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.

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

Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. This paper is devoted to a thorough study on convex analysis approach to d.c. (difierence of convex functions) programming and gives the State of the Art. Main results about d.c. duality, local and global opti- malities in d.c. programming are presented. These materials constitute the basis of the DCA (d.c. algorithms). Its convergence properties have been tackled in detail, especially in d.c. polyhedral programming where it has flnite convergence. Exact penalty, Lagrangian duality without gap, and regularization techniques have beeen studied to flnd appropriate d.c. de- compositions and to improve consequently the DCA. Finally we present the application of the DCA to solving a lot of important real-life d.c. pro- grams.

Convex analysis approach to d.c. programming: Theory, Algorithm and Applications

Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.

DC programming: overview

We investigate the convex–concave procedure, a local heuristic that utilizes the tools of convex optimization to find local optima of difference of convex (DC) programming problems. The class of DC problems includes many difficult problems such as the traveling salesman problem. We extend the standard procedure in two major ways and describe several variations. First, we allow for the algorithm to be initialized without a feasible point. Second, we generalize the algorithm to include vector inequalities. We then present several examples to demonstrate these algorithms.

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Variations and extension of the convex–concave procedure

This paper is devoted to difference of convex functions (d.c.) optimization: d. c. duality, local and global optimality conditions in d. c. programming, the d. c. algorithm (DCA), and its application to solving the trust-region problem. The DCA is an iterative method that is quite different from well-known related algorithms. Thanks to the particular structure of the trust-region problem, the DCA is very simple (requiring only matrix-vector products) and, in practice, converges to the global solution. The inexpensive implicitly restarted Lanczos method of Sorensen is used to check the optimality of solutions provided by the DCA. When a nonglobal solution is found, a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the DCA at this point. It is shown that in the nonconvex case, the DCA converges to the global solution of the trust-region problem, using only matrix-vector products and requiring at most 2m+2 restarts, where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem. Numerical simulations establish the robustness and efficiency of the DCA compared to standard related methods, especially for large-scale problems.

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A D. C. Optimization Algorithm for Solving the Trust-Region Subproblem

Linearly constrained indefinite quadratic problems play an important role in global optimization. In this paper we study d.c. theory and its local approach to such problems. The new algorithm, CDA, efficiently produces local optima and sometimes produces global optima. We also propose a decomposition branch and bound method for globally solving these problems. Finally many numerical simulations are reported.

Solving a Class of Linearly Constrained Indefinite QuadraticProblems by D.C. Algorithms

In this paper we propose a new branch and bound algorithm using a rectangular partition and ellipsoidal technique for minimizing a nonconvex quadratic function with box constraints. The bounding procedures are investigated by d.c. (difference of convex functions) optimization algorithms, called DCA. This is based upon the fact that the application of the DCA to the problems of minimizing a quadratic form over an ellipsoid and/or over a box is efficient. Some details of computational aspects of the algorithm are reported. Finally, numerical experiments on a lot of test problems showing the efficiency of our algorithm are presented.

A Branch and Bound Method via d.c. Optimization Algorithms andEllipsoidal Technique for Box Constrained Nonconvex Quadratic Problems

Numerical solution for optimization over the efficient set by d.c. optimization algorithms

In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for minimizing an indefinite quadratic function over a bounded polyhedral convex set which is not necessarily given explicitly by a system of linear inequalities and/or equalities. It is required that for this set there exists an efficient algorithm to verify whether a point is feasible, and to find a violated constraint if this point is not feasible. The algorithm is based upon the fact that the problem of minimizing an indefinite quadratic form over an ellipsoid can be efficiently solved by some available (polynomial and nonpolynomial time) algorithms. In particular, the d.c. (difference of convex functions) algorithm (DCA) with restarting procedure recently introduced by Pham Dinh Tao and L.T. Hoai An is applied to globally solving this problem. DCA is also used for locally solving the nonconvex quadratic program. It is restarted with current best feasible points in the branch-and-bound scheme, and improved them in its turn. The combined DCA-ellipsoidal branch-and-bound algorithm then enhances the convergence: it reduces considerably the upper bound and thereby a lot of ellipsoids can be eliminated from further consideration. Several numerical experiments are given.

Le Thi Hoai An

Papers

Solving a Class of Linearly Constrained Indefinite QuadraticProblems by D.C. Algorithms

A Branch and Bound Method via d.c. Optimization Algorithms andEllipsoidal Technique for Box Constrained Nonconvex Quadratic Problems

Numerical solution for optimization over the efficient set by d.c. optimization algorithms

A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems

Difference of convex functions optimization algorithms (DCA) for globally minimizing nonconvex quadratic forms on Euclidean balls and spheres