Global optimization algorithm for solving bilevel programming problems with quadratic lower levels
Paul B. Hermanns Nguyen Van Thoai
In this article, we propose a method for finding the global optimum of a class of nonlinear bilevel programming problems. The main idea of this method is to construct iteratively a sequence of points either ending at an optimal solution of the equivalent problem with a complementarity constraint, or converging to an optimal solution. The construction of such a sequence is performed by using a branch-and-bound scheme, together with some relaxation techniques, which are successfully applied in global optimization. Some illustrative examples and results on computational experiments are reported.
keywords: Bilevel programming nonconvex programming branch and bound.
Decomposition branch and bound algorithm for optimization problems over efficient sets
Nguyen Van Thoai
The problem of optimizing some given function over the efficient set is one of the most interesting and important concepts in multicriteria decision making. As the efficient set is in general nonconvex, even for the case of linear multicriteria programming problems, optimizing over the efficient set belongs to a typical problem class of multiextremal optimization problems, which can have local optima different from global optima.
    In this article, we consider the case where the multicriteria programming problem is linear. Characterizing the set of efficient solutions by some constraint of 'reverse convex' type in the space of criteria, we formulate the problem of minimizing a function $f$ over the efficient set as a global optimization problem with a special structure. For the resulting problem, a decomposition branch and bound based algorithm is then proposed, in which the branching procedure is performed in the criteria space. Convergence properties of the algorithm are discussed, and preliminary computational results are reported.
keywords: decomposition. branch and bound multi-criteria optimization reverse convex constraint Nonconvex programming optimization over efficient set
On an inverse problem for fractional evolution equation
Nguyen Huy Tuan Mokhtar Kirane Long Dinh Le Van Thinh Nguyen
In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.
keywords: Fractional evolution equation backward problem regularization
On the blow-up results for a class of strongly perturbed semilinear heat equations
Van Tien Nguyen
We consider in this work some class of strongly perturbed for the semilinear heat equation with Sobolev sub-critical power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. We also classify all possible asymptotic behaviors of the solution when it approaches to singularity. Finally, we describe precisely the blow-up profiles corresponding to these behaviors.
keywords: blow-up profile lower order perturbations. semilinear heat equation asymptotic behavior Finite-time blow-up
A family of extragradient methods for solving equilibrium problems
Thi Phuong Dong Nguyen Jean Jacques Strodiot Thi Thu Van Nguyen Van Hien Nguyen
In this paper we introduce a class of numerical methods for solving an equilibrium problem. This class depends on a parameter and contains the classical extragradient method and a generalization of the two-step extragradient method proposed recently by Zykina and Melen'chuk for solving variational inequality problems. Convergence of each algorithm of this class to a solution of the equilibrium problem is obtained under the condition that the equilibrium function associated with the problem is pseudomonotone and Lipschitz continuous. Some preliminary numerical results are given to compare the numerical behavior of the two-step extragradient method with respect to the other methods of the class and in particular to the extragradient method.
keywords: variational inequalities. two-step extragradient method Extragradient method equilibrium problems

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