International Journal of Advanced and Applied Sciences
Int. j. adv. appl. sci.
EISSN: 2313-3724
Print ISSN:2313-626X
Volume 3, Issue 7 (July 2016), Pages: 89-93
Title: A note on non‐smooth programming problems
Authors: Mohammad Mehdi Mazarei *, Ali Vahidian Kamyad, Ali Asghar Behroozpoor
Affiliation(s):
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, International Campus, Mashhad, Iran
http://dx.doi.org/10.21833/ijaas.2016.07.014
Abstract:
In this paper, we introduce a new approach to obtain a novel numerical solution of nonlinear programming problems (NLP) which the objective function (functions) or constraint function (functions) are non-smooth ones. This technique is based on a new piecewise linearization approach. In fact, we transfer the nonlinear programming problem (NLP) to a variational problem that would reduce the new approximated problem to a linear programming problem (LP). Then, the approximated solution of the original problem would be obtained by the LP problem. Finally, numerical examples are given to show the efficiency of the proposed approach.
© 2016 The Authors. Published by IASE.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Nonlinear programming, Linear programming, Piecewise linearization, Non-smooth function
Article History: Received 15 June 2016, Received in revised form 29 July 2016, Accepted 30 July 2016
Digital Object Identifier: http://dx.doi.org/10.21833/ijaas.2016.07.014
Citation:
Mazarei MM, Vahidian Kamyad A, Behroozpoor AS (2016). A note on non‐smooth programming problems. International Journal of Advanced and Applied Sciences, 3(7): 89-93
http://www.science-gate.com/IJAAS/V3I7/Mazarei.html
References:
| Abhishek K, Leyffer S and Linderoth JT (2010). Modeling without categorical variables: a mixed-integer nonlinear program for the optimization of thermal insulation systems. Optimization and Engineering, 11(2): 185-212. http://dx.doi.org/10.1007/s11081-010-9109-z |
||||
| Bragalli C, D'Ambrosio C, Lee J, Lodi A and Toth P (2006). An MINLP solution method for a water network problem. In European Symposium on Algorithms, Springer Berlin Heidelberg, Lecture Notes in Computer Science, Berlin: Springer, 4168: 696-707 http://dx.doi.org/10.1007/11841036_62 |
||||
| Corsano G, Vecchietti AR and Montagna JM (2011). Optimal design for sustainable bioethanol supply chain considering detailed plant performance model. Computers and Chemical Engineering, 35(8): 1384-1398. http://dx.doi.org/10.1016/j.compchemeng.2011.01.008 |
||||
| Griffith RE and Stewart RA (1961). A nonlinear programming technique for the optimization of continuous processing systems. Management Science, 7(4): 379-392. http://dx.doi.org/10.1287/mnsc.7.4.379 |
||||
| Grossmann IE and Sargent RW (1979). Optimum design of multipurpose chemical plants. Industrial and Engineering Chemistry Process Design and Development, 18(2): 343-348. http://dx.doi.org/10.1021/i260070a031 |
||||
| Guerra A, Newman AM and Leyffer S (2011). Concrete structure design using mixed-integer nonlinear programming with complementarity constraints. SIAM Journal on Optimization, 21(3): 833-863. http://dx.doi.org/10.1137/090778286 |
||||
| Han SP (1976). Superlinearly convergent variable metric algorithms for general nonlinear programming problems. Mathematical Programming, 11(1): 263-282. http://dx.doi.org/10.1007/BF01580395 |
||||
| Han SP (1977). A globally convergent method for nonlinear programming. Journal of optimization theory and applications, 22(3): 297-309. http://dx.doi.org/10.1007/BF00932858 |
||||
| Harjunkoski I, Westerlund T and Pörn R (1999). Numerical and environmental considerations on a complex industrial mixed integer non-linear programming (MINLP) problem. Computers and Chemical Engineering, 23(10): 1545-1561. http://dx.doi.org/10.1016/S0098-1354(99)00310-5 |
||||
| Kamyad AV, Mehne HH and Borzabadi AH (2005). The best linear approximation for nonlinear systems. Applied Mathematics and Computation, 167(2): 1041-1061. http://dx.doi.org/10.1016/j.amc.2004.08.002 |
||||
| Misener R and Floudas CA (2012). Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations. Mathematical Programming, 136(1): 155-182. http://dx.doi.org/10.1007/s10107-012-0555-6 |
||||
| Murray W and Shanbhag UV (2006). A local relaxation approach for the siting of electrical substations. Computational Optimization and Applications, 33(1): 7-49. http://dx.doi.org/10.1007/s10589-005-5957-4 |
||||
| Nesterov Y (2005). Smooth minimization of non-smooth functions. Mathematical Programming, 103(1): 127-152. http://dx.doi.org/10.1007/s10107-004-0552-5 |
||||
| Pardalos PM, Chaovalitwongse W, Iasemidis LD, Sackellares JC, Shiau DS, Carney PR and Yatsenko VA (2004). Seizure warning algorithm based on optimization and nonlinear dynamics. Mathematical Programming, 101(2): 365-385. http://dx.doi.org/10.1007/s10107-004-0529-4 |
||||
| Rockafellar RT (1994). Nonsmooth optimization. Mathematical Programming: State of the Art, University of Michigan Press, Ann Arbor: 248–258. | ||||