Sufficient Conditions In Order Research Articles

Zeros of Dirichlet polynomials via a density criterion

We obtain a necessary and sufficient condition in order that a semi-plane of the form ℜ(s)>r, r∈R, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalisation of a well-known criterion for the truth of the Riemann hypothesis due to Báez-Duarte. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte's one is also established.

Convergence Analysis of the Implicit Euler-discretization and Sufficient Conditions for Optimal Control Problems Subject to Index-one Differential-algebraic Equations

For optimal control problems subject to index-one differential-algebraic equations in semi-explicit form we discuss second order sufficient conditions in form of a coercivity condition taking into account the two-norm discrepancy. Furthermore we introduce a related Riccati-type and Legendre-Clebsch condition which are sufficient for the validity of the coercivity condition. Using the implicit Euler-discretization we approximate the optimal control problem and analyze the convergence of solutions of the local minimum principle for the discretized optimal control problem by applying the general convergence framework of Stetter, which requires the discretization method to be continuous, consistent, and stable.

On a generalized variational inequality problem

In this paper, a sufficient condition in order to have C-udomonotone property for multifunctions is presented. By applying a special minimax theorem and KKM theory some existence results of solutions of a generalized variational inequality problem are established. Some examples in order to illustrate the main results are given. The results of this paper can be considered as extension and improvement of some articles in this area.

A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs

We prove the following result: if a continuous vector field F is Lipschitz when restricted to the hypersurfaces determined by a suitable foliation and a transversal condition is satisfied at the initial condition, then F determines a locally unique integral curve. We also present some illustrative examples and sufficient conditions in order to apply our main result.

Optimal control of normalized SIMR models with vaccination and treatment

We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, \begin{document}$M$\end{document} , to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of steering the system to a specified target. We consider both a fixed time optimal control problem with \begin{document}$L^1$\end{document} cost and the minimum time problem to drive the system to the target. In contrast to the literature, we apply different techniques of optimal control to our problems of interest.Using the direct method, we first solve the fixed time problem and then proceed to validate the computed solutions using both necessary conditions and second order sufficient conditions. Noteworthy, we perform a sensitivity analysis of the solutions with respect to some parameters in the model. We also use the Hamiltonian Jacobi approach to study how the minimum time function varies with respect to perturbations of the initial conditions. Additionally, we consider a multi-objective approach to study the trade off between the minimum time and the social costs of the control of diseases. Finally, we propose the application of Model Predictive Control to deal with uncertainties of the model.

Robust topology optimization under multiple independent unknown-but-bounded loads

The optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable, or even lead to catastrophic failures. A two-level optimization formulation is often used in the Robust Topology Optimization (RTO) under uncertain loads. Various approaches have been reported to identify the critical loads associated with the worst structure responses. Because Convex Model approaches apply convex approximations to the original non-convex model at the lower level, the optimal designs obtained by these methods are greatly dependent on the quality of the approximation. In this paper, a new formulation based on the Wolfe duality for the RTO problems with multiple independent unknown-but-bounded loads is proposed. Following the two-level formulation, the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the Wolfe duality. Both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non-convexity associated with the lower level formulation. Numerical examples are also presented to demonstrate the proposed approach.

Higher order optimality and duality in fractional vector optimization over cones

In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

Diffusion as a strategy for survival in an invasion predator–prey model

AbstractIn some ecosystems, the invasion of species may lead to extinction or displacement of the local species that are in the medium. In this work, we present a theoretical predator–prey model based on reaction diffusion equations, where each species has a strong Allee effect. We use the diffusion coefficients as parameters in a one‐dimensional environment to show that prey can either survive or become extinct from its habitat. When the results are extended to two spatial dimensions, diffusion alone is no longer a sufficient condition in order to assure extinction or survival of the prey species; in this case, the geometry of the propagation front plays an essential role in the dynamics. Finally, a discussion about strategies for survival and extinction of the prey species is considered. The results can be useful in understanding invasion for species such as the Argentine ant Linepithema humile or to help to find control strategies for unwanted species in an ecosystem by introducing a predator.

Algebraic Traveling Wave Solutions, Darboux Polynomials and Polynomial Solutions

In this paper we completely characterize the existence of algebraic traveling wave solutions for the celebrated Kolmogorov–Petrovskii–Piskunov/Zeldovich equation. To do it, we find necessary and sufficient conditions in order that a polynomial linear differential equation has a polynomial solution and we classify all the Darboux polynomials of the planar system \(\dot{x} =y\), \(\dot{y} =-c/d y +f(x)(f'(x)+r)\) where f is a polynomial with \(\deg f \ge 2\), \(c,d>0\) and r are real constants. All results are of interest by themselves.

A map of sufficient conditions for the symmetric nonnegative inverse eigenvalue problem

The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks for necessary and sufficient conditions in order that a list of real numbers be the spectrum of a symmetric nonnegative real matrix. A number of sufficient conditions for the existence of such a matrix are known. In this paper, in order to construct a map of sufficient conditions, we compare these conditions and establish inclusion relations or independence relations between them.

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.

Right saturations and induced pseudofunctors between bicategories of fractions

We fix any bicategory A together with a class of morphisms WA, such that there is a bicategory of fractions A[WA−1] (as described by D. Pronk). Given another such pair (B,WB) and any pseudofunctor F:A→B, we find necessary and sufficient conditions in order to have an induced pseudofunctor G:A[WA−1]→B[WB−1]. Moreover, we give a simple description of G in the case when the class WB is “right saturated”.

Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative

In this paper, using the weighted space method and a fixed point theorem, we investigate the Hyers-Ulam-Rassias stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville derivative on the continuous function space. We obtain some sufficient conditions in order that the nonlinear fractional differential equations are stable on the continuous function space. The results improve and extend some recent results. Finally, we construct some examples to illustrate the theoretical results.

Generic Properties for Semialgebraic Programs

In this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints, in which every problem of the class is obtained by linear perturbations of the objective function, while the “core” objective function and the constraint functions are kept fixed. Assume that the linear independence constraint qualification is satisfied at every point in the constraint set. It is shown that almost all problems in the class are such that (i) the restriction of the objective function on the constraint set is coercive and regular at infinity; (ii) there is a unique optimal solution, lying on a unique active manifold, at which the strict complementarity and second order sufficiency conditions, the quadratic growth condition, and the Hölder type global error bound hold, and (iii) all minimizing sequences converge. Furthermore, the active manifold is constant, and the optimal solution and the optimal value function depend analytically under local perturbations of the objective function. These results are combined with a standard result about the existence of sums of squares certificates to prove that we can build a sequence of semidefinite programs whose solutions give rise to a sequence of points converging to the optimal solution of the original problem in finitely many steps. It is worth emphasizing that the results of this study hold globally and we do not require the problem to be convex or the constraint set to be compact.

Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems

This paper is devoted to studying the robust isolated calmness of the Karush--Kuhn--Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed for the KKT solution mapping to have the Aubin property.

Finite element approximation of sparse parabolic control problems

We study the finite element approximation of an optimal control problem governed by a semilinear partial differential equation and whose objective function includes a term promoting space sparsity of the solutions. We prove existence of solution in the absence of control bound constraints and provide the adequate second order sufficient conditions to obtain error estimates. Full discretization of the problem is carried out, and the sparsity properties of the discrete solutions, as well as error estimates, are obtained.

A Sparsity Preserving Convexification Procedure for Indefinite Quadratic Programs Arising in Direct Optimal Control

Quadratic programs (QP) with an indefinite Hessian matrix arise naturally in some direct optimal control methods, e.g., as subproblems in a sequential quadratic programming scheme. Typically, the Hessian is approximated with a positive definite matrix to ensure having a unique solution; such a procedure is called regularization. We present a novel regularization method tailored for QPs with optimal control structure. Our approach exhibits three main advantages. First, when the QP satisfies a second order sufficient condition for optimality, the primal solution of the original and the regularized problem are equal. In addition, the algorithm recovers the dual solution in a convenient way. Second, and more importantly, the regularized Hessian bears the same sparsity structure as the original one. This allows for the use of efficient structure-exploiting QP solvers. As a third advantage, the regularization can be performed with a computational complexity that scales linearly in the length of the control horizon. We showcase the properties of our regularization algorithm on a numerical example for nonlinear optimal control. The results are compared to other sparsity preserving regularization methods.

The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier's fourth problem

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over k.This occurs for instance if g is quadratic of index 2 with [g,g]≠g and also if g is nilpotent of index at most 2. The converse holds for filiform Lie algebras of type Ln, Qn, Rn and Wn.We show how Dixmier's fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation gΛ. Moreover, Dixmier's statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center.

An interpolation problem on the circle between Lagrange and Hermite problems

This paper is devoted to studying an interpolation problem on the circle, which can be considered an intermediate problem between Lagrange and Hermite interpolation. The difference as well as the novelty is that we prescribe Lagrange values at the 2n roots of a complex number with modulus one and we prescribe values for the first derivative only on half of the nodes. We obtain two types of expressions for the interpolation polynomials: the barycentric expressions and another one given in terms of an orthogonal basis of the corresponding subspace of Laurent polynomials. These expressions are very suitable for numerical computation. Moreover, we give sufficient conditions in order to obtain convergence in case of continuous functions and we obtain the rate of convergence for smooth functions. Finally we present some numerical experiments to highlight the results obtained.

Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that \(\sqrt {Q\left( x \right)} \) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q∇u) = b, we find sufficient conditions in order that \(\sqrt Q \nabla u\) has some first order smoothness. Specifically, if Ω is a bounded open set in Rn, we study when the components of \(\sqrt Q \nabla u\) belong to the first order Sobolev space WQ1,2(Ω) defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of Σi=1nXi′Xiu + b = 0. We do not assume that {Xi} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.