A Method for Converting a Class of Univariate Functions into d.c. Functions

D.c. functions are functions that can be expressed as the sum of a concave function and a convex function (or as the difference of two convex functions). In this paper, we extend the class of univariate functions that can be represented as d.c. functions. This expanded class is very broad including a large number of nonlinear and/or ‘nonsmooth’ univariate functions. In addition, the procedure specifies explicitly the functional and numerical forms of the concave and convex functions that comprise the d.c. representation of the univariate functions. The procedure is illustrated using two numerical examples. Extensions of the conversion procedure for discontinuous univariate functions is also discussed.

[1]  H. Burkill,et al.  A second course in mathematical analysis , 1970 .

[2]  Melvyn W. Jeter Mathematical Programming: An Introduction to Optimization , 1986 .

[3]  E. M. L. Beale Introduction to Optimization , 1988 .

[4]  Hoang Tuy,et al.  D.C. Optimization: Theory, Methods and Algorithms , 1995 .

[5]  R. Fletcher Practical Methods of Optimization , 1988 .

[6]  David K. Smith,et al.  Mathematical Programming: Theory and Algorithms , 1986 .

[7]  C. Heising-Goodman A survey of methodology for the global minimization of concave functions subject to convex constraints , 1981 .

[8]  Singiresu S. Rao,et al.  Optimization Theory and Applications , 1980, IEEE Transactions on Systems, Man, and Cybernetics.

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

[10]  L. E. Scales,et al.  Introduction to Non-Linear Optimization , 1985 .

[11]  Harold P. Benson,et al.  Concave Minimization: Theory, Applications and Algorithms , 1995 .

[12]  E. M. L. Beale,et al.  Nonlinear and Dynamic Programming , 1965 .

[13]  P. Hartman On functions representable as a difference of convex functions , 1959 .

[14]  J. B. Rosen,et al.  Methods for global concave minimization: A bibliographic survey , 1986 .

[15]  Michael J. Panik,et al.  Classical Optimization: Foundations and Extensions , 1976 .

[16]  Bruce W. Lamar A Method for Solving Network Flow Problems with General Nonlinear Arc Costs , 1993 .

[17]  Nikolaos V. Sahinidis,et al.  A branch-and-reduce approach to global optimization , 1996, J. Glob. Optim..

[18]  Hoang Tuy,et al.  Canonical DC programming problem: Outer approximation methods revisited , 1995, Oper. Res. Lett..

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

[20]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

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

[22]  Philip E. Gill,et al.  Practical optimization , 1981 .

[23]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .