Sufficiency Theorem Research Articles

A sublinear functional based approximated equivalence to optimality and duality for multiobjective programming in complex spaces

In this paper, we introduce a new approximation approach for a class of multiobjective programming problems in complex spaces and their duals. Using a sublinear functional, an approximated problem is constructed at a given feasible solution of the original problem. The equivalence between the solution of the considered multiobjective complex programming problem and its approximated problem is established by deriving the sufficiency theorem under the F-convexity assumption. An example showing this equivalence is also presented. Further, corresponding to the approximated problem, the Mond–Weir and Wolfe-type approximated dual problems are formulated. The duality results for the Mond–Weir and Wolfe-type dual problems are derived using the duality results established for the approximated Mond–Weir and Wolfe-type dual problems, respectively. The importance of the optimality conditions and proposed approximation approach is highlighted in a blind equalization problem in a communication system.

Uniqueness and stability for a seismic-type generalized Radon transform

In this paper, we consider a generalized seismic Radon transform that maps a given function to its integrals over a certain family of curves in the plane. Such transforms arise in many areas of mathematics, geophysics, and imaging science. This paper contains new explicit inversion formulas of this generalized seismic Radon transform for various families of curves based on their monotonicity. We derive an analogue of the Fourier slice theorem and present a Sobolev space to obtain a stability estimate for this Radon transform. Finally, we consider local uniqueness results for which we prove a sufficiency theorem.

Coefficients of multiple Fourier-Haar series and variational modulus of continuity

In this paper, we introduce the concept of a variational modulus of continuity for functions of several variables, give an estimate for the sum of the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity, and prove theorems of absolute convergence of series composed of the coefficients of multiple Fourier-Haar series. In this paper, we study the issue of the absolute convergence for multiple series composed of the Fourier-Haar coefficients of functions of several variables of bounded p-variation. We estimate the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity and prove the sufficiency theorem for the condition for the absolute convergence of series composed of the Fourier-Haar coefficients of the considered function class. This paper researches the question: under what conditions, imposed on the variational modulus of continuity of the fractional order of several variables functions, there is the absolute convergence for series composed of the coefficients of multiple Fourier-Haar series.

Solvability of the Quaternary Continuous Classical Boundary Optimal Control Dominated by Quaternary Parabolic System

The purpose of this paper is to study the solvability of the quaternary continuous classical boundary optimal control vector problem dominated by quaternary nonlinear parabolic boundary value problem with state constraints. The existence theorem for a quaternary continuous classical boundary optimal control vector with equality and inequality state constraints is stated and demonstrated under suitable assumptions. The mathematical formulation of the quaternary adjoint eqs. associated with the quaternary nonlinear parabolic boundary value problem with state constraints is discovered. The Fréchet derivative of the cost function and the constraints functions are derived. The necessary and sufficient theorems (conditions) for optimality of the quaternary continuous classical boundary optimal control vector problem are stated and demonstrated under suitable assumptions.

Identification of discontinuous parameters in contaminant convection–reaction–diffusion model of recovered fracturing fluid

The primary goal of this paper is to identify the discontinuous parameters in a complicated contaminant convection–reaction–diffusion model of recovered fracturing fluid (RFF model, for short), which is modeled by a nonlinear stationary incompressible Navier–Stokes equation with multivalued and nonsmooth friction boundary condition coupled with a nonlinear convection–reaction–diffusion equation involving mixed Neumann boundary conditions. First, we impose the parameter identification problem which is, exactly, a nonlinear and nonsmooth inverse problem. Then, we show the local boundedness and weakly relative compactness of solution map to RFF model with respect to the discontinuous parameters. Moreover, the generalized continuity of solution map of RFF model is proved. Finally, via employing the theory of nonsmooth analysis and optimization, a sufficiency theorem of existence of solutions to the inverse problem under consideration is established.

A novel numerical approach and stability analysis for a class of pantograph delay differential equation

The goal of this research is to introduce a numerical technique based on shifted Chebyshev polynomial and collocation method for a second order non-linear Lane–Emden pantograph delay differential equation (PDDE) subject to initial conditions. The proposed formulation is based on converting the initial value problem (IVP) into an equivalent fundamental algebraic equation which reduces computational expense. The convergence analysis of the proposed numerical technique demonstrates the efficiency of the proposed scheme. The uniqueness and existence, regularity and stability analysis of the Lane–Emden PDDE is discussed and investigated thoroughly by providing sufficient theorems. Due to the non-availability of abundant literature, the new approach is implemented and tested against existing approaches on standard and newly constructed nonlinear examples. The comparison demonstrates that the proposed method is more accurate and consumes lesser CPU time to compute the results than the existing methods.

Composition operators on the polydisc

In this paper we study the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc Dn. Studying the volume of sublevel sets we show for which n the necessary conditions obtained by Bayart are sufficient. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.

Global Mittag-Leffler Stability and Global Asymptotic ω-Period for Fractional-Order Cohen–Grossberg Neural Networks with Time-Varying Delays

The dynamic behaviors for fractional-order Cohen–Grossberg neural networks with time-varying delays (FCGNND) are studied in this paper. By introducing the Mittag-Leffler (ML) function, based on properties of fractional calculus, the differential mean-value theorem and Arzela–Ascoli theorem, we give some sufficient theorems to determine the boundedness, global Mittag-Leffler stability (GMLS) and global asymptotical [Formula: see text]-periodicity (GAP) for FCGNND. Finally, a numerical example is given to verify the effectiveness of the theorems.

New sufficient conditions for p-th moment stability of uncertain delay differential equation

Uncertain delay differential system is an important mathematical model. Stability is a basic problem of uncertain delay differential system. Delay and uncertain interference often lead to changes in the stability of the system. Establishing the judgment of the stability of uncertain delay differential system conditions is very important. Based on the strong Lipschitz condition, the judgment of p-th moment stability for uncertain delay differential equations (UDDEs) has been investigated. Actually, the strong Lipschitz condition is assumed that it only relates to the current state, it is difficult to be employed to determine the stability in p-th moment for the UDDEs. In this paper, we consider two kinds of new Lipschitz conditions containing the current state and the past state, which are more weaker than the strong Lipschitz condition. Meanwhile, new sufficient theorems and corollaries under the new Lipschitz conditions as the tools to judge the p-th moment stability for the UDDEs are proved. Some examples explain the rationality of the corresponding theorems and corollaries.

Utility Maximization of Bangladeshi Consumers within Their Budget: A Mathematical Procedure

A consumer is considered as a person or a group of people who uses purchased goods, products, or services only for personal use, and not for manufacturing or resale. Consumers usually purchase valuable and useful commodities or goods by spending all or partial of their income. The property of a commodity that enables it to satisfy human wants is called utility. Producers must be conscious to increase the utility among the consumers. This study has considered the maximization of utility problem of consumers of Bangladesh subject to two constraints; namely, budget constraint and coupon constraint. Consequently, in the study two Lagrange multipliers are used and interpreted these with mathematical analysis. Prediction of consumer behavior will help both producers and consumers to take decision of their future economic productions and consumptions, respectively. This article is ornamented with sufficient theorems and economic analyses. So that all the readers find interest when go through the economic analysis of utility maximization.

On optimality and duality in interval-valued variational problem withB-(p,r)-invexity

In this paper, we consider a class of interval-valued variational optimization problem. We extend the definition ofB-(p,r)-invexity which was originally defined for scalar optimization problem to the interval-valued variational problem. The necessary and sufficient optimality conditions for the problem have been established underB-(p,r)-invexity assumptions. An application, showing utility of the sufficiency theorem in real-world problem, has also been provided. In addition to this, for the interval-optimization problem Mond–Weir and Wolfe type duals are presented and related duality theorems have been proved. Non-trivial examples verifying the results have also been presented throughout the paper.

Stability in mean for uncertain delay differential equations based on new Lipschitz conditions

Uncertain delay differential equations (UDDEs) involving a current state and a certain past state have been proposed to model an uncertain system with a delay time, such as uncertain delay logistic model. Stability of the UDDEs is a vital problem for its applications. Based on the strong Lipschitz condition, the stability in mean for UDDEs have been investigated. Actually, the strong Lipschitz condition is assumed that it only relates to the current state, it is difficult to be employed to determine the stability in mean for the UDDEs. In this paper, we propose two kinds of new Lipschitz conditions containing the current state and the past state, which are more weaker than the strong Lipschitz condition. Meanwhile, two sufficient theorems based on these new Lipschitz conditions as the tools to judge the stability in mean for the UDDEs are verified. For a special class of the UDDEs, which are proved to be stable in mean without any limited condition. Besides, some examples are discussed in this paper.

Weak Measurable Optimal Controls for the Problems of Bolza

Two sufficiency theorems for parametric and a nonparametric problems of Bolza in optimal control are derived. The dynamics of the problems are nonlinear, the initial and final states are free, and the main results can be applied when nonlinear mixed time-state-control inequality and equality constraints are presented. The deviation between admissible costs and optimal costs around the optimal control is estimated by functionals playing the role of the square of some norms.

$\mathcal{A}$-Quasiconvexity, Gårding Inequalities, and Applications in PDE Constrained Problems in Dynamics and Statics

A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimizers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$-local minimizers in the classical $\mathcal{A}={curl}$ case.

Robust energy-to-peak control for Markov jump system with multiple pure time delays

This paper investigates the robust energy-to-peak stability analysis and controller design for Markov jump system with multiple pure time delays. The aim is to get some sufficient conditions such that the controlled system is asymptotically stable with an anti-disturbance performance. Firstly, according to the system transformation, the Markov jump multiple pure time delays system is transformed into a new description, which includes a non-time-delay item and some integral items. Secondly, according to Lyapunov stability theory and linear matrix inequality (LMI) technique, the sufficient theorems are achieved for the Markov jump multiple pure time delays system to be energy-to-peak stability and stabilisation. If those theorems are solvable, controllers can be obtained such that the controlled systems are stable and the peak-values of the controlled outputs are constrained for any energy-bounded external disturbance inputs. Moreover, the uncertain cases are considered, and the robust stability conditions are achieved. Finally, examples are given, and the effectiveness of the obtained methods is illustrated.

On Sufficient Optimality Conditions for Infinite Horizon Optimal Control Problems

We consider the Seierstad sufficiency theorem in comparison with the Mangasarian and Arrow sufficiency theorems for optimal control problems with infinite time horizon. Both finite and infinite values of the objective functional are allowed, since the concepts of overtaking and weakly overtaking optimality are implied. We extend the conditions under which the Seierstad sufficiency theorem can be applied and provide appropriate examples. The sufficient conditions are shown to be both necessary and sufficient when the Hamiltonian is linear with respect to state and control. We obtain a new form of sufficient optimality conditions in the case when the Hamiltonian is neither concave nor differentiable with respect to control.

Necessary and sufficient conditions for the strong local minimality of C1 extremals on a class of non-smooth domains

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for C1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky and Mengesha (2009), where smoothness assumptions are made on the boundary.

On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor

In this article, the sufficiency issues, first integrals and exact solutions for the Uzawa-Lucas model with unskilled labor are investigated. The sufficient conditions are established by utilizing Arrow's Sufficiency theorem. The non-negativeness conditions for the balanced growth path (BGP) are provided and growth rate is explicitly given in terms of parameters of the model. The first integrals are established by the partial Hamiltonian approach. Then first integrals are utilized to construct the exact solutions for all the variables. The growth rates of all variables and graphical representation of exact solutions are provided for the special case when the inverse of the intertemporal elasticity of substitution is the same as the share of physical capital.

A Note on Arrow-Enthoven's Sufficiency Theorem on Quasi-Concave Programming: Corrigendum to: Quasi-Concave Programming

In this note, we reconsider Arrow–Enthoven’s (1961) Sufficiency Theorem 1 onquasi-concave programming [Econometrica, 29(4), p. 783]. One example illustrates thefact that condition (c) of their Theorem 1 should include the twice continuous differen-tiability of the quasi-concave objective function so that the conditions of the theoremare sufficient to identify global optima.

On $$\epsilon $$ ϵ -solutions for robust semi-infinite optimization problems

In this paper, we consider semi-infinite optimization problems involving a convex objective function and infinitely many convex constraint functions with data uncertainty, and give its robust counterpart \(\mathrm{(RSIP)}\). Moreover, we consider approximate solutions (\(\epsilon \)-solutions) for \(\mathrm{(RSIP)}\). Using robust optimization approach (worst-case approach), we establish robust necessary optimality and robust sufficient theorems and give duality results for \(\epsilon \)-solutions for \(\mathrm{(RSIP)}\) under a closed convex cone constraint qualification. Moreover, an example is given to illustrate the obtained duality results.