Fixed-point iteration based algorithm for a class of nonlinear programming problems

• Autorzy: Belegundu A. D.

Identyfikatory

DOI

10.17512/jamcm.2017.2.03

• Języki publikacji: EN

Abstrakty

EN

A fixed-point algorithm is presented for a class of singly constrained nonlinear programming (NLP) problems with bounds. Setting the gradient of the Lagrangian equal to zero yields a set of optimality conditions. However, a direct solution on general problems may yield non-KKT points. Under the assumption that the gradient of the objective function is negative while the gradient of the constraint function is positive, and that the variables are positive, it is shown that the fixed-point iterations can converge to a KKT point. An active set strategy is used to handle lower and upper bounds. While fixed-point iteration algorithms can be found in the structural optimization literature, these are presented without clearly stating assumptions under which convergence may be achieved. They are also problem specific as opposed to working with general functions f, g. Here, the algorithm targets general functions which satisfy the stated assumptions. Further, within this general context, the fixed-point variable update formula is given physical significance. Unlike NLP descent methods, no line search is involved to determine step size which involves many function calls or simulations. Thus, the resulting algorithm is vastly superior for the subclass of problems considered. Moreover, the number of function evaluations remains independent of the number of variables allowing the efficient solution of problems with a large number of variables. Applications and numerical examples are presented.

Słowa kluczowe

EN

fixed-point iteration; resource allocation; nonlinear programming; optimality criteria

PL

iteracja punku stałego; alokacja zasobów; programowanie nieliniowe; kryteria optymalizacji dwukierunkowych torów wodnych; algorytm punktu stałego

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2017
• Tom: Vol. 16, nr 2
• Strony: 29--41
• Opis fizyczny: Bibliogr. 13 poz., rys.

Twórcy

autor

Belegundu A. D.

• Department of Mechanical and Nuclear Engineering The Pennsylvania State University, University Park, PA 16802 USA

Bibliografia

• [1] Venkayya V.B., Design of optimum structures, Computers and Structures 1971, 1, 265-309.
• [2] Berke L., An efficient approach to the minimum weight design of deflection limited structures, AFFD1-TM-70-4-FDTR, Flight Dynamics Laboratory, Wright Patterson AFB, OH 1970.
• [3] Khot N.S., Berke L., Venkayya V.B., Comparison of optimality criteria algorithms for minimum weight design of structures, AIAA Journal 1979, 17(2), 182-190.
• [4] Dobbs M.W., Nelson R.B., Application of optimality criteria to automated structural design, AIAA Journal 1975, 14(10), 1436-1443.
• [5] Khan M.R., Willmert K.D., Thornton W.A., Optimality criterion method for large-scale structures, AIAA Journal 1979, 17(7), 753-761.
• [6] McGee O.G., Phan K.F., A robust optimality criteria procedure for cross-sectional optimization of frame structures with multiple frequency limits, Computers and Structures 1991, 38(5/6), 485--500.
• [7] Yin L., Yang W., Optimality criteria method for topology optimization under multiple constraints, Computers and Structures 2001, 79, 1839-1850.
• [8] Belegundu A.D., A general optimality criteria algorithm for a class of engineering optimization problems, Engineering Optimization 2015, 47(5), 674-688.
• [9] Patriksson M., A survey on the continuous nonlinear resource allocation problem, European Journal of Operational Research 2008, 185(1), 1-46.
• [10] Jose J.A., Klein C.A., A note on multi-item inventory systems with limited capacity, Operations Research Letters 1988, 7(2), 71-75.
• [11] Atkinson K.E., An Introduction to Numerical Analysis, John Wiley, 1978.
• [12] Bryant V., Metric Spaces: Iteration and Application, Cambridge University Press, 1985.
• [13] Belegundu A.D., Chandrupatla T.R., Optimization Concepts and Applications in Engineering, 2nd edition, Cambridge University Press, 2011.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).