Pham Dinh Tao

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

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

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

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.

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

In this paper we outline a d.c. optimization scheme and use it for (locally) maximizing a concave, a convex or a quadratic function f over the efficient set of a multiple objective convex program. We also propose a decomposition method for globally solving this problem with f concave. Numerical experiences are discussed.

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

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.

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

We present DCA for globally minimizing quadratic forms on Euclidean balls and spheres. Since these problems admit at most one local-nonglobal minimizer, DCA converges in general to a solution for these problems. Numerical simulations show robustness, stability and efficiency of DCA with respect to related standard methods.

A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem

AbstractWe present a numerical method for solving the d.c. programming problem

Methods for optimizing over the efficient and weakly efficient sets of an affine fractional vector optimization program

c^* = \min \{ \langle c,x\rangle s.t. f_i (x) \leqslant 0, i = 1,...,m, x \in D\}

Fuel Mixture Nonconvex Optimization Problem: Solution Methods and Numerical Simulations

wherefi, i=1,...,m are d.c. (difference of two convex functions) and D is a convex set in ℝn. An (ɛ, η)-solutionx(ɛ, η) satisfying