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The purpose of this paper is to apply second order -approximation method introduced to optimization theory by Antczak [2] to obtain a new second order -saddle point criteria for vector optimization problems involving second order invex functions. Therefore, a second order -saddle point and the second order -Lagrange function are defined for the second order -approximated vector optimization problem constructed in this approach. Then, the equivalence between an (weak) efficient solution of the...
We propose new projection method for nonsmooth convex minimization problems. We present some method of subgradient selection, which is based on the so called residual selection model and is a generalization of the so called obtuse cone model. We also present numerical results for some test problems and compare these results with some other convex nonsmooth minimization methods. The numerical results show that the presented selection strategies ensure long steps and lead to an essential acceleration...
Il est démontré par Mentagui [ESAIM : COCV 9 (2003) 297-315] que, dans le cas des espaces de Banach généraux, la convergence d’Attouch-Wets est stable par une classe d’opérations classiques de l’analyse convexe, lorsque les limites des suites d’ensembles et de fonctions satisfont certaines conditions de qualification naturelles. Ceci tombe en défaut avec la slice convergence. Dans cet article, nous établissons des conditions de qualification uniformes assurant la stabilité de la slice convergence...
Il est démontré par Mentagui [ESAIM: COCV9 (2003) 297-315] que,
dans le cas des espaces de Banach généraux, la convergence
d'Attouch-Wets est stable par une classe d'opérations classiques de
l'analyse convexe, lorsque les limites des suites d'ensembles et de
fonctions satisfont certaines conditions de qualification naturelles. Ceci
tombe en défaut avec la slice convergence. Dans cet article, nous
établissons des conditions de qualification uniformes assurant la
stabilité de la slice convergence...
In this work, an alternative for sliding surface design based on linear and bilinear matrix inequalities is proposed. The methodology applies for reduced and integral sliding mode control, both continuous- and discrete-time; it takes advantage of the Finsler's lemma to provide a greater degree of freedom than existing approaches for sliding subspace design. The sliding surfaces thus constructed are systematically found via convex optimization techniques, which are efficiently implemented in commercially...
Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family involves the concept of cumulant transformation and a standard homogenization procedure.
There has been much interest in studying symmetric cone complementarity problems. In this paper, we study the circular cone complementarity problem (denoted by CCCP) which is a type of nonsymmetric cone complementarity problem. We first construct two smoothing functions for the CCCP and show that they are all coercive and strong semismooth. Then we propose a smoothing algorithm to solve the CCCP. The proposed algorithm generates an infinite sequence such that the value of the merit function converges...
We consider general convex large-scale optimization problems in finite dimensions. Under usual assumptions concerning the structure of the constraint functions, the considered problems are suitable for decomposition approaches. Lagrangian-dual problems are formulated and solved by applying a well-known cutting-plane method of level-type. The proposed method is capable to handle infinite function values. Therefore it is no longer necessary to demand the feasible set with respect to the non-dualized...
Nachdem der Begriff des sphärischen Bildes der Menge und der Begriff von sphärisch äquivalenten Mengen eingeführt wurde, werden verschiedene Zusammenhänge zwischen der Menge und ihrem sphärischen Bild untersucht und zwar unter verschiedenen Voraussetzung über (z. B. ihre Beschränkheit, Unbeschränkheit, strenge Konvexität). Die bewiesene Tatsache, dass die Menge und ihre -Umgebung sphärisch äquivalent sind, kann - sowie andere Ergebnisse der Arbeit - in der Theorie der konvexen parametrischen...
We are concerned with two-level optimization problems called strongweak
Stackelberg problems, generalizing the class of Stackelberg problems in the
strong and weak sense. In order to handle the fact that the considered two-level
optimization problems may fail to have a solution under mild assumptions, we
consider a regularization involving ε-approximate optimal solutions in the lower
level problems. We prove the existence of optimal solutions for such regularized
problems and present some approximation...
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