January  2012, 8(1): 189-227. doi: 10.3934/jimo.2012.8.189

Convex optimization on mixed domains

1. 

Izmir University of Economics, Department of Mathematics, 35330, Balcova, Izmir, Turkey

2. 

Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906

Received  April 2011 Revised  August 2011 Published  November 2011

This paper aims to study convex analysis on some “generalized domains,” in particular, the domain of the product of closed subsets of reals. We introduce the basic concepts and derive analytic properties regarding convex subsets of mixed domains and convex functions defined on convex sets in mixed domains. The results obtained may open an avenue for modeling and solving a new type of optimization problems that involve both discrete and continuous variables at the same time.
Citation: Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial and Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189
References:
[1]

M. Adıvar and Y. N. Raffoul, Existence of resolvent for Volterra integral equations on time scales, Bull. of Aust. Math. Soc., 82 (2010), 139-155. doi: 10.1017/S0004972709001166.

[2]

M. Adıvar and Y. N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications, 58 (2009), 264-272. doi: 10.1016/j.camwa.2009.03.065.

[3]

M. Adıvar and Y. N. Raffoul, Existence results for periodic solutions of integro-dynamic equations on time scales, Annali di Matematica Pura ed Applicata (4), 188 (2009), 543-559. doi: 10.1007/s10231-008-0088-z.

[4]

M. Adıvar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 7519-7531.

[5]

D. R. Anderson, R. J. Krueger and A. C. Peterson, Delay dynamic equations with stability, Advances in Difference Equations, 2006, Article ID 94051, 1-19. doi: 10.1155/ADE/2006/94051.

[6]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics, Mathematical and Computer Modelling, 43 (2006), 718-726. doi: 10.1016/j.mcm.2005.08.014.

[7]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, "Nonlinear Programming Theory and Algorithms," 3rd edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0471787779.

[8]

M. Bohner and A. C. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Inc., Boston, MA, 2001.

[9]

M. Bohner and A. C. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Inc., Boston, MA, 2003.

[10]

M. Bohner and R. P. Agarwal, Oscillation and boundedness of solutions to first and second order forced dynamic equations with mixed nonlinearities, Australian Journal of Mathematical Analysis and Applications, 5 (2008), 12 pp.

[11]

M. Bohner, L. Erbe and A. C. Peterson, Oscillation for nonlinear second order dynamic equations on time scales, J. Math. Anal. Appl., 301 (2005), 491-507. doi: 10.1016/j.jmaa.2004.07.038.

[12]

V. I. Danilov and G. A. Koshevoĭ, Discrete convexity, Journal of Mathematical Sciences, 133 (2006), 1418-1421. doi: 10.1007/s10958-006-0057-2.

[13]

C. Dinu, Convex functions on time scales, Annals of the University of Craiova Ser. Math. Inform., 35 (2008), 87-96.

[14]

Y. Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales, Journal of Industrial and Management Optimization, 5 (2009), 1-10.

[15]

S. Hilger, Analysis on measure chains--a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56.

[16]

R. Hilscher, "Optimality Conditions for Time Scale Variational Problems," DSc dissertation, Masaryk University, Brno, 2008.

[17]

R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Analysis, 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025.

[18]

T. Kulik and C. C. Tisdell, Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains, Int. J. Difference Equ., 3 (2008), 103-133.

[19]

K. Murota, Discrete convex analysis, Mathematical Programming, 83 (1998), 313-371. doi: 10.1007/BF02680565.

[20]

R. T. Rockafellar, "Convex Analysis," Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997.

[21]

M. Z. Sarıkaya, N. Aktan, H. Yıldırım and K. İlarslan, Partial $\Delta$-differentiation for multivariable functions on $n$-dimensional time scales, Journal of Mathematical Inequalities, 3 (2009), 277-291.

[22]

M. Z. Sarıkaya, N. Aktan, H. Yıldırım and K. İlarslan, Directional $\nabla$-derivative and curves on $n$-dimensional time scales, Acta. Appl. Math, 105 (2009), 45-63. doi: 10.1007/s10440-008-9264-9.

[23]

C. C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal., 68 (2008), 3504-3524. doi: 10.1016/j.na.2007.03.043.

[24]

R. Oberste-Vorth, The Fell topology for dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9 (2009), 407-414.

show all references

References:
[1]

M. Adıvar and Y. N. Raffoul, Existence of resolvent for Volterra integral equations on time scales, Bull. of Aust. Math. Soc., 82 (2010), 139-155. doi: 10.1017/S0004972709001166.

[2]

M. Adıvar and Y. N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications, 58 (2009), 264-272. doi: 10.1016/j.camwa.2009.03.065.

[3]

M. Adıvar and Y. N. Raffoul, Existence results for periodic solutions of integro-dynamic equations on time scales, Annali di Matematica Pura ed Applicata (4), 188 (2009), 543-559. doi: 10.1007/s10231-008-0088-z.

[4]

M. Adıvar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 7519-7531.

[5]

D. R. Anderson, R. J. Krueger and A. C. Peterson, Delay dynamic equations with stability, Advances in Difference Equations, 2006, Article ID 94051, 1-19. doi: 10.1155/ADE/2006/94051.

[6]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics, Mathematical and Computer Modelling, 43 (2006), 718-726. doi: 10.1016/j.mcm.2005.08.014.

[7]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, "Nonlinear Programming Theory and Algorithms," 3rd edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0471787779.

[8]

M. Bohner and A. C. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Inc., Boston, MA, 2001.

[9]

M. Bohner and A. C. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Inc., Boston, MA, 2003.

[10]

M. Bohner and R. P. Agarwal, Oscillation and boundedness of solutions to first and second order forced dynamic equations with mixed nonlinearities, Australian Journal of Mathematical Analysis and Applications, 5 (2008), 12 pp.

[11]

M. Bohner, L. Erbe and A. C. Peterson, Oscillation for nonlinear second order dynamic equations on time scales, J. Math. Anal. Appl., 301 (2005), 491-507. doi: 10.1016/j.jmaa.2004.07.038.

[12]

V. I. Danilov and G. A. Koshevoĭ, Discrete convexity, Journal of Mathematical Sciences, 133 (2006), 1418-1421. doi: 10.1007/s10958-006-0057-2.

[13]

C. Dinu, Convex functions on time scales, Annals of the University of Craiova Ser. Math. Inform., 35 (2008), 87-96.

[14]

Y. Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales, Journal of Industrial and Management Optimization, 5 (2009), 1-10.

[15]

S. Hilger, Analysis on measure chains--a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56.

[16]

R. Hilscher, "Optimality Conditions for Time Scale Variational Problems," DSc dissertation, Masaryk University, Brno, 2008.

[17]

R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Analysis, 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025.

[18]

T. Kulik and C. C. Tisdell, Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains, Int. J. Difference Equ., 3 (2008), 103-133.

[19]

K. Murota, Discrete convex analysis, Mathematical Programming, 83 (1998), 313-371. doi: 10.1007/BF02680565.

[20]

R. T. Rockafellar, "Convex Analysis," Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997.

[21]

M. Z. Sarıkaya, N. Aktan, H. Yıldırım and K. İlarslan, Partial $\Delta$-differentiation for multivariable functions on $n$-dimensional time scales, Journal of Mathematical Inequalities, 3 (2009), 277-291.

[22]

M. Z. Sarıkaya, N. Aktan, H. Yıldırım and K. İlarslan, Directional $\nabla$-derivative and curves on $n$-dimensional time scales, Acta. Appl. Math, 105 (2009), 45-63. doi: 10.1007/s10440-008-9264-9.

[23]

C. C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal., 68 (2008), 3504-3524. doi: 10.1016/j.na.2007.03.043.

[24]

R. Oberste-Vorth, The Fell topology for dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9 (2009), 407-414.

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