A test problem generator, by means of neural networks nonlinear function approximation capability, is given in this paper which provides test problems, with many predetermined local minima and a global minimum, to eva...
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A test problem generator, by means of neural networks nonlinear function approximation capability, is given in this paper which provides test problems, with many predetermined local minima and a global minimum, to evaluate nonlinear programming algorithms that are designed to solve the problem globally.
Certain shortcomings are pointed out in the recent work of (S. Nanda, L.N. Das, European Journal of Operational Research 88 (1996) 572-577) and appropriate modifications are suggested for studying duality under pseudo...
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Certain shortcomings are pointed out in the recent work of (S. Nanda, L.N. Das, European Journal of Operational Research 88 (1996) 572-577) and appropriate modifications are suggested for studying duality under pseudo-invexity assumptions. (C) 2000 Elsevier Science B.V. All rights reserved.
When Kuhn and Tucker proved the Kuhn-Tucker theorem in 1950 they launched the theory of nonlinear programming. However, in a sense this theorem had been proven already: In 1939 by W. Karush in a master's thesis, w...
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When Kuhn and Tucker proved the Kuhn-Tucker theorem in 1950 they launched the theory of nonlinear programming. However, in a sense this theorem had been proven already: In 1939 by W. Karush in a master's thesis, which was unpublished;in 1948 by F. John in a paper that was at first rejected by the Duke Mathematical Journal;and possibly earlier by Ostrogradsky and Farkas. The questions of whether the Kuhn-Tucker theorem can be seen as a multiple discovery and why the different occurences of the theorem were so differently received by the mathematical communities are discussed on the basis of a contextualized historical analysis of these works. The significance of the contexts both mathematically and socially for these questions is discussed, including the role played by the military in the shape of Office of Naval Research (ONR) and operations research (OR). (C) 2000 Academic Press MSC 1991 subject classification: 01A60;49-03;52-03;90-03;90C30.
The paper extends prior work by the authors on LUQO, an interior point algorithm for nonconvex nonlinear programming. The specific topics covered include primal versus dual orderings and higher order methods, which at...
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The paper extends prior work by the authors on LUQO, an interior point algorithm for nonconvex nonlinear programming. The specific topics covered include primal versus dual orderings and higher order methods, which attempt to use each factorization of the Hessian matrix more than once to improve computational efficiency. Results show that unlike linear and convex quadratic programming, higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming, but that a variant of Mehrotra's predictor-corrector algorithm can definitely improve performance.
A dual algorithm based on the smooth function proposed by Polyak (1988) is constructed for solving nonlinear programming problems with inequality constraints. It generates a sequence of points converging locally to a ...
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We show that the quadratic growth condition and the Mangasarian-Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points. As a result, when started suffici...
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We show that the quadratic growth condition and the Mangasarian-Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an L-infinity exact penalty sequential quadratic programming algorithm will induce at least R-linear convergence of the iterates to such a local minimum. We construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the MFCQ but for which no positive semidefinite augmented Lagrangian exists. We present numerical results obtained using several nonlinear programming packages on this example and discuss its implications for some algorithms.
Environmental protection, shortage of fresh-water and rising costs for wastewater treatment are all convincing motives for reducing fresh-water consumption and wastewater discharge of the chemical, petrochemical, petr...
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Environmental protection, shortage of fresh-water and rising costs for wastewater treatment are all convincing motives for reducing fresh-water consumption and wastewater discharge of the chemical, petrochemical, petroleum refining and other process industries. Maximizing water reuse, regeneration re-use, and regeneration recycling within the chemical plant, as well as optimal distribution of waste streams for end-of-pipe treatment can reduce fresh-water usage and wastewater discharge, while they are also significant in shrinking capital investment in wastewater treatment systems. Optimal assignment and design of water consuming, regenerating, and treatment systems is a complicated task that can be mathematically formulated as mixed integer non-linear programming (MINLP). In the present article the superstructure based 'Cover and Eliminate' approach with NLP is applied with the tools of the GAMS/MINOS/CONOPT package and compared to previous results. After introducing the problem in the context of chemical process synthesis, a mathematical model is described and the use of the methodology is explained. Experience with the use of GAMS is discussed. Several case studies are solved including basic examples from the literature and their variants. The main conclusion is that the application of the mathematical programming for the optimal water allocation problem is essential owing to the broad variety of the specification opportunities. The complex nature of re-use, regeneration re-use, and recycling with multiple pollutants and multiple treatment processes cannot be simultaneously taken into account by conceptual approaches. It is also shown that the assumption on the independency of contamination rates, generally applied in earlier works, are nor necessarily valid;and the NLP approach can deal with the more reliable specifications.
An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regio...
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An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented.
Operation of a pumped storage system is dictated by the time dependent price of electricity and capacity limitations of the generating plants. This thesis considers the optimization of the Smith Mountain Lake-Leesvill...
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Operation of a pumped storage system is dictated by the time dependent price of electricity and capacity limitations of the generating plants. This thesis considers the optimization of the Smith Mountain Lake-Leesville Pumped Storage-Hydroelectric facility. The constraints include the upper and lower reservoir capacities, downstream channel capacity and flood stage, in-stream flow needs, efficiency and capacity of the generating and pumping units, storage-release relationships, and permissible fluctuation of the upper reservoir water surface elevation to provide a recreational environment for the lake shore property owners.
Two formulations are presented: (1) a nonlinear mixed integer program and (2) a discretized linear mixed integer program. These formulations optimize the operating procedure to generate maximum revenue from the facility. Both formulations are general and are applicable to any pumped storage system. The nonlinear program retains the physical aspects of the system as they are but suffers from non-convexity related issues. The linear formulation uses a discretization scheme to approximate the nonlinear efficiency, pump, turbine, spillway discharge, tailrace elevation-discharge, and storage-elevation relationships. Also, there are binary unit dispatch and either/or constraints accommodating spill and gated release.
Both formulations are applied to a simplified scheme of the Smith Mountain Lake and Leesville pumped storage system. The simplified scheme uses a reduced number of generating and pumping units at the upper reservoir to accommodate the software limitations. Various sensitivity analyses were performed to test the formulations. The linear formulation consistently performs better than the nonlinear. The nonlinear solution requires a good starting point for optimization. It is most useful as a verification tool for the solution from the linear program on all occasions. The formulations yield the best schedules for generating and pumping. A coa
Recently in Burer et al. (Mathematical programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n x n matrix-valued function of a...
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Recently in Burer et al. (Mathematical programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n x n matrix-valued function of a certain form into the positivity constraint on n scalar variables while keeping the number of variables unchanged. Based on this transformation, they proposed a first-order interior-point algorithm for solving a special class of linear semidefinite programs. In this paper, we extend this approach and apply the transformation to general linear semidefinite programs, producing nonlinear programs that have not only the n positivity constraints, but also n additional nonlinear inequality constraints. Despite this complication, the transformed problems still retain most of the desirable properties. We propose first-order and second-order interior-point algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the first-order method are also presented.
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