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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

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