ORIGINAL_ARTICLE
Approximate Pareto Optimal Solutions of Multi objective Optimal Control Problems by Evolutionary Algorithms
In this paper an approach based on evolutionary algorithms to find Pareto optimal pair of state and control for multi-objective optimal control problems (MOOCP)'s is introduced. In this approach, first a discretized form of the time-control space is considered and then, a piecewise linear control and a piecewise linear trajectory are obtained from the discretized time-control space using a numerical method. To do that, a modified version of two famous evolutionary genetic algorithm (GA) and particle swarm optimization (PSO) to obtain Pareto optimal solutions of the problem is employed. Numerical examples are presented to show the efficiency of the given approach.
https://mathco.journals.pnu.ac.ir/article_2033_6025868ddbdd4dc87ad0fcbab8275cf9.pdf
2016-08-01
1
19
Multi-objective optimal control problem
Pareto solution
Evolutionary algorithm
Discretization
Approximation
Akbar
Hashemi Borzabadi
borzabadi@du.ac.ir
1
damghan university
LEAD_AUTHOR
Manije
Hasanabadi
manijehasanabadi@yahoo.com
2
Damghan University
AUTHOR
Navid
Sadjadi
sadjadi@autom.uva.es
3
University of Valladolid
AUTHOR
[1] Agnieszka B. M., Delfim F. M. T. (2007) " Nonessential functionals in multi-objective optimal control problems ", Proceedings of the Estonian Academy of Sciences, series Physics and Mathematics and Chemistry, 56, 336-346.
1
[2] Ahmad I., Sharma S. (2010) " Sufficiency and duality for multi-objective variational control problems with generalized $(F,alpha ,rho ,theta )$-V-convexity ", Nonlinear Analysis, 72, 2564-2579.
2
[3] Andersson J. (2000) " A survey of multi-objective optimization in engineering design ", Technical report LiTH-IKP-R-1097, Department of Mechanical Engineering, Linkِping University, Linkِping, Sweden.
3
[4] Bonnel H., YalçnKaya C. (2010) " Optimization over the efficient set of multi-objective convex optimal control problems", Journal of Optimization: Theory and Applications, 147, 93-112.
4
[5] Coello C. A., Lechuga M.S. (2002) " MOPSO: A proposal for multiple objective particle swarm optimization ", In Proceeding Congress on Evolutionary Computation (CEC'2002), Honolulu, HI., 1, 1051-1056.
5
[6] Dahleh M. A., Diaz-Bobillo I. J. (1995) " Control of uncertain systems: A linear programming approach ", Englewood Cliffs, NJ.: Prentice-Hall.
6
[7] Deb K., Pratap A., Agarwal S., Meyarivan T. (2002) " A fast and elitist multi-objective genetic algorithm: NSGA-II ", Transactions on Evolutionary Computation, 6.
7
[8] Eberhart R.C., Simpson P., Dobbins R. (1996) " Computational intelligence PC tools ", Academic Press Professional, San Diego, CA., 212-226.
8
[9] El-Kady M. M., Salim M. S., El-Sagheer A. M. (2003) " Numerical treatment of multi-objective optimal control problems ", Automatica, 39, 47--55.
9
[10] Gambier A., Bareddin E. (2007) " Multi-objective optimal control: An overview ", IEEE Conference on Control Applications, CCA., Singapore, 170-175.
10
[11] Gambier A., Jipp M. (2011) " Multi-objective optimal control: An introduction ", Proceedings of the 8th Asian Control Conference (ASCC'11), Kaohsiung, Taiwan, 15-18.
11
[12] Guzma'n M. A., Delgado A., Carvalho J. D. (2010) "A novel multi-objective optimization algorithm based On bacterial chemotaxis ", Engineering Applications of Artificial Intelligence, 23, 292-301.
12
[13] Hiroyasu T., Miki M., Kamiura J., Watanabe S., Hiroyasu H. (2002) " Multi-objective optimization of Diesel engine emissions and fuel economy using genetic algorithms and phenomenological model ", Society of Automotive Engineer Inc., 1-12.
13
[14] Hu, Z., Salcudean S. E., Loewen D. (1998) "A Numerical solution of multi-objective control problems ", Journal of Optimal Control Applications and Methods, 19, 411-422.
14
[15] Kennedy J., Eberhart R. C. (1995) " Particle swarm optimization ", Proceedings of 4th IEEE International Conference on Neural Networks. Journal of IEEE, Piscataway, NJ., 1942-1948.
15
[16] Knowles J. D., Corne D.W. (2000) " Approximating the non-dominated front using the Pareto archived evolution strategy ", Journal of Evolutionary Computation, 8, 149-172.
16
[17] Kumar A., Vladimirsky A. (2010) " An efficient method for multi-objective optimal control and optimal control subject to integral constraints ", Journal of Computational Mathematics, 28, 517-551.
17
[18] Kundu D., Suresh K., Ghosh S., Das S., Panigrahi B. K. (2011) " Multi-objective optimization with artificial weed colonies ", Journal of Information Science, 181, 2441-2454.
18
[19] Lin J. G. (1976) " Multi-objective problems: Pareto-optimal solutions by method of proper equality constraints ", IEEE Transactions on Automatic Control, 21, 641-650.
19
[20] Liu G. P., Yang J. B., Whidborne J. F. (2003) " Multi-objective optimization and control ", Research Studies Press Ltd., Exeter.
20
[21] Lozovanu D., Pickl S. (2007) " Algorithms for solving multi-objective discrete control problems and dynamic c-games on networks ", Journal of Discrete Applied Mathematics, 155, 1846-1857.
21
[22] Maity k., Maiti M. (2005) " Numerical approach of multi-objective optimal control problem in imprecise environment ", Fuzzy Optimization and Decision Making, 4, 313-330.
22
[23] Milano F., Caٌizares C. A., Invernizzi M. (2003) " Multi-objective optimization for pricing system security in electricity markets ", IEEE Transactions on Power Systems,18, 596-604.
23
[24] Rangan S., Poolla K. (1997) " Weighted optimization for multi-objective full information control problems ", Journal of Systems and Control Letters, 31, 207-213.
24
[25] Shi Y., Eberhart R. C. (1998) " A modified particle swarm optimizer ", in Proceedings of IEEE International Conference on Evolutionary Computation, Piscataway, NJ., 69-73.
25
[26] Srinivas N., Deb K. (1995) " Multi objective function optimization using non-dominated sorting genetic algorithms ", Journal of Evolutionary Computation, 2, 221-248.
26
[27] Wajge R. M., Gupta S. K. (1994) " Multi-objective dynamic optimization of a nonvaporizing nylon 6 batch reactor ", Polymer Engineering and Science, 34, 1161-1174.
27
[28] Wang N. F., Tai K. (2010) " Target matching problems and an adaptive constraint strategy for multi-objective design optimization using genetic algorithms ", Journal of Computers Structures, 88, 1064- 1076.
28
[29] Zarei H., Kamyad A. V., Effati S. (2010) " Multi-objective optimal control of HIV dynamics ", Journal of Mathematical Problems in Engineering, 1-15.
29
[30] Zitzler E., Deb K., Thiele L. (2000) " Comparison of multi-objective evolutionary algorithms: Empirical results ", Journal of Evolutionary Computation, 8, 173-195.
30
[31] Zitzler E., Laumanns M., Thiele L. (2001) " SPEA2: Improving the strength Pareto evolutionary algorithm ", Zurich, Switzerland: Swiss Federal Institute Technology.
31
[32] Zitzler E., Thiele L. (1999) " Multi-objective evolutionary algorithms: A comparative case study and the strength Pareto approach ", IEEE Transactions on Evolutionary Computation, 3, 257-271.
32
ORIGINAL_ARTICLE
Regularity Conditions for Non-Differentiable Infinite Programming Problems using Michel-Penot Subdifferential
In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are locally Lipschitz. Necessary optimality conditions and regularity conditions are given. Our approach are based on the Michel-Penot subdifferential.
https://mathco.journals.pnu.ac.ir/article_2036_b4c59900a48266b0ad2b6385eac8fc8b.pdf
2016-08-01
21
30
Programming problem
Regularity conditions
Optimality condition
Michel-Penot subdifferential
Nader
Kanzi
nad.kanzi@gmail.com
1
payame Noor university of Yazd
LEAD_AUTHOR
[1] Borwein J. M., Zhu Q. J. (1999) " A Survey of subdifferential calculus with applications '', Nonlinear Analysis, 38, 687-773.
1
[2] Hiriart-Urruty J. B., Lemarechal C. (1991) " Convex analysis and minimization algorithms, I & II ", Springer, Berlin, Heidelberg.
2
[3] Giorgi J., Gwirraggio A., Thierselder J. (2004) " Mathematics of optimization; smooth and non-smooth cases ", Elsivier.
3
[4] Michel P., Penot J. P. (1984) " Calculsous differentiel pour des fonctions lipschitziennes et non-lipschitziennes '', Academic Sciences Paris (I); Mathematics, 12, 269-272.
4
[5] Michel P., Penot J. P. (1992) " A generalized derivative for calm and stable functions '', Differential and Integral Equations, 5, 433- 454.
5
[6] Mordukhovich B. S., Nghia T. T. A. (2013) " Constraint qualification and optimality conditions in semi-infinite and infinite programming '', Mathematical Programming, 139, 271-300.
6
[7] Mordukhovich B. S., Nghia T. T. A. (2012) " Nonsmooth cone-constrained optimization with applications to semi-infinite programming '', Optimization, online, 3/3396.
7
[8] Mordukhovich B. S., Nghia T. T. A. (2011) " Sub-differentials of non-convex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data '', Optimization, online, 12/3261.
8
ORIGINAL_ARTICLE
On Efficiency of Non-Monotone Adaptive Trust Region and Scaled Trust Region Methods in Solving Nonlinear Systems of Equations
In this paper we run two important methods for solving some well-known problems and make a comparison on their performance and efficiency in solving nonlinear systems of equations. One of these methods is a non-monotone adaptive trust region strategy and another one is a scaled trust region approach. Each of methods showed fast convergence in special problems and slow convergence in other ones; we try to categorize these problems and find out that which method has better numerical behavior. The robustness of methods is demonstrated by numerical experiments.
https://mathco.journals.pnu.ac.ir/article_2035_11095e0649e2c164939bca8bed4acb50.pdf
2016-08-01
31
40
Non-monotone adaptive
Scaled trust region
Nonlinear systems of equations
Numerical comparison
Rasoul
Hekmati
rhekmati@math.uh.edu
1
University of Houston
LEAD_AUTHOR
[1] Bellavia S., Macconi M., Morini, B. (2004) " STRSCNE: A scaled trust region solver for constrained nonlinear equations '', Computational Optimization and Applications, 28, 31-50.
1
[2] Bellavia S., Macconi M., Morini B. (2013) " An affine scaling trust-region method approach to bound-constrained nonlinear systems '', Applied Numerical Mathematics, 44, 257-280.
2
[3] Fletcher R. (1987) " Practical methods of optimizations '', New York: John wiley & sons.
3
[4] Fletcher R. (1980) " Practical method of optimization, unconstrained Optimization '', Vol. 1, John Wiley, New York.
4
[5] Floudas C. A., Pardalos P. M., Adjiman C., Esposito W. R., Gümüs Z. H., Harding S. T., Klepeis J. L., Meyer C. A., Schweiger C. A. (1991) " Handbook of test problems in local and global optimization, nonconvex optimization and its applications ", Vol. 33, Kluwer Academic, Dordrecht.
5
[6] Hong-Wei L. (2011) " A non-monotone adaptive trust region algorithm for nonlinear equations '', The tenth International Symposium on Operation Research and its Applications (ISORA), Dunhuang, China, 28-31. [7] Kanzow C. (2001) " An active set-type newton method for constrained nonlinear systems'', Complementarity: Applications, Algorithms and Extensions Applied Optimization, 50, 179-200.
6
[8] Ferris O. L., Mangasarian J. S., Pang J. S. " Complementarity: Applications, algorithms and extensions ", Kluwer Academic, Dordrecht, Springer.
7
[9] Levenberge K. (1944) "A method for the solution of certain nonlinear problem in least squares '', Quarterly Journal of Applied Mathematics, 2, 2, 164-166.
8
[10] Marquardt D. W. (1963) " An algorithm for least-squares estimation of nonlinear inequalities '', SIAM Journal on Applied Mathematics, 11, 2, 431-441.
9
[11] More J. J. (1983) " Recent developments in algorithms and software for trust region methods, mathematical programming ", The State of the Art, (Edited by A. Bachem, M. Gortchel and B. Korte) Springer-Verlag, Berlin, 258-287.
10
[12] More J. J., Garbow B. S., Hillstorm K. E. (1981) " Testing unconstrained optimization software '', ACM Transaction on Mathematical Software, 7, 17-41.
11
[13] Nocedal J., Wright S. J. (1999) " Numerical optimization '', Springer Series in Operation Research, Springer, New York.
12
[14] Yuan Y. (1996) " On the convergence of the trust region algorithms '', Mathematica Numerica Sinica, 16333-346.
13
[15] Yuan Y., Sun W. (1997) " Optimization theory and algorithm '', Scientific Publishing House, China.
14
ORIGINAL_ARTICLE
A New Measure for Evaluating the Efficiency of Human's Resources in University
In this paper we try to introduce a new approach and study the notion of efficiency under a multi objectives linear programming problem in the university by using analysis of hierarchy process (AHP). To this end, we first extract some effective parameters due to efficiency offices in university and then prioritized these parameters by the AHP method. Hence, we could classify the most important factors of people's dissatisfaction in the offices and could underlie further studies in related offices to evaluate the efficiency and also effective factors for increasing the efficiency. More clearly, a mathematical model is suggested to calculate the amount of efficiency under a multi objectives linear programming problem and then it is solved by using the existing methods. Note that in order to examine the approach's performance, the Payame Noor University of Mashhad (PNUM) is selected as a case study. Numerical experiments are included to illustrate the effectiveness of the proposed approach.
https://mathco.journals.pnu.ac.ir/article_2031_67e30f9b118bc22227c768a31ee37fdb.pdf
2016-08-01
41
53
Efficiency
AHP Method
Multi Objectives Linear Programming Problem
Aghile
Heydari
a_heidari@pnu.ac.ir
1
Payame Noor university
AUTHOR
Hamid Reza
Yousefzadeh
usefzadeh.math@pnu.ac.ir
2
Payame Noor university
LEAD_AUTHOR
[1] Akhundi R. N. (2007) " Mathematical structure of fuzzy for AHP method ", MSc Thesis, Ferdowsi University of Mashhad.
1
[2] Bessent A., Bessent W., Kennington J., and Reagan B. (1982) " An application of mathematical programming to assess productivity ", Houston Independent school District, INFORMS.
2
[3] Dezgahi A. (1376) " The study of fuzzy multi objectives linear programming ", Msc. Thesis, Ferdowsi University of Mashhad.
3
[4] GhodsiPour H. (2009) " Analysis of hierarchy process (AHP) ", Amir Kabir University Publications, (in Persian).
4
[5] GhodsiPour H. (2006) " Topics in multi-objectives programming multi-criterions decision making ", Amir Kabir University Publications.
5
[6] Ghomi S. (2010) " Creating intelligent feedback for categorizing fuzzy multi-criterions AHP " ,MSc. Thesis, Ferdowsi University of Mashhad.
6
[7] Katta G. (1983) " Linear programming ", John Wiley & Sons.
7
[8] Olsen D. (2008) " Multiple-criteria decision- making methods ", Marandiz Publications.
8
[9] Oreiee K. (2009) " Productivity in industry ", Jahad Daneshgahi Publications.
9
[10] Pirayesh R. (2008) " Productivity management ", Zanjan University Publications.
10
[11] Putti J. (1986) " Understanding productivity ", Federal Publications.
11
[12] Sabeghi S. (2013) " Increasing productivity of human resources in Payame Noor University of Mashhad ", MSc. Thesis.
12
[13] Sakhaee B. (2012) " Using fuzzy sets of type 2 in AHP method ", Msc. Thesis, Ferdowsi University of Mashhad.
13
[14] Zibaee M. (2006) " Supporting systems for AHP via fuzzy logic ", Master Thesis, Ferdowsi University of Mashhad.
14
ORIGINAL_ARTICLE
Solving Linear Semi-Infinite Programming Problems Using Recurrent Neural Networks
Linear semi-infinite programming problem is an important class of optimization problems which deals with infinite constraints. In this paper, to solve this problem, we combine a discretization method and a neural network method. By a simple discretization of the infinite constraints,we convert the linear semi-infinite programming problem into linear programming problem. Then, we use a recurrent neural network model, with a simple structure based on a dynamical system to solve this problem. The portfolio selection problem and some other numerical examples are solved to evaluate the effectiveness of the presented model.
https://mathco.journals.pnu.ac.ir/article_2034_0faff11932ef505dc435167738955b71.pdf
2016-08-01
55
67
Linear semi-infinite programming
Recurrent neural network
Dynamical system
Discretization
Linear programming
Alaeddin
Malek
mala@modares.ac.ir
1
Tarbiat Modarres university
AUTHOR
Ghasem
Ahmadi
g.ahmadi@pnu.ac.ir
2
Payame Noor university
LEAD_AUTHOR
Seyyed Mehdi
Mirhoseini Alizamini
seyyedmehdi_mirhosseini@yahoo.com
3
Payame Noor university
AUTHOR
[1] Abbe L. (2001) " Two logarithmic barrier methods for convex semi-infinite programming '', In M. A. Goberna and M. A. Lopez, (editors) Semi-infinite Programming Recent Advances, Nonconvex Optimization and Its Applications 57, 169-195.
1
[2] Alipour M., Rostamy D., Malek, A. (2011) " A recurrent neural network for nonlinear convex optimization with application to a class of variational inequalities problems '', Australian Journal of Basic and Applied Sciences, 5, 5, 894-909.
2
[3] Anderson E. J., Levis A. S. (1989) " An extension of the simplex algorithm for semi-infinite linear programming '', Mathematical Programming 44, 247-269.
3
[4] Chen Y. H., Fang S. C. (1998) " Solving convex programming with equality constraints by neural networks '', Computers & Mathematics with Applications, 36, 41-68.
4
[5] Ferris M. C., Philpott A., B. (1989) " An interior point algorithm for semi-infinite linear programming '', Mathematical Programming 43, 257-276.
5
[6] Floudas C. A., Stein O. (2007) " The adaptive convexification algorithm: A feasible point method for semi-infinite programming '', SIAM Journal on Optimization, 18, 4, 1187-1208.
6
[7] Goberna M. A. (2007) " Semi-infinite programming '', European Journal of Operational Research, 180, 491-518.
7
[8] Gustafson S. (1979) " On numerical analysis in semi-infinite programming '', In Semi-Infinite Programming (Lecture Notes in Control and Information Sciences), 15, 51-65.
8
[9] Hettich R., Kortanek K. O. (1993) " Semi-infinite programming: Theory, methods and applications '', SIAM Review, 35, 3, 380-429.
9
[10] Lopez M., Still G.} (2007) " Semi-infinite programming '', European Journal of Operational Research, 180, 491-518.
10
[11] Malek A., Yari A.(2005) " Primal dual solution for the linear programming problems using neural networks '', Applied Mathematics and Computation, 167, 198-211.
11
[12] Malek A., Yashtini M. (2010) " Image fusion algorithms for color and gray level images based on LCLS method and novel artificial neural network '', Neuro computing, 73, 937-943.
12
[13] Reemsten R., Gorner S. (1998) " Numerical methods for semi-infinite programming: A survey In R. Reemsten and J. Rueckmann (editors), Semi-infinite Programming, Non-convex Optimization and Its Applications, 25, 101, 195-275.
13
[14] Stein O., Still G. (2003) " Solving semi-infinite optimization problems with interior point techniques '', SIAM Journal on Control and Optimization, 42, 3, 769-788.
14
[15] Tank D. W., Hopfield J. J. (1986) " Simple neural optimization network: An A/D convertor, signal decision circuit, and a linear programming circuit '', IEEE Transactions on Circuits and Systems, 35, 533-541.
15
[16] Vazquez F. G., Ruckmann J. J., Stein O., Still G. (2008) " Generalized semi-infinite programming: A tutorial '', Journal of Computational and Applied Mathematics 217, 394-419.
16
[17] Vaz F. I. A., Fernandes E. M. G. P., Gomes M. P. S. F. (2003) " Robot trajectory planning with semi-infinite programming '', European Journal of Operational Research, 153, 607–617.
17
[18] Vaz F. I. A., Ferreira E. C. (2009) "Air pollution control with semi-infinite programming '', Applied Mathematical Modelling 33, 1957-1969.
18
[19] Xia Y., Leung H., Wang J.} (2002) " A Projection neural network and its application to constrained optimization problems '', IEEE Transactions on Circuits and Systems—I: Fundamental theory and applications, 49, 4, 447-458.
19
[20] Xia Y., Wang J. (2004) "A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints'', IEEE Transactions on Circuits and Systems-I: Regular Papers, 51, 7, 1385-1394.
20
ORIGINAL_ARTICLE
Solving Fully Fuzzy Linear Programming Problems with Zero-One Variables by Ranking Function
Jahanshahloo has suggested a method for the solving linear programming problems with zero-one variables. In this paper we formulate fully fuzzy linear programming problems with zero-one variables and a method for solving these problems is presented using the ranking function and also the branch and bound method along with an example is presented.
https://mathco.journals.pnu.ac.ir/article_2032_bddde463bc52ea73351058285d33fa3e.pdf
2016-08-01
69
78
Fuzzy set
Fuzzy number
Ranking function
Triangular fuzzy number
Zero-one triangular fuzzy number
Aminalah
Alba
alba_1359@yahoo.com
1
Teacher
LEAD_AUTHOR
[1] Bellman R. E., Zadeh L. A. (1970) " Decision making in a fuzzy environment '', Management Sciences, 17, 141-164.
1
[2] Campos L., Verdegay J. L. (1989) " Linear programming problems and ranking of fuzzy numbers '', Fuzzy Set and Systems, Elsevier Science Publishers, 32, 1-11.
2
[3] Ebrahimnejad A., Nasseri Sh., Lotfi F. H., Soltanifar M. (2010) " A primal- dual method for linear programming problems with fuzzy variables '', European Journal of Industrial Engineering, 4, 189-209.
3
[4] Ganesan K., Veeramani P. (2006) " Fuzzy linear programs with trapezoidal fuzzy numbers '', Annals Of Operation Research-incl, 143, 305-315.
4
[5] Jahanshahloo Gh. (2003) " Operations research 2 '', Pnu, 74, Tehran.
5
[6] Liou T. S., Wang M. J. (1992) " Ranking fuzzy numbers with integral value '', Fuzzy Sets and Systems, Elsevier Science Publishers, 50, 247-255.
6
[7] Maleki H. R. (2002) " Ranking function and their application to fuzzy linear programming '', Far East Journal of Mathematics Sciences, 4, 283-301.
7
[8] Maleki H. R., Tata M., Mashinchi M. (2000) " Linear programming with fuzzy variables '', Fuzzy Set and System, Elsevier Science Publishers, 109, 21-33.
8
[9] Tanaka H., Okada T., Asai K. (1973) " On fuzzy mathematical programming '', Journal of Cybernetics Systems, 3, 37-46.
9
[10] Zimmerman H. J. (1978) " Fuzzy programming and linear programming with several objective function '', Fuzzy Set and Systems, Elsevier Science Publishers, 1, 45-55.
10