Skip to main content
SpringerLink
Log in

The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The DC programming and its DC algorithm (DCA) address the problem of minimizing a function f=g−h (with g,h being lower semicontinuous proper convex functions on R n) on the whole space. Based on local optimality conditions and DC duality, DCA was successfully applied to a lot of different and various nondifferentiable nonconvex optimization problems to which it quite often gave global solutions and proved to be more robust and more efficient than related standard methods, especially in the large scale setting. The computational efficiency of DCA suggests to us a deeper and more complete study on DC programming, using the special class of DC programs (when either g or h is polyhedral convex) called polyhedral DC programs. The DC duality is investigated in an easier way, which is more convenient to the study of optimality conditions. New practical results on local optimality are presented. We emphasize regularization techniques in DC programming in order to construct suitable equivalent DC programs to nondifferentiable nonconvex optimization problems and new significant questions which have to be answered. A deeper insight into DCA is introduced which really sheds new light on DCA and could partly explain its efficiency. Finally DC models of real world nonconvex optimization are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • An, Le Thi Hoai. (1994). “Analyse numérique des algorithmes de l’optimisation DC. Approches locale et globale. Codes et simulations numériques en grande dimension. Applications.” Thèse de Doctorat de l’Université de Rouen.

  • An, Le Thi Hoai. (1997). “Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, algorithmes et applications.” Habilitation à Diriger des Recherches, Université de Rouen.

  • An, Le Thi Hoai. (2000). “An Efficient Algorithm for Globally Minimizing a Quadratic Function under Convex Quadratic Constraints.” Math. Programming Ser. A 87(3), 401–426.

    Google Scholar 

  • An, Le Thi Hoai. (2003). “Solving Large Scale Molecular Distance Geometry Problem by a Smoothing Technique via the Gaussian Transform an DC Programming.” J. Global Optim. 27, 375–397.

    Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (1997). “Solving a Class of Linearly Constrained Indefinite Quadratic Problems by DC Algorithms.” J. Global Optim. 11, 253–285.

    Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (1996). “Numerical Solution for Optimization over the Efficient Set by DC Optimization Algorithm.” Oper. Res. Lett. 19, 117–128.

    Article  Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (1998). “A Branch-and-Bound Method via DC Optimization Algorithm and Ellipsoidal Techniques for Box Constrained Nonconvex Quadratic Programming Problems.” J. Global Optim. 13, 171–206.

    Article  Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (1998). “A Combined DC Optimization – Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems.” J. Combin. Optim. 2(1), 9–28.

    Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (1999). “Exact Penalty in DC Programming.” Vietnam J. Math. 27(2), 169–178.

    Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (1999). “DCA Revisited and DC Models of Real World Nonconvex Optimization Problems.” Technical Report, LMI, Insa-Rouen.

  • An, Le Thi Hoai and Pham Dinh Tao. (2000a). “DC Programming Approach for Large-Scale Molecular Optimization via the General Distance Geometry Problem.” In Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches. Nonconvex Optimization and Its Applications, Vol. 40. Kluwer Academic, pp. 301–339.

  • An, Le Thi Hoai and Pham Dinh Tao. (2000b). “Large Scale Molecular Conformation via the Exact Distance Geometry Problem.” In Optimization. Lecture Notes in Economics and Mathematical Systems, Vol. 481. Heidelberg: Springer, pp. 260–277.

    Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (2001a). “A Continuous Approach for Large-Scale Linearly Constrained Quadratic Zero-One Programming.” Optim. 50(1,2), 93–120.

    Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (2001b). “DC Optimization Approaches via Markov Models for Restoration of Signals (1-D) and (2-D).” In Advances in Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications, Vol. 54. Kluwer Academic, pp. 300–317.

  • An, Le Thi Hoai and Pham Dinh Tao. (2001c). “DC Programming Approach and Solution Algorithm to the Multidimensional Scaling Problem.” In From Local to Global Optimization. Nonconvex Optimization and Its Applications, Vol. 53. Kluwer Academic, pp. 231–276.

  • An, Le Thi Hoai and Pham Dinh Tao. (2002a). “DC Programming Approach for Multicommodity Network Optimization Problems with Step Increasing Cost Functions.” J. Global Optim. 22(1–4), 204–233. Special Issue, dedicated to Professor R. Horst on the occasion of his 60th birthday.

    Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (2002b). “DC Programming. Theory, Algorithms, Applications: The State of the Art.” In First International Workshop on Global Constrained Optimization and Constraint Satisfaction. Valbonne-Sophia Antipolis, France, October 2–4, 2002. Research Report, 28 pages. Laboratory of Modeling, Optimization & Operations Research, Insa-Rouen, France (2002).

  • An, Le Thi Hoai and Pham Dinh Tao. (2003a). “Large Scale Global Molecular Optimization from Exact Distance Matrices by a DC Optimization Approach.” SIAM J. Optim. 14(1), 77–114.

    Article  Google Scholar 

  • An, Le Thi Hoai and Pham Dinh Tao. (2003b). “A New Algorithm for Solving Large Scale Molecular Distance Geometry Problems, Applied Optimization.” In Hight Performance Algorithms and Software for Nonlinear Optimization. Kluwer Academic, pp. 276–296.

  • An, Le Thi Hoai, Pham Dinh Tao, and Nguyen Van Thoai. (2002). “Combination between Local and Global Methods for Solving an Optimization Problem over the Efficient Set.” European J. Oper. Res. 142, 257–270.

    Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (2003a). “Simplicially Constrained DC Optimization over the Efficient Set and Weakly Efficient Sets.” J. Optim. Theory Appl. 117(3), 503–531.

    Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (2003b). “A DC Optimization Approach to Mathematical Programming Problems with Linear Equilibrium Constraints.” Submitted.

  • An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2002). “Towards Tikhonov Regularization of Nonlinear Ill-Posed Problems: A DC Programming Approach.” C.R. Acad. Sci. Paris Ser. I 335, 1073–1078.

    Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2003). “Solving Inverse Problems for an Elliptic Equations by D.C. (Difference of Convex Functions) Programming.” J. Global Optim. 25(4), 407–423.

    Google Scholar 

  • An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2004a). “On the Ill-Posedness of the Trust Region Subproblem.” J. Inverse Ill-Posed Problems, to appear.

  • An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2004b). “DC Programming Approach to Tikhonov Regularization for Nonlinear Ill-Posed Problems.” Submitted.

  • Conn, A.R., N.I.M. Gould, and P.L. Toint. (2000). Trust Region Methods. MPS–SIAM Series on Optimization.

  • Hiriat Urruty, J.B. (1989). “Conditions nécessaires et suffisantes d’optimalité globule en optimisation de différences de deux fonctions convexes.” C.R. Acad. Sci. Paris Ser. I 309, 459–462.

    Google Scholar 

  • Hiriat Urruty, J.B. and C. Lemarechal. (1993). Convex Analysis and Minimization Algorithms. Berlin: Springer.

    Google Scholar 

  • Horst, R., P.M. Pardalos, and Nguyen Van Thoai. (1995). Introduction to Global Optimization. Kluwer Academic.

  • Horst, R. and Nguyen Van Thoai. (1999). “DC Programming: Overview.” J. Optim. Theory Appl. 2103(1), 1–43.

    Google Scholar 

  • Horst, R. and Hoang Tuy. (1996). Global Optimization (Deterministic Approaches), 3rd ed. Springer.

  • Konno, H., Phan Thien Thach and Hoang Tuy. (1999). Optimization on Low Rank Nonconvex Structures. Kluwer Academic.

  • Laurent, P.J. (1972). Approximation et optimisation. Paris: Hermann.

    Google Scholar 

  • Mahey, P. and Pham Dinh Tao. (1993). “Partial Regularization of the Sum of Two Maximal Monotone Operators.” Math. Modell. Numer. Anal. (M 2 AN) 27, 375–395.

    Google Scholar 

  • Mahey, P. and Pham Dinh Tao. (1995). “Proximal Decomposition of the Graph of Maximal Monotone Operator.” SIAM J. Optim. 5, 454–468.

    Article  Google Scholar 

  • More, J.J. and Z. Wu. (1997). “Global Continuation for Distance Geometry Problems.” SIAM J. Optim. 8, 814–836.

    Google Scholar 

  • More, J.J. and Z. Wu. (1996). “Distance Geometry Optimization for Protein Structures.” Preprint MCS-P628-1296, Argonne National Laboratory, Argonne, IL.

  • Moreau, J.J. (1965). “Proximité et dualité dans un espace Hilbertien.” Bull. Soc. Math. France 93, 273–299.

    Google Scholar 

  • Pardalos, P.M. and J.B. Rosen. (1987). Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Sciences, Vol. 268. Springer.

  • Pschenichny, B.N. (1971). Contrôle optimal et jeux differentiels. Rocquencourt: Cahiers de l’IRIA.

    Google Scholar 

  • Rockafellar, R.T. (1970). Convex Analysis. Princeton: Princeton University.

    Google Scholar 

  • Rockafellar, R.T. (1976). “Monotone Operators and the Proximal Point Algorithm.” SIAM J. Control Optim. 14, 877–898.

    Google Scholar 

  • Tao, Pham Dinh. (1975). “Elements homoduaux relatifs à un couple de normes (φ,ψ). Applications au calcul de S φψ(A).” Technical Report, Grenoble.

  • Tao, Pham Dinh. (1976). “Calcul du maximum d’une forme quadratique définie positive sur la boule unité de la norme du max.” Technical Report, Grenoble.

  • Tao, Pham Dinh. (1981). “Contribution à la théorie de normes et ses applications à l’analyse numérique.” Thèse de Doctorat d’Etat Es Science, Université Joseph Fourier, Grenoble.

  • Tao, Pham Dinh. (1984). “Convergence of Subgradient Method for Computing the Bound Norm of Matrices.” Linear Algebra Appl. 62, 163–182.

    Article  Google Scholar 

  • Tao, Pham Dinh. (1985). “Algorithmes de calcul d’une forme quadratique sur la boule unité de la norme maximum.” Numer. Math. 45, 377–440.

    Google Scholar 

  • Tao, Pham Dinh. (1986). Algorithms for Solving a Class of Non Convex Optimization Problems. Methods of Subgradients. Fermat Days 85. Mathematics for Optimization, Elsevier Science Publishers, B.V. North-Holland.

  • Tao, Pham Dinh. (1988). “Duality in D.C. (Difference of Convex Functions) Optimization. Subgradient Methods.” In Trends in Mathematical Optimization, International Series of Numerical Mathematics, Vol. 84. Birkhäuser, pp. 277–293.

  • Tao, Pham Dinh and Le Thi Hoai An. (1994). “Stabilité de la dualité lagrangienne en optimisation DC (différence de deux fonctions convexes).” C.R. Acad. Sci. Paris Sér. I 318, 379–384.

    Google Scholar 

  • Tao, Pham Dinh and Le Thi Hoai An. (1995). “Lagrangian Stability and Global Optimality in Nonconvex Quadratic Minimization over Euclidean Balls and Spheres.” J. Convex Anal. 2, 263–276.

    Google Scholar 

  • Tao, Pham Dinh and Le Thi Hoai An. (1998). “DC Optimization Algorithms for Solving the Trust Region Subproblem.” SIAM J. Optim. 8, 476–505.

    Article  Google Scholar 

  • Tao, Pham Dinh and Le Thi Hoai An. (1997). “Convex Analysis Approach to DC Programming: Theory, Algorithms and Applications.” Acta Math. Vietnamica 22(1), 287–355.

    Google Scholar 

  • Thach, Phan Thien. (1993a). “DC Sets, D.C. Functions and Nonlinear Equations.” Math. Programming 58, 415–428.

    Article  Google Scholar 

  • Thach, Phan Thien. (1993b). “Global Optimality Criteria and Duality with Zero Gap in Nonconvex Optimization Problems.” SIAM J. Math. Anal. 24, 1537–1556.

    Article  Google Scholar 

  • Thach, Phan Thien. (1994). “A Nonconvex Duality with Zero Gap and Applications.” SIAM J. Optim. 4, 44–64.

    Article  Google Scholar 

  • Thach, Phan Thien. (1996). “On the Degree and Separability of Nonconvexity and Applications to Optimization Problems.” Math. Programming 77, 23–47.

    Google Scholar 

  • Toland, J.F. (1978). “Duality in Nonconvex Optimization.” J. Math. Anal. Appl. 66, 399–415.

    Article  Google Scholar 

  • Toland, J.F. (1979). “On Subdifferential Calculus and Duality in Nonconvex Optimization.” Bull. Soc. Math. France, Mémoire 60, 177–183.

    Google Scholar 

  • Tuy, Hoang. (1964). “Concave Programming under Linear Constraints.” Translated Soviet Mathematics 5, 1437–1440.

    Google Scholar 

  • Tuy, Hoang. (1998). Convex Analysis and Global Optimization. Kluwer Academic.

  • Tuy, Hoang. (2000). “Monotonic Optimization: Problems and Solution Approaches.” SIAM J. Optim. 2, 464–494.

    Google Scholar 

  • Wu, Z. (1996). “The Effective Energy Transformation Scheme as a Special Continuation Approach to Global Optimization with Application to Molecular Conformation.” SIAM J. Optim. 6, 748–768.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de I’Informatique Théorique et Appliquée, UFR MIM, Université de Metz, Ile du Saulcy, 57045, Metz, France

    Le Thi Hoai An

  2. Laboratory of Modelling, Optimization and Operations Research, National Institute for Applied Sciences-Rouen, BP 8, F 76 131, Mont Saint Aignan Cedex, France

    Pham Dinh Tao

Authors
  1. Le Thi Hoai An
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Dinh Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Dinh Tao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

An, L.T.H., Tao, P.D. The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems. Ann Oper Res 133, 23–46 (2005). https://doi.org/10.1007/s10479-004-5022-1

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-004-5022-1

Keywords

Search

Navigation