55 research outputs found
On the Necessity of the Sufficient Conditions in Cone-Constrained Vector Optimization
The object of investigation in this paper are vector nonlinear programming
problems with cone constraints. We introduce the notion of a Fritz John
pseudoinvex cone-constrained vector problem. We prove that a problem with cone
constraints is Fritz John pseudoinvex if and only if every vector critical
point of Fritz John type is a weak global minimizer. Thus, we generalize
several results, where the Paretian case have been studied.
We also introduce a new Frechet differentiable pseudoconvex problem. We
derive that a problem with quasiconvex vector-valued data is pseudoconvex if
and only if every Fritz John vector critical point is a weakly efficient global
solution. Thus, we generalize a lot of previous optimality conditions,
concerning the scalar case and the multiobjective Paretian one.
Additionally, we prove that a quasiconvex vector-valued function is
pseudoconvex with respect to the same cone if and only if every vector critical
point of the function is a weak global minimizer, a result, which is a natural
extension of a known characterization of pseudoconvex scalar functions.Comment: 12 page
Second-Order Karush-Kuhn-Tucker Optimality Conditions for Vector Problems with Continuously Differentiable Data and Second-Order Constraint Qualifications
Some necessary and sufficient optimality conditions for inequality
constrained problems with continuously differentiable data were obtained in the
papers [I. Ginchev and V.I. Ivanov, Second-order optimality conditions for
problems with C\sp{1} data, J. Math. Anal. Appl., v. 340, 2008, pp.
646--657], [V.I. Ivanov, Optimality conditions for an isolated minimum of order
two in C\sp{1} constrained optimization, J. Math. Anal. Appl., v. 356, 2009,
pp. 30--41] and [V. I. Ivanov, Second- and first-order optimality conditions in
vector optimization, Internat. J. Inform. Technol. Decis. Making, 2014, DOI:
10.1142/S0219622014500540].
In the present paper, we continue these investigations. We obtain some
necessary optimality conditions of Karush--Kuhn--Tucker type for scalar and
vector problems. A new second-order constraint qualification of Zangwill type
is introduced. It is applied in the optimality conditions.Comment: 1
Second-order optimality conditions for problems with C1 data
AbstractIn this paper we obtain second-order optimality conditions of Karush–Kuhn–Tucker type and Fritz John one for a problem with inequality constraints and a set constraint in nonsmooth settings using second-order directional derivatives. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable, but their gradients are not necessarily locally Lipschitz. In the sufficient conditions for a global minimum x¯ we assume that the objective function is differentiable at x¯ and second-order pseudoconvex at x¯, a notion introduced by the authors [I. Ginchev, V.I. Ivanov, Higher-order pseudoconvex functions, in: I.V. Konnov, D.T. Luc, A.M. Rubinov (Eds.), Generalized Convexity and Related Topics, in: Lecture Notes in Econom. and Math. Systems, vol. 583, Springer, 2007, pp. 247–264], the constraints are both differentiable and quasiconvex at x¯. In the sufficient conditions for an isolated local minimum of order two we suppose that the problem belongs to the class C1,1. We show that they do not hold for C1 problems, which are not C1,1 ones. At last a new notion parabolic local minimum is defined and it is applied to extend the sufficient conditions for an isolated local minimum from problems with C1,1 data to problems with C1 one
First-Order Conditions for Optimization Problems with Quasiconvex Inequality Constraints
2000 Mathematics Subject Classification: 90C46, 90C26, 26B25, 49J52.The constrained optimization problem min f(x), gj(x) ≤ 0 (j = 1,…p) is considered, where f : X → R and gj : X → R are nonsmooth functions with domain X ⊂ Rn. First-order necessary and first-order sufficient optimality conditions are obtained when gj are quasiconvex functions. Two are the main features of the paper: to treat nonsmooth problems it makes use of Dini derivatives; to obtain more sensitive conditions, it admits directionally dependent multipliers. The two cases, where the Lagrange function satisfies a non-strict and a strict inequality, are considered. In the case of a non-strict inequality pseudoconvex functions are involved and in their terms some properties of the convex programming problems are generalized. The efficiency of the obtained conditions is illustrated on examples
Unusual magnetoelectric effect in paramagnetic rare-earth langasite
Violation of time reversal and spatial inversion symmetries has profound
consequences for elementary particles and cosmology. Spontaneous breaking of
these symmetries at phase transitions gives rise to unconventional physical
phenomena in condensed matter systems, such as ferroelectricity induced by
magnetic spirals, electromagnons, non-reciprocal propagation of light and spin
waves, and the linear magnetoelectric (ME) effect - the electric polarization
proportional to the applied magnetic field and the magnetization induced by the
electric field. Here, we report the experimental study of the holmium-doped
langasite, HoLaGaSiO, showing a puzzling combination
of linear and highly non-linear ME responses in the disordered paramagnetic
state: its electric polarization grows linearly with the magnetic field but
oscillates many times upon rotation of the magnetic field vector. We propose a
simple phenomenological Hamiltonian describing this unusual behavior and derive
it microscopically using the coupling of magnetic multipoles of the rare-earth
ions to the electric field.Comment: 8 pages, 3 figure
Electrospun magnetic composite poly-3-hydroxybutyrate/magnetite scaffolds for biomedical applications: composition, structure, magnetic properties, and biological performance
Magnetically responsive composite polymer scaffolds have good potential for a variety of biomedical applications. In this work, electrospun composite scaffolds made of polyhydroxybutyrate (PHB) and magnetite (Fe3O4) particles (MPs) were studied before and after degradation in either PBS or a lipase solution. MPs of different sizes with high saturation magnetization were synthesized by the coprecipitation method followed by coating with citric acid (CA). Nanosized MPs were prone to magnetite-maghemite phase transformation during scaffold fabrication, as revealed by Raman spectroscopy; however, for CA-functionalized nanoparticles, the main phase was found to be magnetite, with some traces of maghemite. Submicron MPs were resistant to the magnetite-maghemite phase transformation. MPs did not significantly affect the morphology and diameter of PHB fibers. The scaffolds containing CA-coated MPs lost 0.3 or 0.2% of mass in the lipase solution and PBS, respectively, whereas scaffolds doped with unmodified MPs showed no mass changes after 1 month of incubation in either medium. In all electrospun scaffolds, no alterations of the fiber morphology were observed. Possible mechanisms of the crystalline-lamellar-structure changes in hybrid PHB/Fe3O4 scaffolds during hydrolytic and enzymatic degradation are proposed. It was revealed that particle size and particle surface functionalization affect the mechanical properties of the hybrid scaffolds. The addition of unmodified MPs increased scaffolds' ultimate strength but reduced elongation at break after the biodegradation, whereas simultaneous increases in both parameters were observed for composite scaffolds doped with CA-coated MPs. The highest saturation magnetization-higher than that published in the literature-was registered for composite PHB scaffolds doped with submicron MPs. All PHB scaffolds proved to be biocompatible, and the ones doped with nanosized MPs yielded faster proliferation of rat mesenchymal stem cells. In addition, all electrospun scaffolds were able to support angiogenesis in vivo at 30 days after implantation in Wistar rats
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