Canonical DC programming problem: Outer approximation methods revisited

For solving the canonical DC programming problem (i.e. a convex program with an additional reverse convex constraint) several cutting plane procedures have been developed in the past few years which suffer from serious defects regarding efficiency or convergence. We discuss a general conceptual framework of outer approximation unifying various conjunctive cutting plane approaches and on this basis propose modifications to overcome these defects.

[1]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[2]  A. A. Goldstein,et al.  Newton's method for convex programming and Tchebycheff approximation , 1959, Numerische Mathematik.

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

[4]  Sihem BenSaad.  A new cutting plane-algorithm for a class of reverse convex 0–1 integer programs , 1992 .

[5]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

[6]  James E. Falk,et al.  A Successive Underestimation Method for Concave Minimization Problems , 1976, Math. Oper. Res..

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

[8]  Arthur F. Veinott,et al.  The Supporting Hyperplane Method for Unimodal Programming , 1967, Oper. Res..

[9]  T. V. Thieu.  A finite method for globally minimizing a concave function over an unbounded polyhedral convex set and its applications , 1988 .

[10]  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..

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

[12]  Panos M. Pardalos,et al.  Recent Advances in Global Optimization , 1991 .

[13]  S. Bensaad,et al.  Comments on a reverse convex programming algorithm , 1994, J. Glob. Optim..

[14]  János Fülöp.  A finite cutting plane method for solving linear programs with an additional reverse convex constraint , 1990 .

[15]  Stephen E. Jacobsen,et al.  A level set algorithm for a class of reverse convex programs , 1990 .

[16]  S. E. Jacobsen,et al.  Reverse convex programming , 1980 .

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

[18]  Pierre Hansen,et al.  On-line and off-line vertex enumeration by adjacency lists , 1991, Oper. Res. Lett..