zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A deterministic global optimization algorithm for generalized geometric programming. (English) Zbl 1105.65335
Summary: A deterministic global optimization algorithm is proposed for generalized geometric programming (GGP). By utilizing some transformations, the initial non-convex problem is reduced to a reverse convex programming (RCP), where the objective function and constraint functions are convex. Then a linear relaxation of the problem (RCP) is obtained based on the linear lower bounding functions of the convex constraint functions and the linear upper bounding functions of the reverse convex constraint functions inside some hyperrectangle region. A cutting-plane method is proposed to add some effective linear constraints to the linear relaxation programming based on the famous arithmetic-geometric mean inequality, then derive a tighter linear relaxation programming. The proposed global optimization algorithm which connects the branch and bound method with the cutting-plane method successfully is convergent to the global minimum through the successive refinement of the linear relaxation of the feasible region of the objective function and the solutions of a series of linear relaxation problems. Finally a numerical experiment is given to illustrate the feasibility and the robust stability of the present algorithm.

65K05Mathematical programming (numerical methods)
90C26Nonconvex programming, global optimization
90C57Polyhedral combinatorics, branch-and-bound, branch-and-cut
Full Text: DOI
[1] Avriel, M.; Williams, A. C.: An extension of geometric programming with applications in engineering optimization. Journal of engineering mathematics 5, No. 3, 187-199 (1971)
[2] Jefferson, T. R.; Scott, C. H.: Generalized geometric programming applied to problems of optimal control: I. Theory. Jota 26, 117-129 (1978) · Zbl 0369.90120
[3] Nand, K. J.: Geometric programming based robot control design. Computers and industrial engineering 29, No. 1-4, 631-635 (1995)
[4] Das, K.; Roy, T. K.; Maiti, M.: Multi-item inventory model with under imprecise objective and restrictions: a geometric programming approach. Production planning & control 11, No. 8, 781-788 (2000)
[5] Chul, C. Jae; Dennis, L. Bricker: Effectiveness of a geometric programming algorithm for optimization of machining economics models. Computers & operations research 23, No. 10, 957-961 (1996) · Zbl 0873.90043
[6] Barmi, H. Ei; Dykstra, R. L.: Restricted multinomial maximum likelihood estimation based upon Fenchel duality. Statistics and probability letters 21, 121-130 (1994) · Zbl 0801.62033
[7] D.L. Bricker, K.O. Kortanek, L. Xu, Maximum likelihood estimates with order restrictions on probabilities and odds ratios: A geometric programming aproach, Applied Mathematical and Computational Sciences, The University of IA, Iowa City, IA 52242, 1995. · Zbl 0953.62027
[8] Jagannathan, R.: A stochastic geometric programming problem with multiplicative recourse. Operations research letters 9, 99-104 (1990) · Zbl 0703.90069
[9] Sönmez, A. I.; Baykasoglu, A.; Dereli, T.; Filiz, I. H.: Dynamic optimization of multipass milling operations via geometric programming. International journal of machine tools and manufacture 39, 297-320 (1999)
[10] Scott, C. H.; Jefferson, T. R.: Allocation of resources in project management. International journal on systems science 26, 413-420 (1995) · Zbl 0821.90069
[11] Maranas, C. D.; Floudas, C. A.: Global optimization in generalized geometric programming. Computers and chemical engineering 21, No. 4, 351-369 (1997)
[12] Rijckaert, M. J.; Martens, X. M.: Analysis and optimization of the Williams-otto process by geometric programming. Aiche journal 20, No. 4, 742-750 (1974)
[13] Passy, U.: Generalized weighted mean programming. SIAM journal on applied mathematics 20, 763-778 (1971) · Zbl 0233.90021
[14] Passy, U.; Wilde, D. J.: Generalized polynomial optimization. Journal on applied mathematics 15, No. 5, 1344-1356 (1967) · Zbl 0171.18002
[15] Kortanek, K. O.; Xu, X.; Ye, Y.: An infeasible interior-point algorithm for solving primal and dual geometric programs. Mathematical programming 76, 155-181 (1996) · Zbl 0881.90106
[16] Duffin, R. J.; Peterson, E. L.: Geometric programming with signomial. Journal of optimization theory and applications 11, No. 1, 3-35 (1973) · Zbl 0238.90069
[17] Horst, R.; Tuy, H.: Global optimization, deterministic approaches. (1990) · Zbl 0704.90057
[18] Zhang, K.: Geometric programming and optimal design. (1990)