REFERENCES

[1] R.K. Ahuja, T.L. Magnanti and J.B. Orlin (1993): Networks Flows: The­ory, Algorithms and Applications, Prentice Hall, Englewood ClifIs, N.J.

[2] A.D. Alexandrov (1949): On surfaces which may be represented by a dif­ference of convex functions (in Russian), lzvestiya Akademii Navk Kaza­khskoj SSR, Seria Fiziko-Matematicheskikh, 3, 3-20.

[3] A.D. Alexandrov (1950): On surfaces which may be represented by dif­ferences of convex functions (in Russian), Doklady Akademii Navk SSR, 72, 613-616.

[4] F.A. AI-Khayyal (1989): Jointly constrained bilinear programa and re­lated problems. An overview, Computers and Mathematics with Applica­tions, 19, 53-62.

[5] F.A. Al-Khayyal and J.E. Falk (1983): Jointly constrained biconvex pro­gramming, Mathematics of Operations Research, 8, 273-286.

[6] F. A. Al-Khayyal, C. Larsen and T. Van Voorhis (1995): A relaxation method for nonconvex quadratically constrained quadratic programs, Journal of Global Optimization, 6, 215-230.

[7] LP. Androulakis, C.D. Maranas and C.A. Floudas {1995}: aBB: A global optimization method for general constrained nonconvex problems, Jour­nal of Global Optimization, 7, 337-363.

[8] E. Asplund (1973): Differentiability of the metric projection in finite dimensional Euclidean space, Proceedings of the American Mathematical Society, 38, 218-219.

[9] M. Avriel and A.C. Williams (1970): Complementary Geometric Pro­gramming, SIAM Joumal on Applied Mathematics, 19, 125-141.

[10] E. Balas (1971): Intersection cuts- a new type of cutting planes for integer programming, Operations Research, 19, 19-39.

319

320 CONVEX ANALYSIS AND GLOBAL OPTIMIZATION

[11J E. Balas (1972): Integer programming and convex analysis: intersection cuts and outer polars, Mathematical Programming, 2, 330-382.

[12J E. Balas and C.A. Burdet (1973): Maximizing a convex quadratic func­tion subject to linear constraints, Management Science Research Report No 299, Carnegie-Melon University.

[13J P.P. Bansal and S.E. Jacobsen (1975): An algorithm for optimizing net­work flow capacity under economies of scale, Joumal of Optimization Theory and Applications, 15, 565-586.

[14J H.P. Benson (1995): Concave minimization: theory, applications and algorithms, in R. Horst and P. Pardalos (eds.), Handbook of Global Opti­mization, Kluwer Academic Publishers, 43-148.

[15J H. P. Benson (1996): Deterministic algorithms for constrained concave minimization: a unified critical survey, Naval Research Logistics, 43, 765-795.

[16J H.P.· Benson and R. Horst (1991): A branch and bound - outer ap­proximation for concave minimization over a a convex set, Joumal of Computers and Mathematics with Applications, 21, 67-76.

[17J H. Benson and S. Sayin (1994): A finite concave minimization algorithm using branch and bound and neighbor generation, Joumal of Global Op­timization, 5, 1-14.

[18J A. Ben-TaI, G. Eiger and V. Gershovitz (1994): Global minimization by reducing the duality gap, Mathematical Programming, 63, 193-212.

[19J J.L. Bentley and M.I. Shamos (1978): Divide and conquer for linear expected time, In! Proc. Lett. 7,87-91.

[20J L. Bittner (1970): Some representation theorems for functions and sets and their applications to nonlinear programming', Numerische Mathe­matik, 16, 32-51.

[21J A. Bui (1993) : Etude analytique d'algorithmes distribues de routage, These de doctorat, Universite Paris VII.

[22J V.P. Bulatov (1977): Embedding methods in optimization problems, Nauka, Novosibirsk (in Russian)

[23J V.P. Bulatov (1990): Methods for solving multiextremal problems (global search), Annals of Operations Research, 25, 253-278.

REFERENCES 321

[24] P.C. Chen, P. Hansen and B. Jaumard (1991): On-line and off-line vertex enumeration by adjacency lists, Operations Research Letters, 10, 403-409.

[25] P.-C. Chen, P. Hansen, B. Jaumard and H. Tuy (1992): Weber's problem with attraction and repulsion, Joumal of Regional Science, 32, 467-486.

[26] P.-C. Chen, P. Hansen, B. Jaumard and H. Tuy (1992): Solution of the multifacility Weber and conditional Weber problems by D.C. Program­ming, Cahier du GERAD G-92-35, Ecole Polytechnique, Montreal.

[27] L. Cooper (1963): Location-allocation problems, Operations Research, 11, 331-343.

[28] E.W. Cheney and A.A. Goldstein (1959): Newton's method for convex programming and Tchebycheff approximation, Numerishe Matematik, 1, 253-268.

[29] G.B. Dantzig (1963): Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey.

[30] L.L. Dines (1941): On themappingofquadraticforms, Bull. Amer. Math. Soc., 47, 494-498.

[31] L.C.W. Dixon and G.P. Szego .(eds.) (1975): Towards global optimization, Vol.I. North-Holland, Amsterdam.

[32] L.C.W. Dixon and G.P. Szego (eds.) (1978): Towards global optimiiation, Vol.II. North-Holland, Amsterdam.

[33] R.J. Duffin, E.L. Peterson and C. Zener (1967): Geometric Programming- Theory and Application, Wiley, New York.

[34] B.C. Eaves and W. 1. Zangwill (1971): Generalized cutting plane algo­rithms, SIAM Joumal on Control, 9, 529-542.

[35] R. Ellaia (1984): Contribution ă,l'analyse et l'optimisation de differences de fonctions convexes, These de trosieme cyc1e, Universite de Toulouse.

[36] J.E. Falk (1969): Lagrange multipliers and nonconvex problems, SIAM J. Control, 7, 534-545.

[37] J.E. Falk (1973a): Conditions for global optimality in nonlinear program­. ming, Operations Research, 21, 337-340.

[38] J.E. Falk (1973b): A linear max-min problem, Mathematical Program­ming, 5, 169-188.

322 CONVEX ANALYSIS AND GLOBAL OPTIMIZATION

[39] J.E. Falk (1974): Sharper bounds on nonconvex programs, Operations Research, 21, 337-340.

[40] J .E. Falk and K.L. Hoffman (1976): A successive underestimation method for concave minimization problems, Math. of Operations Research, 1, 251-259.

[41] J.E. Falk and K.L. Hoffman (1986): Concave minimization via collapsing polytopes,Operations Research, 34, 919-929.

[42] J. E. Falk and R.M. Soland (1969): An algorithm for separablenonconvex programming problems, Management Science, 15, 550-569.

[43] P. Finsler (1937): Uber das Vorkommen definiter und semidefiniter For­men in Scharen quadratischer Formen, Commentarii Mathematici Hel­vetici, 9, 188-192.

[44] C.A. Floudas and A. Aggarwal (1990): A decomposition strategy for global optmum search in the pooling problem, ORSA Joumal on Com­puting, 2(3):225-235.

[45] C.A. Floudas and P.M. Pardalos (1990): A Collection of Test Problems for Constrained Global Optimization Problems, Lecture Notes in Com­puter Science 455, Springer-Verlag, Berlin.

[46] C.A. Floudas and V. VlSweswaran (1990): A Global Optimization AIgo­rithm for Certain Classes of Nonconvex NLPs -I. Theory, Computers and Chemical Engineering, 14, 1397.

[47] C. Floudas and V. Visweswaran (1995): Quadratic Optimization, in R. Horst and P.Pardalos (eds.), Handbook of Global Optimization, Kluwer Academic Publishers, 217-269

[48] F. Forgo (1975): The solution of a special quadratic problem (in Hungar­ian), Szigma, 53-59.

[49] F. Forgo (1988): Nonconvex Programming, Akademiai Kiad6, Budapest.

[50] L.R. Foulds, D. Haugland and K. Jomsten (1992): A bilinear approach to the pooling problem, Optimization, 24, 165-180.

[51] M.L. Fredman and R.E. Tarjan (1984): Fibonacci heaps and their uses in improved network optimization algorithms, Joumal of the Association for computing Machinery, 34, 596-615.

REFERENCES 323

[52] J. Fiilop (1992): An outer approximation method for solving canonical d.c. problems', in P. Gritzmann, R. Tettich, R. Horst and E. Sachs (eds.), Operations Research 91, Physica-Verlag, Heidelberg, 95-98.

[53] O. Fujiwara and D.B. Khang (1990): A two-phase decomposition method for optimal design of looped water distribution networks, Water Re­sources Research, 23, 977-982.

[54] R. Gabasov and F.M. Kirillova (1980): Linear Programming Methods, Part 3 (Special Problems) (in Russian), Minsk.

[55] P. Gale (1981): An algorithm for exhaustive generation of building floor plans, Common ACM, 24, 813-825.

[56] P. Gahinet and A. Nemirovski (1997): The projective method for solving linear matrix inequalities, Mathematical Programming, 11, 163-190.

[57] S. Ghannadan, A. Migdalas, H. Tuy and P. Vărbrand (1994): Heuris­tics based on tabu search and Lagrangian relaxation for the concave production-transportation problem, Studies in Regional and Urban Plan­ning, issue 3, 127-141.

[58] R.P. Ge and Y.F. Qin (1987): A Class of Filled Functions for Finding Global Minimizers of a Function of Sever al Variables, Journal of Opti­mization Theory and Applications, 54, 241-252.

[59] F. Glover (1972): Cut search methods in integer programming, Mathe­matical Programming, 3, 86-100.

[60] F. Glover (1973a): Convexity cuts and cut search, Operations Research, 21, 123-124.

[61] F. Glover (1973b): Concave programming applied to a special class of 0-1 integer programs, Operations Research, 21, 135-140.

[62] K.C. Goh, M. G. Safonov and G.P. Papavassilopoulos (1995): Global op­timization for the Biaffine Matrix Inequality Problem, Journal of Global Optimization, 1, 365-380.

[63] R.L. Graham (1972): An efficient algorithm for determining the convex hulI of a finite planar set, In! Proc. Lett. 1, 132-133.

[64] A. Groch, L. Vidigal and S. Director (1985): A new global optimization method for electronic circuit design, IEEE Transactions on Circuits and Systems, 32, 160-179.

324 CONVEX ANALYSIS AND GLOBAL OPTIMIZATION

[65) G.M. Guisewite and P.M. Pardalos (1992): A polynomial time solvable concave network fiow problem', Network, 23, 143-148.

[66) M.J. Haims and H. Freeman {1970}: A multistage solution of the template-Iayout problem,IEEE Transactions Syst. Science and Cybemet­ics, 6, 145- 151.

[67) M. Hamami and S.E. Jacobsen {1988}: Exhaustive non-degenerate con­ical processes for concave minimization on convex polytopes, Math. of Operations Research, 13,479-487.

[68) P. Hansen, B. Jaumard and H. 'fuy (1995) 'Global optimization in loca­tion', in Facility Location {Zvi Dresner, ed.}, Springer-Verlag, 43-68.

[69) P. Hansen, B. Jaumard, C. Meyer and H. 'fuy (1997): Generalized convex multiplicative programming via quasiconcave minimization, to appear in Joumal of Global Optimization.

[70) P. Hartman (1959): On functions representable as a difference of convex functions, Pacific Joumal of Mathematics, 9, 707-713.

[71) R.J. Hillestad and S.E. Jacobsen (1980a): Reverse Convex Programming, Applied Mathematics and Optimization, 6, 63-78

[72) R.J. Hillestad and S.E. Jacobsen (1980b): Linear programs with an addi­tional reverse convex constraint', Applied Mathematics and Optimization, 6,257-269.

[73) J.-B. Hiriart-Urruty (1985): Generalized diffetentiability, duality and optimization for problems dealing with differences of convex functions, Lecture Notes in Economics and Mathematical Systems, 256, 37-69, Springer-Verlag, Berlin.

[74) J.-B. Hiriart-Urruty (1989a): From convex optimization to nonconvex optimization, Part I: Necessary and sufficient conditions for global opti­mality, in F.H. Clarke, V.F. Demyanov, F. Giannessi (eds.},Nonsmooth optimization and related topics, Plenum, New York.

[75) J.-B, Hiriart-Urruty (1989b): Conditions necessaires et suffisantes d'optimalite globale en optimisation de differences de deux fonctions con­vexes, C.R. Acad. Sci. Paris, 309, Serie I, 459- 462.

[76] K.L. Hoffman (1981): A method for globally minimizing concave func­tions over convex sets, Mathematical Programming, 20, 22-32.

REFERENCES 325

[77] K. Holmberg and H. Tuy (1995): A production-transportation problem with stochastic demands and concave production cost', Preprint, Depart­ment of Mathematics, Linkoping University.

[78] R. Horst (1976): An algorithm for nonconvex problems, Mathematical Programming, 10, 312-321.

[79] R. Horst (1988): Deterministic global optimization with partition sets whose feasibility is not known. Application to concave minimization, re­verse convex constraints, d.c. programming and Lipschitz optimization', Joumal of Optimization Theory and Application, 58, 11-37.

[80] R. Horst (1990): Deterministic global optimization: some recent advances and new fields of application', Naval Research Logistics, 37, 433-471.

[81] R. Horst, P.M. Pardalos and N.V. Thoai: 1995, lntroduction to Global Optimization, Kluwer Academic Publishers, Dordrecht/Boston/London.

[82] R. Horst, N.V. Thoai and H.P. Benson (1991): Concave minimization via conical partitions and polyhedral outer approximation, Mathematical Programming, 50, 259-274.

[83] R. Horst, N.V. Thoai and J. de Vries (1988): On finding new vertices and redundant constraints in cutting plane algorithms for global opti­mization', Operations Research Letters, 7, 85-90.

[84] R. Horst and H. Tuy (1996): Global Optimization (Deterministic Ap­proaches), third edition, Springer-Verlag, Berlin/New York.

[85] B. Jaumard and C.Meyer (1996): On the convergence of cone splitting algorithms with w-subdivisions, GERAD, Ecole des Hautes Etudes Com­merciales, Montreal.

[86] B. Kalantari and J.B. Rosen (1987): An algorithm for global minimiza­tion of linearly constrained concave quadratic problems, Math. of Oper­ations Research, 12, 544-561.

[87] N. Karmarkar (1990): An interior-point approach for NP-complete prob­lems, Contemporary Mathematics, 114, 297-308.

[88] J.E. Kelley (1960): The cutting plane methof for solving convex pr<r grams, Joumal SIAM, 8, 703-712.

[89] B. Klinz and H. Tuy (1993): Minimum concave-cost network flow prob­lems with a single nonlinear arc cost, in P. Pardalos and Dingzhu Du (eds.),Network Optimization Problems, World Scientific, 125-143.

326 CONVEX ANALYSIS AND GLOBAL OPTIMIZATION

!90] H. Konno (1976a): A cutting plane for solving bilinear programs, Math­ematical Programming, 11,14-27.

!91] H. Konno (1976b): Maximization of a convex function subject to linear constraints, Mathematical Frogramming, 11, 117-127.

!92] H. Konno and T. Kuno (1990): Generalized linear and fractional pro­gramming, Annals of Operations Research, 25, 147-162.

!93] H. Konno, Y. Yajima and T. Matsui (1991): Parametric simplex algo­rithms for solving a special class of nonconvex minimization problems, Journal of Global Optimization, 1,65-82.

!94] H. Konno and T. Kuno (1992): Linear multiplicative programming, Mathematical Programming, 56, 51-64

!95] H. Konno, T. Kuno and Y. Yajima (1992): Parametric simplex algorithms for a class of NP-complete problems whose average numbers of steps are polynomial, Computational Optimization and Applications, 1, 227-239.

!96] H. Konno, T. Kuno and Y. Yajima (1994): Global minimization of a gen­eralized convex multiplicative function, Journal of Global Optimization, 4,47-62.

!97] H. Konno and T.Kuno (1995): Multiplicative programming problems, in R. Horst and P. Pardalos (eds.) , Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht/Boston/London, 369-405.

!98] H. Konno, P.T. Thach and H. Thy (1997): Optimization on low rank nonconvex structures, Kluwer Academic Publishers, Dor­drecht/Boston/London.

!99] P.L. Kough (1979): The indefinite quadratic programming problem, Op­erations Research, 27, 516-533.

!100] T. Kuno (1993): Globally determining a minimum-area rectangle en­closing the projection of a higher dimensional set, Operations Research Letters, 13, 295-305.

!101] T. Kuno, Y. Yajima and H. Konno (1991): An outer approximation method for minimizing the product of sever al convex functions on a con­vex set, Joumal of Global Optimizzation, 3 325-335.

!102] T. Kuno and H. Konno (1991): A parametric successive underestima­tion method for convex multiplicative programming problems', Joumal of Global Optimization, 1, 267-286 .

REFERENCES 327

[103] T. Kuno, H. Konno and Y. Yamamoto (1992): A parametric succes­sive underestimation method for convex programming problems with an additional convex multiplicative constraint', Journal of the Operations Researeh Soeiety of Japan, 35, 290-299.

[104] E.M. Landis (1951): an functions representable as the difference of two convex functions, Dokl. Akad. Nauk SSSR, 80, 9-11.

[105] A. L. Levy and A. Montalvo (1988): The tunneling algorithm for the global minimization of functions, SIAM Journal Sei. Stat. Comp., 6, 15-29.

[106] K. Maling, S. H. NuelIer, and W.R. Heller (1982): an finding optimal rectangle package plans, Proeeedings 19th Design Automation Confer­enee, 663-670.

[107] O.L. Mangasarian (1969): Nonlinear programming, MacGraw-Hill, New York.

[108] J.M. Martinez (1994): Local minimizers of quadratic functions on Eu­clidean balIs and spheres, SIAM J. Optimization, 4, 159-176.

[109] T. Matsui (1996): NP-hardness of linear multiplicative programming and related problems, Journal of Global Optimization, 9, 113--119.

[110] D.Q.Mayne and E. Polak (1984): Outer approximation algorithms for non-differentiable optimization, Journal of Optimization: Theory and Applieations, 42, 19-30.

[111] G.P. McCormick (1972): Attempts to calculate global solutions of prob­lems that may have local minima, in F. Lootsma (ed.), Numerieal Meth­ods for Nonlinear Optimization, Academic Press, London and New York, 209-221.

[112] G.P. McCormick (1982) Nonlinear Programming: Theory, Algorithms and Applications, John Wiley and Sons, New York.

[113] N. Megiddo and K.J. Supowwit (1984): an the complexity of some com­mon geometric location problems, SIAM Journal on Computing, 13, 182-196.

[114] D. Melzer (1986): an the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions, Math­ematical Programming Study, 29, 118-134.

328 CONVEX ANALYSIS AND GLOBAL OPTIMIZATION

[115] C. Meyer (1996): Algorithmes coniques pour la minimisation quasicon­cave, Ph.D. Thesis, Ecole Polytechnique, Montreal.

[116] K.G. Murty (1983): Linear Programming, Wiley & Sons, New York.

[117] K.G. Murty and S.N. Kabadi (1987): Some NP-complete problems in quadratic aud nonlinear programming, Math. Programming, 39,117-130.

[118] L.D. Muu (1985): A convergent algorithm for solving linear programs with au additional reverse convex constraint, Kybernetika, 21, 428-435.

[119] L.D. Muu and W. Oettli (1991): Method for minimizing a convex con­cave function over a convex set', Joumal of Optimization Theory and Applications, 70, 377-384.

[120] L.D. Muu (1993): An algorithm for solving convex programs with an additional convex-concave constraint, Math. Programming, 61, 75-87.

[121] L.D. Muu (1996): Models and methods for solving convex-concave pro­grams, Rabilitation thesis, Institute of Mathematics, Ranoi.

[122] N.D. Nghia aud N.D. Rieu (1986): A Method for solving reverse convex programming problems, Acta Mathematica Vietnamica, 11, 241-252.

[123] V.R. Nguyen aud J.J. Strodiot (1992): Computing a global optimal 80-

lution to a design centering problem', Mathematical Programming, 53, 111-123.

[124] J.M. Ortega and W.C. Rheinboldt (1970): Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.

[125] P.M. Pardalos (1988): Enumerative techniques for solving some noncon­vex global optimization problems, Operations Research Spektrum, 10, 29-35

[126] P.M. Pardalos and J.B. Rosen (1987): Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268, Springer-Verlag, Berlin.

[127] P.M. Pardalos aud S.A. Vavasis (1991): Quadratic programming with one negative eigenvalue is NP-hard, Journal of Global Optimization, 21, 843-855.

[128] U. pferschy aud H. Thy (1994): Linear programs with an additional rank two reverse convex constraint, Journal of Global Optimization, 4, 347-366.

REFERENCES 329

[129] D.T. Pham and L.T. Hoai-An {1996}: DC optimization algorithms for solving the trust region subproblem, tQ appear in SI AM J. Optimization.

[130] A.T. Phillips and J.B. Rosen {1988}: A parallel algorithm for constrained concave quadratic global minimization, Math. Progmm'TTling, 42, 421-448.

[131] M. Pincus {1968} : A close form solution of certain programming prob­lems, Opemtions Research, 16, 690-694.

[132] J. Pinter, J. Szabo and L. Somlyody {1986}, Multiextremal optimization for calibrating water resources models, Environmental Software, 1, 98-105.

[133] J. Pinter {1990}: Solving nonlinear equations systems via global partition and search, Computing, 43, 309-323.

[134] J. Pinter {1996}: Global optimization in action., Kluwer Academic Pub­lishers, Dordrecht/Boston/London.

[135] M. V. Ramana {1993}: An algorithmic analysis of multiquadmtic and semi-definite progmmming problems, Ph.D. Thesis, Johns Hopkins Uni­versity, Baltimore.

[136] H. Ratschek and J. Rokne{1988}: New Computer Methods for Global Optimization, Horwood, Chichester.

[137] R. G. Reeves {1975}: Global minimization in nonconvex all-quadratic programming, Management Science, 22, 76-86.

[138] A.W.Roberts and D.E. Varberg {1973}: Convex Functions, Academic Press, New York.

[139] R.T. Rockafellar {1970}: Convex Analysis, Princeton University Press, Princeton.

[140] J.B. Rosen {1983}: Global minimization of a linearly constrained concave function by partition of feasible domain, Math. of Opemtions Research, 8,215-230.

[141] J.B. Rosen and P.M. Pardalos {1986}: Global minimization of large­scale constrained concave quarlratic problems by separable programming, Mathematical Progmmming, 34, 163-174.

[142] S. Sahni {1974}: Computationally related problems. SIAM J. on Corn­put., 3, 262-279.

330 CONVEX ANALYSIS AND GLOBAL QpTIMIZATION

[143] A. Shapiro (1983): On functions representable as a difference of two convex functions in inequality constrained optimization, Research Report, University of South Africa.

[144] J.P. Shectman and N.V. Sahinidis (1996): A finite algorithmfor for global minimization of separable concave programs, in C.A. Floudas and P.M. Pardalos (eds.), State of the Art in Global Optimization: Computational Methods and Applications, Kluwer, 303-309.

[145] H.O. Sherali and C.H. Tuncbilek (1992): A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique, Joumal of Global Optimization, 2, 101-112.

[146] H.O. Sherali and C.H. Tuncbilek (1995): A reformulation-convexification approach to solving nonconvex quadratic programming problems, Joumal of Global Optimization, 7, 1-3I.

[147] N.Z. Shor and S.I. Stetsenko (1989): Quadmtic extremal problems and nondifferentiable optimization (in Russian), Naukova Oumka, Kiev.

[148] N.Z. Shor (1991): Oual Estimates in Multiextremal Problems, Joumal of Global Optimization, 2, 411-418.

[149] M. Sion (1958): On general minimax theorems, Pacific J. Math., 8, 171-176.

[150] M. Sniedovich (1986): C-programming and the minimization of pseu­dolinear and additive concave functions, Opemtions Research Letters 5, 185-189.

[151] R.M. Soland (1974): Optimal facility location with concave costs, Oper­ations Research, 22, 373-382.

[152] A. C. Strekalovski (1987): On the global extremum problem (in Russian), Soviet Doklady, 292, 1062-1066.

[153] A. C. Strekalovski (1990): On problems of global extremum in nonconvex extremal problems (in Russian), Izvestya Vuzov, ser. Matematika, 8, 74-80.

[154] A. C. Strekalovski (1993): On search for global maximum of convex func­tions on a constraint set (in Russian) , J. of Comp. Mathematics and Math. Physics, 33, 349-363.

REFERENCES 331

[155] R.G. Strongin (1984): Numerical methods for multiextremal nonlinear programming problems with nonconvex constraints, in V. F. Demyanov and D.Pallaschke (eds.), NondijJerentiable Optimization: Motivations and Applications, Proceedings IIASA Workshop, Sopron 1984, IIASA, Lax­enburg.

[156] K. Swarup (1966a): Programming with indefinite quadratic function with linear constraints, Cahiers du Centre d 'Etude de Recherche Opera­tionnelle, 8, 132-136.

[157] K. Swarup (1966b): Indefinite quadratic prorgamming, Cahiers du Centre d'Etude de Recherche Operationnelle, 8, 217-222.

[158] K. Swarup (1966c): Quadratic programming, Cahiers du Centre d'Etude de Recherche Operationnelle, 8, 223-234.

[159] T. Tanaka, P.T. Thach and S. Suzuki (1991): An optimal ordering policy for jointly replenished products by nonconvex minimization techniques, J. of the Operations Research Bociety of Japan, 34, 109-124.

[160] P.T. Thach (1988): The design centering problem as a d.c. programming problem, Mathematical Programming, 41, 229-248.

[161] P.T. Thach {1990}: A decomposition method for the minconcave cost fiow problem with a staircase structure, Japan Journal of Applied Math­ematics, 7, 103-120.

[162] P.T. Thach (1991): Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and applications, Journal of Mathematical Analysis and Applications, 159, 299-322.

[163] P.T. Thach {1992}: New partitioning method for a class of nonconvex optimization problems, Mathematics of Operations Research, 17, 43-69.

[164] P.T. Thach {1993a}: D.c. sets, d.c. functions and nonlinear equations, Mathematical Programming, 58, 415-428.

[165] P.T. Thach {1993b}: Global optimality criteria and duality with zero gap in nonconvex optimization problems, BlAM J. Math. Anal.24, 1537-1556.

[166] P.T. Thach {1994}: A nonconvex duality with zero gap and applications, BlAM J. Optimization, 4, 44-64.

[167] P.T. Thach, R.E. Burkard and W. Oettli {1991}: Mathematical programs with a two dimensional reverse convex constraint, Journal of Global Op­timization, 1, 145-154.

332 CONVEX ANALYSIS AND GLOBAL OPTIMIZATION

[168] P.T. Thach and H. Konno (1996): On the degree and separability of nonconvexity and applications to optimization problems, Mathematical Programming, 77, 23-47.

[169] P.T. Thach and H. Thy (1990): The relief indicator method for con­strained global optimization, Naval Research Logistics, 37,473-497

[170] L.S. Thakur (1991): Domain contraction in nonlinear programming: min­imizing a quadraticconcave function over a polyhedron, Math. of Opera­tions Research, 16, 390-407.

[171] T.V. Thieu (1989): Improvement and implementation of some algorithms for nonconvex optimization problems, in Optimization - Fifth French Ger­man Conference, Castel Nove11988, Lecture Notes in Mathematics 1405, Springer, 159-170.

[172] T.V. Thieu (1991): A variant of Tuy's decomposition algorithm for solv­ing a class of concave minimization problems, Optimization, 22, 607-620.

[173] T.V. Thieu (1992): A linear programming approach to solving a jointly contrained bilinear programming problem with special structure, Van tru va He thong, 44, Institute of Mathematics, Hanoi, 113-121.

[174] T.V. Thieu, B.T. Tam and V.T. Ban (1983): An outer approximation method for globally minimizing a concave function over a compact convex set, Acta Mathematica Vietnamica, 8, 21-40.

[175] N.V. Thoai (1988): A modified version of Thy's method for solving d.c. programming problems, Optimization, 19, 665-674.

[176] N.V. Thoai and H. Tuy (1980): Convergent algorithms for minimizing a concave function, Mathematics of Operations Research, 5, 556-566.

[177] N.V. Thuong and H. Thy (1984): A finite algorithm for solving linear programs with an additional reverse convex constraint, in V. Demyannov and H. Pallaschke (eds.), Nondifferentiable Optimization, Lecture Notes in Econonmics and Mathematical Systems, 225, Springer, 291-302.

[178] J. F. Toland (1978): Duality in nonconvex optimization, Joumal of Math­ematical Analysis and Applications, 66, 399-415.

[179] G.T. Toussaint (1983): Solving geometric problems with the rotating calipers, Proc. of IEEE MELECON 83.

[180] H.D. Tuan (1996): A comment on an alternative theorem for quadratic forms and extensions, Preprint, Department of Electronic-Mechanical En­gineering, Nagoya University.

REFERENCES 333

[181] H. Tuy {1964}: Concave programming under linear constraints, Soviet Mathematics, 5, 1437-1440.

[182] H. Tuy {1974}: On a general minimax theorem, Soviet Math. Dokl., 15, 1689-1693.

[183] H. Tuy {1983}: On outer approximation methods for solving concave minimization problems, Acta Mathematica Vietnamica, 8,3-34.

[184] H. Tuy {1984}: Global minimization of a difference of two convex func­tions, in Lecture Notes in Economics and Mathematical Systems 226, Springer-Verlag, Berlin, 98-108,

[185] H. Tuy {1986}: A general deterministic approach to global optimization via d.c. programming, in J.B. Hiriart-Urruty {ed.} , Fermat Days 1985: Mathematics for Optimization, North-Holland, Amsterdam, 137-162.

[186] H. Tuy {1987a}: Convex Programs with an additional reverse convex constraint, Joumal of Optimization Theory and Applications, 52, 463-486.

[187] H. Tuy {1987b}: Global minimization of a difference of two convex func­tions, Mathematical Programming Study, 30, 150-182

[188] H. Tuy {1990}: On polyhedral annexation method for concave minimiza­tion, in Lev J. Leifman and J.B. Rosen {eds.}, Functional Analysis, Opti­mization and Mathematical Economics, Oxford University Press, 248-260.

[189] H. Tuy {1991a}: Normal conical algorithm for concave minimization over polytopes, Mathematical Programming, 51, 229-245.

[190] H. Tuy {1991b}: Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms, Joumal of Global Op­timization, 1, 23-36.

[191] H. Tuy {1991c}: Polyhedral annexation, dualization and dimension reduc­tion technique in global optimization, Joumal of Global Optimization, 1, 229-244.

[192] H. Tuy {1992a}: The complementary convex structure in global optimiza­tion, Joumal of Global Optimization, 2, 21-40.

[193] H. Tuy {1992b}: On nonconvex optimization problems with separated nonconvex variables, Joumal of Global Optimization, 2, 133-144.

334 CONVEX ANALYSIS AND GLOBAL QpTIMIZATION

[194] H. Tuy (1992c): Introduction to Global Optimization, Ph.D. Course (manuscript), Department of Mathematics, Linkoping University. AIso GERAD, Ecole Polytechnique, Montreal, 1994.

[195] H. Tuy (1995a): D.C. Optimization: Theory, Methods and Algorithms, in R. Horst and P. Pardalos {eds.} , Handbook of Global Optimization, Kluwer Academic Publishers, 149-216.

[196] H. Tuy {1995b}: Canonical d.c. programming: outer approximation methods revisited, Operations Research Letters, 18, 99-106.

[197] H. Tuy {1996}: A general d.c. approach to location problems, in C. Floudas and P. Pardalos (eds.), State of the Art in Global Optimization: Computational Methods and Applications, Kluwer 413-432.

[198] H. Tuy and F.A. AI-Khayyal {1992}: Global optimization of a noncon­vex single facility location problem by sequential unconstrained convex minimization, Journal of Global Optimization, 2, 61-71.

[199] H.Tuy, Faiz AI-Khayyal and Fangjun Zhou {1995}: A d.c. optimization method for single facility location problems, Joumal of Global Optimiza­tion, 7, 209-227.

[200] H. Tuy, N.D. Dan and S. Ghannadan {1993}: Strongly polynomial time algorithm for certain concave minimization problems on networks, Oper­ations Research Letters, 14, 99-109.

[201] H. Tuy, S. Ghannadan, A. Migdalas and P. Viirbrand: (1993): Strongly polynomial algorithm for a production-transportation problem with con­cave production cost, Optimization, 27, 205-227.

[202] H. Tuy, S. Ghannadan, A. Migdalas and P. Viirbrand: {1994}: Strongly polynomial algorithm for two special minimum concave cost network flow problems, Optimization, 32, 23-44.

[203] H. Tuy, S. Ghannadan, A. Migdalas and P. Viirbrand: {1995}: The mini­mum concave cost flow problem with fixed numbers of nonlinear arc costs and sources, Journal of Global Optimization, 6, 135-151.

[204] H. Tuy, S. Ghannadan, A. Migdalas and P. Viirbrand (1996): Strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables, Mathematical Programming, 72, 229-258.

REFERENCES 335

[205] H. Thy and R. Horst (1988): Convergence and restart in branch aud bound algorithms for global optimization. Application to concave min­imization and d.c. optimization problems, Mathematical Programming, 42, 161-184.

[206] H. Thy and R. Horst (1991): The geometric complementarity problem and transcending stationarity problem in global optimization, DIMACS Series in Discrete Mathematics aud Computer Science, VoI. 4, Applied Geometry and Discrete Mathematics, The Victor Klee Festschrift, 341-353.

[207] H. Thy, A. Migdalas aud P. Varbrand (1993): A global optimization ap­proach for the linear two-Ievel program, Journal of Global Optimization, 3,1-23.

[208] H. Thy, A. Migdalas aud P. Varbrand (1994): A quasiconcave minimiza­tion method for solving linear two level programs, Journal of Global Op­timization, 4, 243-264.

[209] H. Thy and W. Oettli (1994): an necessary and sufficient conditions for global optimization, Matematicas Aplicadas, 15, 39-4l.

[210] H. Thy and B.T. Tam (1992): An efficient solution method for rank two quasiconcave minimization problemst Optimization, 24, 43-56.

[211] H. Thy aud B.T. Tam (1995): Polyhedral annexation vs outer approxima­tion methods for decomposition of monotonic quasiconcave minimization, Acta Mathematica Vietnamica, 20, 99-114.

[212] H. Thy, B.T. Tam and N.D. Dan (1994): Minimizing the sum of a convex function and a specially structured nonconvex function, Optimization, 28, 237-248.

[213] H. Thy and P.T. Thach (1990): The relief indicator method as a new approach to constrained global optimization, in Bystem Modelling and Optimization, Proceedings 14th IFIP Conference, Leipzig, Lecture Notes in Control Information Sciences, 143, 219-233.

[214] H. Thy and N.V. Thuong (1988): an the global minimization of a convex function under general nonconvex constraints, Applied Mathematics and Optimization, 18, 119-142.

[215] L. Vandenberge and S. Boyd (1996): Semidefinite programming, BlAM Review, 38, 49-95

336 CONVEX ANALYSIS AND GLOBAL QpTIMIZATION

[216] T. Van Voorhis (1997): The quadratically constrained quadratic programs, Ph.D. Thesis, Georgia Institute of Technology, Atlanta.

[217] A.F. Veinott (1967): The supporting hyperplane method for unimodal programming, Operations Research, 15, 147-152.

[218] L. Vidigal and S. Director (1982): A design centering algorithm for non­convex regions of acceptability, IEEE 1ransactions on Computer-Aided Design of Integrated Circuits and Systems, 13-24.

[219] V: Visweswaran and C.A. Floudas (1990): A global optimization aIgo­rithm for certain classes of nonconvex NLPs--II. Application and test problems, Computers and Chemical Engineering, 14, 1419.

[220] V: Visweswaran and C.A. Floudas (1992): Unconstrained and constrained global optimization of polynomial functions in one variable, Journal of Global Optimization, 2, 73-100.

[221] Y. Yajima and H. Konno (1992): Efficient algorithms for solving rank two and rank three bilinear programming problems, Journal of Global Optimization, 1, 155-171.

[222] V.A. Yakubovich (1977): The S-procedure in nonlinear control theory, Vestnik Leningrad Univ., 4, 73-93.

[223] Y. Ye (1992): On affine scaling algorithms for nonconvex quadratic pro­gramming, Mathematical Programming, 56, 285-300.

[224] N. ZOOeh (1974): On building minimum cost communication networks over time, Networks, 4, 19-34.

[225] Q. Zheng (1985): Optimality conditions for global optimization (I) and (II),Acta Mathematicae Applicatae Sinica, English Series, 2, 66-78 and 118-134.

[226] F. Zhou (1997): Nonmonotone methods in optimization and d.c. opti­mization of location problems, Ph.D. Thesis, Georgia Institute of Tech­nology, Atlanta.

[227] S.I. Zukhovitzki, R.A. Polyak and M.E. Primak (1968): On a class of con­cave programming problems (in Russian),Ekonomika i Matematitcheskie Methody, 4, 431-443.

[228] P.B. Zwart (1974): Global maximization of a convex function with linear inequality constraints, Operations Research, 22, 602-609.

Index

Index

affine function 41 affine set; manifold 3 affinely independent 5 approximate subdifferential 69 e-approximate optimal

solution 166 barycentric coordinates 6 basic concave programming

problem (BCP) 133 basic outer approximation

theorem 180 bilinear programming 290 bisection 141; exact 141

of ratio a 141 branch and cut 155 branch and bound 155 branch and select 153 cone 7,143 canonical cone 139 canonical d.c. programming

problem 118 Caratheodory core 16 Caratheodory's Theorem 14 elosed function 53 elosure of a convex function 53 elf. 53 coercive 200 combined algorithm 190 complementary convex set 84, 117 complementary convex structure 83 concave function 41 concave majorant 100 concave production-transportation

problem 262 contave quadratic programming

problem 287 concavity cut 136 conical subdivision 143

conjugate 72 coruştancy space 52 convex envelope 48 convex function 41 convex hulI 13; of a set 13

of a function 48 convex inequality 56 convex minorant 100 convg 48 convex set 6 convex-concave 77

337

CS restart algorithm for BCP 148 for LRCP 169

cutting plane method 177 DC feasiblity problem 123 d.c. function 83 d.c. programming 117 d.c. representation 88 d.c set 95 d.c. structure 83 decomposition 223; by projection 227

by polyhedral annexation 233 decoupling relaxation 301 dimension of a convex set 10

of a convex function 53 directional derivative 65 direction of recession 11 direction in which f is affine 53 distance function 42, 64 distinguished point 154, 177

member 154 dual DC feasibility problem 193 dual problema 202 dualization by polyhedral

annexation 233 edge 33 edge property 164 effective domain 41 epigraph 41 exact bisection 141 exact selection 155 exhaustive fl.lter 141

338 CONVEX ANALYSIS AND GLOBAL QpTIMIZATION

exhaustive subdivision process 143 'Y-extension 138; 287 extreme direction 24 extreme point 24 face 23; 31 facet 32 faclorable function 87 filter 141; 153 full dimension 10 gauge 10 general concave minimization

- problem 181 general nonconvex quadratic

problem 293 generalized convex multiplicative

programming problems 250 generalized linear program 127; 285 global 74; - maximizer 74

- minimizer 74 - c-optimal solution 123

geometric programming 126; 285 halfspace 6; closed, open 6 indefinite quadratic program 291 indicator function 42 infimal convolution 48 Lagrange dual problem 309 Lagrangian 309 level set (lower, upper ) 50 lineality 26

- of a convex set 26 - of a convex function 53

linear matrix inequality (LMI) 129 linear multiplicative program 240 linear program under LMI

constraints 316 locally convexifiable 301 locally d.c. 86 local optimal solution 109 . local minimizer 109 location-allocation problem 112 lower linearizable 296 low rank nonconvex problem 223

lower semi-continuous (l.s.c.) 51 l.s.c. hulI 53 minimax theorem 78 minimum concave cost network

flow problem MCNFP 262 Minkowski functional 10 Minkowski's Theorem 27 modulus of strong convexity 71 monotonie functions 223 multiextremal 109 monotonie reverse convex problem 255 net 151 network constraints 261 noncanonical d.c. problems 206 nonconvex quadratic programming 277 nonregular problems 167 norm 9; Tchebychev 9

Euclidean 9; dual 35 normal 22; normal cone 22 NS (normal subdivision) rule 150 normal procedure DC 151 outer approximation 177 optimal visible point problem 209 optimal visible point algorithm 209 parametric approach 236 parametric d.c .. feasibility problem 169 parametric d.c. inclusion 134,205 partition via (v, j) 160 phase (global, local) 124 polar 28 polyhedral convex set 30 polyhedral annexation 192 polyhedron 30 polytope 34 positively homogeneous 063 preprocessing 125 procedure DC 148; DC* 195 problem PT P(2) 271 projection of a point

on a convex set 23 prototype branch and select

algorithm 154

Index

quadratic function 044 quadratic minimization over

ellipsoids 281 qualified member 154 quadratic problem with

quadratic constraints 293 quasiconcave function 077 quasiconcave minimization 116 quasiconvex function 50 quasiconvex maximization 116 quasiconjugate function 199 radial partition (subdivision) 141 rank 30

of a convex function 53 of a quadratic form 279

recession (cone, direction) 12 rect angular subdivision 169 reformulation-linearization 304 refinement of a subdivision 143 regular problem 166 regularity asumption 167 relative interior 10 relative boundary 10 relief indicator method 212 representation theorem 26 restart 152 reverse convex 56

constraint 116 - inequality 84; 164 - programming 116; 164

robust set 119 saddle-function 77; saddle-point 77 second quasiconjugate 201 semi-continuous (lower, upper) 51 semi-definite program 130 separable basic concave program 161 separation theorem (first 19; second 20) separator 97 Shapley-Folkman's Theorem 015 simple outer approximation 181 simple OA for CDC 189

339

simplicial subdivision 141 special concave program CPL(I) 269 S-shaped functions 93 stochastic transportation-Iocation

problem 264 strictly convex 74 strictly quasiconcave function 238 strongly convex 71 strongly separated 20 subdifferential 62 subdifferentiable 62 subdivision processes 140; 153 w-subdivision 149 subdivision (partition) via (v,j) 160 subgradient 62 g-subgradient 70 support function 29 42 supporting hyperplane 21

- hafspace 21 tight convex minorant 298 transcending the incumbent 119 transcending stationarity 122 trust region subproblem 282 two phase scheme 124 unary matrix 294 unary program 294 upper envelope 46 ')'-valid concavity cut 138 vertex 033 visibility assumption 207 weakly convex 105 Weber problem with attraction

and repulsion 130