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In this paper, several Turán-type inequalities for the more generalized Mittag-Leffler function are proved. In addition, we also gave affirmative answers to two open problems posed by Mehrez and Sitnik.

The purpose of this paper is to study the approximate optimality condition for composite convex optimization problems with a cone-convex system in locally convex spaces, where all functions involved are not necessarily lower semi-continuous. By using the properties of the epigraph of conjugate functions, we introduce a new regularity condition and give its equivalent characterizations. Under this new regularity condition, we derive necessary and sufficient optimality conditions of \(\varepsilon \)-optimal solutions for the composite convex optimization problem. As applications of our results, we derive approximate optimality conditions to cone-convex optimization problems. Our results extend or cover many known results in the literature.

Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.

In this paper, we consider robust optimal solutions for a convex optimization problem in the face of data uncertainty both in the objective and constraints. By using the properties of the subdifferential sum formulae, we first introduce a robust-type subdifferential constraint qualification, and then obtain some completely characterizations of the robust optimal solution of this uncertain convex optimization problem. We also investigate Wolfe type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Moreover, we show that our results encompass as special cases some optimization problems considered in the recent literature.

The control of the hypersonic glide vehicle (HGV) during re-entry is confronted with the input saturation and unknown dynamical uncertainty problem. A nonlinear anti-windup (AW) compensation control method is designed to deal with the nonlinear control problem caused by the saturation of the control surfaces of the HGV. The scheme transforms the problem of saturation control into the optimization problem through nonlinear matrix inequality (NMIL) constraints, which is solved by particle swarm optimization (PSO). The saturation situation can be relieved within a short time and the satisfactory control performance is attained by the method. Then, a novel sigmoid function tracking differentiator based disturbance observer (STDDO) is proposed to estimate parameter uncertainties and external disturbances for the problem of dynamical uncertainties. This method does not need the priori information about the bounds of disturbances and has global fast convergence property. Finally, the simulation results show the effectiveness and robustness of the proposed control scheme.

The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider a DC infinite programming problem (P) with inequality constraints. By using the properties of the epigraph of the conjugate functions, we introduce some new notions of regularity conditions for (P). Under these new regularity conditions, we completely characterize the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for (P). Similarly, we also completely characterize the Farkas-type results and the stable Farkas-type results for (P). As applications, we obtain the corresponding results for conic programming problems.

This paper is concerned with the boundedness of solutions of a new chaotic system. For this system, the global exponential attractive set and positively invariant set are derived based on generalized Lyapunov function theory and the extremum principle of function. Furthermore, we can conclude that the rate of the trajectories of the system going from the exterior of the attractive set ? ? to the interior of the attractive set ? ? is an exponential rate. The rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme.

In this paper, by using the properties of the epigraph of the conjugate functions, we introduce some closedness conditions and investigate some characterizations of these closedness conditions. Then, by using these closedness conditions, we obtain some Farkas-type results for a constrained fractional programming problem with DC functions. We also show that our results encompass as special cases some programming problems considered in the recent literature.

In this article, gap functions for a generalized vector equilibrium problem (GVEP) with explicit constraints are investigated. Under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are investigated by considering the different perturbations to GVEP. Then, gap functions for GVEP are established by using the weak and strong duality results. As application, the proposed approach is applied to construct gap functions for a vector optimization problem and a generalized vector variational inequality problem.

This paper is concerned with generalized second-order contingent epiderivatives of frontier and solution maps in parametric vector optimization problems. Under some mild conditions, we obtain some formulas for computing generalized second-order contingent epiderivatives of frontier and solution maps, respectively. We also give some examples to illustrate the corresponding results.

In this paper, we present a duality theory for fractional programming problems in the face of data uncertainty via robust optimization. By employing conjugate analysis, we establish robust strong duality for an uncertain fractional programming problem and its uncertain Wolfe dual programming problem by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. We show that our results encompass as special cases some programming problems considered in the recent literature. Moreover, we also show that robust strong duality always holds for linear fractional programming problems under scenario data uncertainty or constraint-wise interval uncertainty, and that the optimistic counterpart of the dual is tractable computationally.

In this paper, we deal with extended Ky Fan inequalities (EKFI) with DC functions. Firstly, a dual scheme for (EKFI) is introduced by using the method of Fenchel conjugate function. Under suitable conditions, weak and strong duality assertions are obtained. Then, by using the obtained duality assertions, some Farkas-type results which characterize the optimal value of (EKFI) are given. Finally, as applications, the proposed approach is applied to a convex optimization problem (COP) and a generalized variational inequality problem (GVIP).

In this paper, by virtue of lower Studniarski derivatives of set-valued maps, relationships between lower Studniarski derivative of a set-valued map and its profile map are discussed. Some results concerning sensitivity analysis are obtained in parametrized vector optimization.

By using the formula of theε-subdifferential for the sum of a convex function with a composition of convex functions, some necessary and sufficient optimality conditions for a DC programming problem involving a composite function are obtained. As applications, a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator are examined at the end of this paper.

In this paper, a DC infinite programming problem with inequality constraints is considered. By using the method of Fenchel conjugate functions, a dual scheme for the DC infinite programming problem is introduced. Then, under suitable conditions, weak and strong duality assertions are obtained. Moreover, by using the obtained duality assertions, some Farkas-type results which characterize the optimal value of the DC infinite programming problem are given. As applications, the proposed approach is applied to conic programming problems.

A novel theory, namely, Fourier mode coupling (FMC) theory for fiber Bragg gratings (FBGs) is proposed in this paper. During analyzing coupled modes of FBGs, the Fourier transform relations among the amplitude coefficients of coupled modes are found for the first time. The general expressions of reflective and transmissive spectra of FBGs are deduced from the combination of Fourier transform with the well-known coupled-mode theory. In the proposed FMC theory, the spectral characteristics of the FBG are achieved by the calculation of coupled modes in the spatial domain spectrum, which is the Fourier transform result of refractive index perturbation in the FBG. The FBG spectrum based on the FMC theory is simulated here, and compared with those obtained from the coupled mode theory and pure Fourier transform. The comparison shows that the FMC theory for and the derived spactra of FBGs are in accordance with the coupled mode theory and the practical spectra of the FBG respectively. The FMC theory has many features, these being simple, clear, direct, accurate and fast, which could be used as a universal tool for fast spectrum analysis of any FBG with an arbitrary distribution of refractive index perturbation along the fiber axis.

A novel theory, namely, Fourier mode coupling (FMC) theory for long-period fiber gratings (LPFGs) is proposed in this paper. During analyzing the co-propagating coupling between the core mode and cladding modes in LPFGs, the Fourier transform relations among the amplitude coefficients of co-propagating coupled-modes are found for the first time, to the best of authors’ knowledge. The general expressions of the coupling and transmission spectra of LPFGs are also deduced from the combination of Fourier transform with the well-known coupled-mode theory. In the proposed FMC theory, the spectral characteristics of the LPFGs without over-coupling are derived from the calculation of co-propagating mode coupling in the spatial domain spectrum, which is the Fourier transform result of refractive index perturbation in the LPFG. According to the FMC theory, the spectra of the LPFGs in different perturbation amplitudes and lengths are numerically simulated here. A measured transmission spectrum is also compared with the calculated transmission spectra based on the FMC theory and the coupled mode theory, respectively. The comparison shows that the FMC theory and the derived spectra for LPFGs are in consistance with the coupled-mode theory and the practical spectra of LPFGs respectively. The FMC has many features, these being simple, fast and accurate, which could be employed for spectrum analysis of any LPFG with an arbitrary distribution of refractive index perturbation along the fiber axis.

In this paper, we introduce a notion of Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems in terms of weakly efficient solutions. We obtain some metric characterizations of Levitin-Polyak well-posedness for this problem. We derive the relations between the Levitin-Polyak well-posedness and the upper semi-continuity of approximate solution maps for generalized semi-infinite multiobjective programming problems. Examples are given to illustrate our main results.