differential Game of Transboundary Industrial Pollution

This paper presents a cooperative differential game of transboundary industrial pollution. A noted feature of the game model is that the industrial sectors remain competitive among themselves while the governments cooperate in pollution abatement. It is the first time that time consistent solutions are derived in a cooperative differential game on pollution control with industries and governments being separate entities. A stochastic version of the model is presented and a subgame-consistent cooperative solution is provided. This is the first study of pollution management in a stochastic differential game framework. This research was supported by the Research Grant Council of Hong Kong Grant HKBU2103/04H and Hong Kong Baptist University Grant FRG/05-06/II22.     

Heuristic reliability optimization by tabu search

A new heuristic algorithm, based on the tabu search methodology, is proposed for constrained redundancy optimization in series and in complex systems. It has the advantage of not being blocked as soon as a local optimum is found. Results given by the new method are compared with those of previous heuristics on a series of examples.We are grateful to R. Bulfin for making the code for reliability optimization of series systems he wrote with C.-Y. Liu available to us.Work of the first author was supported by NSERC Grant No. GP0105574, FCAR Grant No. 92EQ1048 and AFOSR Grant No. 90-0008 to Rutgers University.Work of the second author was partly supported by AFOSR Grant No. 90-0008 to Rutgers University while he was a graduate student.     

Suboptimal shape control for quasi-static distributed-parameter systems

We propose an on-line control approach which will adjust the steady-state shape of a large antenna arbitrarily close to any achievable desired profile. The method makes use of distributed-parameter system theory and allows refocusing using a limited number of control actuators and sensors.The controller gains are calculated by approximating the solution to an infinite-dimensional optimal quasi-static control problem. The controller gain calculation is computationally simpler than that proposed in a companion paper. The Galerkin (finite element) approximation method is used for model reduction. We prove that both gain and state convergence can be achieved by using the proposed approximation scheme.This work was partially supported by the Air Force Office of Scientific Research, Grant No. AFOSR 83-0124, and by the National Aeronautics and Space Administration, Grant No. NAG-1-515.