On solving general reverse convex programming problems by a sequence of linear programs and line searches

Many multiextremal global optimization problems can be formulated as problems of minimizing a linear function over the intersection of a convex set with the complement of a convex set (so-called canonical d.c. programs or general reverse convex programming problems). In this paper it is shown that these general reverse convex programming problems can be solved by a sequence of linear programs and univariate convex minimization problems (line searches).

[1]  Hoang Tuy,et al.  On the global minimization of a convex function under general nonconvex constraints , 1988 .

[2]  Panos M. Pardalos,et al.  GLOBAL OPTIMIZATION ALGORITHMS FOR LINEARLY CONSTRAINED INDEFINITE QUADRATIC PROBLEMS , 1991 .

[3]  K. Grasse,et al.  A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization , 1986 .

[4]  H. Tuy,et al.  A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set , 1985, Math. Oper. Res..

[5]  R. Horst.  On consistency of bounding operations in deterministic global optimization , 1989 .

[6]  N. Thoai,et al.  A modified version of TUY’s method for solving d.c. programming problems , 1988 .

[7]  Karla L. Hoffman,et al.  A method for globally minimizing concave functions over convex sets , 1981, Math. Program..

[8]  Reiner Horst,et al.  Deterministic methods in constrained global optimization: Some recent advances and new fields of application , 1990 .

[9]  S. E. Jacobsen,et al.  An algorithm for optimizing network flow capacity under economies of scale , 1975 .

[10]  Nguyen V. Thoai,et al.  Concave minimization via conical partitions and polyhedral outer approximation , 1991, Math. Program..

[11]  Reiner Horst,et al.  An algorithm for nonconvex programming problems , 1976, Math. Program..

[12]  Hoang Tuy,et al.  A class of exhaustive cone splitting procedures in conical algorithms for concave minmization , 1987 .

[13]  Panos M. Pardalos,et al.  Constrained Global Optimization: Algorithms and Applications , 1987, Lecture Notes in Computer Science.

[14]  Nguyen V. Thoai,et al.  Convergent Algorithms for Minimizing a Concave Function , 1980, Math. Oper. Res..

[15]  Michel Sermange,et al.  Non-convex methods for computing free boundary equilibria of axially symmetric plasmas , 1982 .

[16]  H. Tuy,et al.  Global optimization under Lipschitzian constraints , 1987 .

[17]  H. Tuy.  A General Deterministic Approach to Global Optimization VIA D.C. Programming , 1986 .

[18]  J. Toland.  Duality in nonconvex optimization , 1978 .

[19]  S. E. Jacobsen,et al.  Linear programs with an additional reverse convex constraint , 1980 .

[20]  R. Horst,et al.  Outer approximation by polyhedral convex sets , 1987 .

[21]  Reiner Horst.  A New Branch and Bound Approach for Concave Minimization Problems , 1975, Optimization Techniques.

[22]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[23]  J. Toland.  A duality principle for non-convex optimisation and the calculus of variations , 1979 .

[24]  Hoang Tuy,et al.  A FINITE ALGORITHM FOR SOLVING LINEAR PROGRAMS WITH AN ADDITIONAL REVERSE CONVEX CONSTRAINT , 1985 .

[25]  R. Horst,et al.  On the global minimization of concave functions , 1984 .

[26]  H. Tuy.  Convex programs with an additional reverse convex constraint , 1987 .

[27]  Phan Thien Thach,et al.  The design centering problem as a D.C. programming problem , 1988, Math. Program..

[28]  J. Hiriart-Urruty.  Generalized Differentiability / Duality and Optimization for Problems Dealing with Differences of Convex Functions , 1985 .

[29]  Giulio Maier,et al.  Optimal excavation profile for a pipeline freely resting on the sea floor , 1979 .

[30]  Harold P. Benson,et al.  On the convergence of two branch-and-bound algorithms for nonconvex programming problems , 1982 .

[31]  R. Horst,et al.  On an outer approximation concept in global optimization , 1989 .

[32]  R. Horst,et al.  On the convergence of global methods in multiextremal optimization , 1987 .

[33]  R. Horst,et al.  On finding new vertices and redundant constraints in cutting plane algorithms for global optimization , 1988 .

[34]  Reiner Horst,et al.  Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and D.C. Optimization problems , 1988, Math. Program..

[35]  R. J. Hillestad.  Optimization Problems Subject to a Budget Constraint with Economies of Scale , 1975, Oper. Res..

[36]  Reiner Horst,et al.  A branch and bound-outer approximation algorithm for concave minimization over a convex set , 1991 .

[37]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[38]  H. Tuy.  Global Minimization of a Difference of Two Convex Functions , 1987 .

[39]  R. Hoprst,et al.  Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization , 1988 .

[40]  Reiner Horst,et al.  A note on the convergence of an algorithm for nonconvex programming problems , 1980, Math. Program..