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

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 Rn) 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.

[1]  Jorge J. Moré,et al.  Distance Geometry Optimization for Protein Structures , 1999, J. Glob. Optim..

[2]  Phan Thien Thach.  D.c. sets, d.c. functions and nonlinear equations , 1993, Math. Program..

[3]  P. T. Thach,et al.  Optimization on Low Rank Nonconvex Structures , 1996 .

[4]  J. Moreau.  Proximité et dualité dans un espace hilbertien , 1965 .

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

[6]  Hoang Tuy,et al.  Monotonic Optimization: Problems and Solution Approaches , 2000, SIAM J. Optim..

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

[8]  Phan Thien Thach.  A Nonconvex Duality with Zero Gap and Applications , 1994, SIAM J. Optim..

[9]  Zhijun Wu,et al.  The Eeective Energy Transformation Scheme as a General Continuation Approach to Global Optimization with Application to Molecular Conformation , 2022 .

[10]  Le Thi Hoai An,et al.  Towards Tikhonov regularization of non-linear ill-posed problems: a dc programming approach , 2002 .

[11]  Pham Dinh Tao,et al.  D.C. Optimization Approaches via Markov Models for Restoration of Signal (1-D) and (2-D) , 2001 .

[12]  Thi Hoai An Le,et al.  D.C. Programming Approach for Large-Scale Molecular Optimization via the General Distance Geometry Problem , 2000 .

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

[14]  Le Thi Hoai An,et al.  A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems , 1998, J. Comb. Optim..

[15]  Le Thi Hoai An,et al.  A new algorithm for Solving Large Scale Molecular Distance Geometry Problems , 2003 .

[16]  Jorge J. Moré,et al.  Global Continuation for Distance Geometry Problems , 1995, SIAM J. Optim..

[17]  Phan Thien Thach.  Global optimality criterion and a duality with a zero gap in nonconvex optimization , 1993 .

[18]  Le Thi Hoai An,et al.  Solving an Inverse Problem for an Elliptic Equation by d.c. Programming , 2003, J. Glob. Optim..

[19]  Le Thi Hoai An,et al.  A Branch and Bound Method via d.c. Optimization Algorithms and Ellipsoidal Technique for Box Constrained Nonconvex Quadratic Problems , 1998, J. Glob. Optim..

[20]  Le Thi Hoai An,et al.  D.C. programming approach for multicommodity network optimization problems with step increasing cost functions , 2002, J. Glob. Optim..

[21]  P. D. Tao.  Lagrangian Stability and Global Optimality in Nonconvex Quadratic Minimization Over Euclidean Balls and Spheres , 1995 .

[22]  Le Thi Hoai An.  Solving Large Scale Molecular Distance Geometry Problems by a Smoothing Technique via the Gaussian Transform and D.C. Programming , 2003, J. Glob. Optim..

[23]  Le Thi Hoai An,et al.  D.C. Programming Approach to the Multidimensional Scaling Problem , 2001 .

[24]  Pham Dinh Tao,et al.  Duality in D.C. (Difference of Convex functions) Optimization. Subgradient Methods , 1988 .

[25]  Hoai An Le Thi,et al.  STABILITE DE LA DUALITE LAGRANGIENNE EN OPTIMISATION D.C. (DIFFERENCE DE DEUX FONCTIONS CONVEXES) , 1994 .

[26]  El Bernoussi Souad,et al.  Algorithms for Solving a Class of Nonconvex Optimization Problems. Methods of Subgradients , 1986 .

[27]  Le Thi Hoai An,et al.  Solving a Class of Linearly Constrained Indefinite Quadratic Problems by D.C. Algorithms , 1997, J. Glob. Optim..

[28]  Hiroshi Konno,et al.  On the degree and separability of nonconvexity and applications to optimization problems , 1997, Math. Program..

[29]  R. Horst,et al.  DC Programming: Overview , 1999 .

[30]  Le Thi Hoai An,et al.  An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints , 2000, Math. Program..

[31]  L. Thi,et al.  Analyse numérique des algorithmes de l'optimisation D. C. . Approches locale et globale. Codes et simulations numériques en grande dimension. Applications , 1994 .

[32]  Hoai An Le Thi,et al.  Large Scale Molecular Conformation via the Exact Distance Geometry Problem , 2000 .

[33]  P. D. Tao,et al.  Partial regularization of the sum of two maximal monotone operators , 1993 .

[34]  Hoai An Le Thi,et al.  A continuous approch for globally solving linearly constrained quadratic , 2001 .

[35]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[36]  L. Muu,et al.  Simplicially-Constrained DC Optimization over Efficient and Weakly Efficient Sets , 2003 .

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

[38]  T. P. Dinh,et al.  Convex analysis approach to d.c. programming: Theory, Algorithm and Applications , 1997 .

[39]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[40]  H. Tuy.  Convex analysis and global optimization , 1998 .

[41]  Le Thi Hoai An,et al.  Combination between global and local methods for solving an optimization problem over the efficient set , 2002, Eur. J. Oper. Res..

[42]  P. Laurent.  Approximation et optimisation , 1972 .

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

[44]  Tao Pham Dinh,et al.  Proximal Decomposition on the Graph of a Maximal Monotone Operator , 1995, SIAM J. Optim..

[45]  P. D. Tao.  Convergence of a subgradient method for computing the bound norm of matrices , 1984 .

[46]  Le Thi Hoai An,et al.  A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem , 1998, SIAM J. Optim..

[47]  Pham Dinh Tao.  Algorithmes de calcul du maximum des formes quadratiques sur la boule unité de la norme du maximum , 1984 .

[48]  Le Thi Hoai An,et al.  Large-Scale Molecular Optimization from Distance Matrices by a D.C. Optimization Approach , 2003, SIAM J. Optim..

[49]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[50]  Le Thi Hoai An,et al.  Numerical solution for optimization over the efficient set by d.c. optimization algorithms , 1996, Oper. Res. Lett..

[51]  Hoai An Le Thi,et al.  On the ill-posedness of the trust region subproblem , 2003 .