Nguyen Luan - Academia.edu (original) (raw)

Papers by Nguyen Luan

Journal of Optimization Theory and Applications, 2021

We consider the conic linear program given by a closed convex cone in an Euclidean space and a ma... more We consider the conic linear program given by a closed convex cone in an Euclidean space and a matrix, where vector on the right-hand-side of the constraint system and the vector defining the objective function are subject to change. Using the strict feasibility condition, we prove the locally Lipschitz continuity and obtain some differentiability properties of the optimal value function of the problem under right-hand-side perturbations. For the optimal value function under linear perturbations of the objective function, similar differentiability properties are obtained under the assumption saying that both primal problem and dual problem are strictly feasible. Keywords Conic linear programming • Primal problem • Dual problem • Optimal value function • Lipschitz continuity • Differentiability properties • Increment estimates Mathematics Subject Classification (2000) 49K40 • 90C31 • 90C25 • 90C30 1 Introduction If the feasible region or the objective function of a mathematical programming problem depends on a parameter, then the optimal value of the problem is a func-⋆ Dedicated to Professor Franco Giannessi on the occasion of his 85th birthday.

Optimization, 2019

It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many... more It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra in locally convex Hausdorff topological vector spaces. Our results develop those of X. Y. Zheng (Set-Valued Anal., Vol. 17, 2009, 389-408), which were established in a Banach space setting. Applications of the representation formulas to proving solution existence theorems for generalized linear programming problems and generalized linear vector optimization problems are shown.

Numerical Functional Analysis and Optimization, 2017

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hau... more Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential, are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized via the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.

Journal of Global Optimization, 2019

Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vecto... more Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions, weak and strong duality theorems. In particular, we show that the dual problem has the same structure as the primal problem, and the strong duality relation holds under three different sets of conditions.

Journal of Optimization Theory and Applications, 2021

We consider the conic linear program given by a closed convex cone in an Euclidean space and a ma... more We consider the conic linear program given by a closed convex cone in an Euclidean space and a matrix, where vector on the right-hand-side of the constraint system and the vector defining the objective function are subject to change. Using the strict feasibility condition, we prove the locally Lipschitz continuity and obtain some differentiability properties of the optimal value function of the problem under right-hand-side perturbations. For the optimal value function under linear perturbations of the objective function, similar differentiability properties are obtained under the assumption saying that both primal problem and dual problem are strictly feasible. Keywords Conic linear programming • Primal problem • Dual problem • Optimal value function • Lipschitz continuity • Differentiability properties • Increment estimates Mathematics Subject Classification (2000) 49K40 • 90C31 • 90C25 • 90C30 1 Introduction If the feasible region or the objective function of a mathematical programming problem depends on a parameter, then the optimal value of the problem is a func-⋆ Dedicated to Professor Franco Giannessi on the occasion of his 85th birthday.

Optimization, 2019

It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many... more It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra in locally convex Hausdorff topological vector spaces. Our results develop those of X. Y. Zheng (Set-Valued Anal., Vol. 17, 2009, 389-408), which were established in a Banach space setting. Applications of the representation formulas to proving solution existence theorems for generalized linear programming problems and generalized linear vector optimization problems are shown.

Numerical Functional Analysis and Optimization, 2017

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hau... more Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential, are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized via the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.

Journal of Global Optimization, 2019

Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vecto... more Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions, weak and strong duality theorems. In particular, we show that the dual problem has the same structure as the primal problem, and the strong duality relation holds under three different sets of conditions.