Unified global optimality conditions for smooth minimization problems with mixed variables

Vaithilingam Jeyakumar; Sivakolundu Srisatkunarajah; Nguyen Quang Huy

RAIRO - Operations Research (2008)

  • Volume: 42, Issue: 3, page 361-370
  • ISSN: 0399-0559

Abstract

top
In this paper we establish necessary as well as sufficient conditions for a given feasible point to be a global minimizer of smooth minimization problems with mixed variables. These problems, for instance, cover box constrained smooth minimization problems and bivalent optimization problems. In particular, our results provide necessary global optimality conditions for difference convex minimization problems, whereas our sufficient conditions give easily verifiable conditions for global optimality of various classes of nonconvex minimization problems, including the class of difference of convex and quadratic minimization problems. We discuss numerical examples to illustrate the optimality conditions

How to cite

top

Jeyakumar, Vaithilingam, Srisatkunarajah, Sivakolundu, and Huy, Nguyen Quang. "Unified global optimality conditions for smooth minimization problems with mixed variables." RAIRO - Operations Research 42.3 (2008): 361-370. <http://eudml.org/doc/250403>.

@article{Jeyakumar2008,
abstract = { In this paper we establish necessary as well as sufficient conditions for a given feasible point to be a global minimizer of smooth minimization problems with mixed variables. These problems, for instance, cover box constrained smooth minimization problems and bivalent optimization problems. In particular, our results provide necessary global optimality conditions for difference convex minimization problems, whereas our sufficient conditions give easily verifiable conditions for global optimality of various classes of nonconvex minimization problems, including the class of difference of convex and quadratic minimization problems. We discuss numerical examples to illustrate the optimality conditions },
author = {Jeyakumar, Vaithilingam, Srisatkunarajah, Sivakolundu, Huy, Nguyen Quang},
journal = {RAIRO - Operations Research},
keywords = {Nonconvex optimization; global optimization; optimality conditions; discrete constraints; box constraints; difference of convex functions; quadratic minimization.; nonconvex optimization; global optimization; box constraints; quadratic minimization},
language = {eng},
month = {8},
number = {3},
pages = {361-370},
publisher = {EDP Sciences},
title = {Unified global optimality conditions for smooth minimization problems with mixed variables},
url = {http://eudml.org/doc/250403},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Jeyakumar, Vaithilingam
AU - Srisatkunarajah, Sivakolundu
AU - Huy, Nguyen Quang
TI - Unified global optimality conditions for smooth minimization problems with mixed variables
JO - RAIRO - Operations Research
DA - 2008/8//
PB - EDP Sciences
VL - 42
IS - 3
SP - 361
EP - 370
AB - In this paper we establish necessary as well as sufficient conditions for a given feasible point to be a global minimizer of smooth minimization problems with mixed variables. These problems, for instance, cover box constrained smooth minimization problems and bivalent optimization problems. In particular, our results provide necessary global optimality conditions for difference convex minimization problems, whereas our sufficient conditions give easily verifiable conditions for global optimality of various classes of nonconvex minimization problems, including the class of difference of convex and quadratic minimization problems. We discuss numerical examples to illustrate the optimality conditions
LA - eng
KW - Nonconvex optimization; global optimization; optimality conditions; discrete constraints; box constraints; difference of convex functions; quadratic minimization.; nonconvex optimization; global optimization; box constraints; quadratic minimization
UR - http://eudml.org/doc/250403
ER -

References

top
  1. A. Beck and M. Teboulle, Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optim.11 (2000) 179–188.  Zbl0990.90089
  2. E. Cela, The quadratic assignment problem: theory and algorithms. Kluwer Academic Publishers (1998).  Zbl0909.90226
  3. P. De Angelis, P. Pardalos and G. Toraldo, Quadratic programming with box constraints, in Developments in global optimization, edited by Bomze I. et al. Kluwer Acad. Publ., Dordrecht (1997) 73–93.  Zbl0886.90130
  4. C.A. Floudas and P.M. Pardalos, Optimization in computational chemistry and molecular biology: Local and global approaches. Kluwer Academic Publishers (2000).  Zbl0936.00053
  5. N.Q. Huy, V. Jeyakumar and G.M. Lee, Sufficient global optimality conditions for multi-extremal smooth minimization problems with bounds and linear matrix inequality constraints, The ANZIAM J.47 (2006) 439–450.  Zbl1113.90147
  6. N.Q. Huy, V. Jeyakumar and G.M. Lee, Globally minimizing smooth functions with simple constraints: Necessary, and sufficient optimality conditions (submitted).  
  7. V. Jeyakumar and N.Q. Huy, Global minimization of difference of quadratic and convex functions over box or binary constraints, To appear in Optimization Letters, DOI: (2008).  Zbl1161.90465DOI10.1007/s11590-007-0053-6
  8. V. Jeyakumar, A.M. Rubinov and Z.Y. Wu, Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints. J. Global Optim.36 (2006) 461–468.  Zbl1131.90060
  9. V. Jeyakumar, A.M. Rubinov and Z.Y. Wu, Nonconvex quadratic minimization with quadratic constraints: Global optimality conditions. Math. Program.110 (2007) 521–541.  Zbl1206.90178
  10. M. Junger, A. Martin, G. Reinelt and R. Weismantel, 0/1 optimization and a decomposition approach for the placement of electronic circuits. Math. Program.63 (1994) 257–279.  Zbl0801.90079
  11. R.F. Marcia, J.C. Mitchell and J.B. Rosen, Iterative convex quadratic approximation for global optimization in protein docking, Comput. Optim. Appl.32 (2005) 285–297.  Zbl1125.90396
  12. P.M. Pardalos and G.P. Rodgers, Computational aspects of quadratic zero-one programming, Computing45 (1990) 131–144.  Zbl0721.65034
  13. M.C. Pinar, Sufficient global optimality conditions for bivalent quadratic optimization, J. Optim. Theor. Appl.122 (2004) 433–440.  Zbl1091.90059
  14. M.C. Pinar and M. Teboulle, On semidefinite bounds for maximization of a non-convex quadratic objective over l1 unit ball. RAIRO Oper. Res.40 (2006) 253–265.  Zbl1180.90222
  15. Z. Wu, V. Jeyakumar, A.M. Rubinov, Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints. J. Optim. Theor. Appl.133 (2007) 123–130.  Zbl1144.90458

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.