Contingent Derivatives Research Articles

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Viability and Filippov-type Lemma for Stieltjes Differential Inclusions

We prove a viability result for differential inclusions involving the Stieltjes derivative with respect to a left-continuous non-decreasing function with time dependent state constraints. A tangential condition using a generalized notion of the contingent derivative is imposed. Classical viability results (for usual differential inclusions) are thus generalized and, at the same time, the gate to new viability results for difference inclusions, impulsive differential inclusions or dynamic inclusions on time scales is open. As a consequence, in the particular case where the state constraint is a tube, a Filippov-type lemma is obtained for this very general setting, of differential problems driven by Stieltjes derivatives.

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions.

Strict efficiency conditions for nonsmooth optimization with inclusion constraint under Hölder directional metric subregularity

In this paper, we investigate the higher-order optimality conditions for strict efficient solutions to a nonsmooth optimization problem subject to inclusion constraint. A concept of higher-order contingent derivative type for set-valued maps, some main calculus rules of which are obtained, is proposed and employed with the Robinson–Ursescu open mapping theorem to get a Karush–Kuhn–Tucker condition under assumptions of Hölder direction metric subregularity. Sufficient conditions for these solutions are established without convexity assumptions and possibly without the existence of derivatives. As an application, we extend some optimality conditions for a nonsmooth optimization problem subject to generalized inequality constraint. Another application is presented for necessary and sufficient conditions for robust local strict efficient solutions in uncertain vector optimization. Some examples are provided to illustrate our theorems as well. Our results are new and improve the existing ones in the literature substantially.

Short-Term Interest Rate Estimation by Filtering in a Model Linking Inflation, the Central Bank and Short-Term Interest Rates

We consider the model of Antonacci, Costantini, D’Ippoliti, Papi (arXiv:2010.05462 [q-fin.MF], 2020), which describes the joint evolution of inflation, the central bank interest rate, and the short-term interest rate. In the case when the diffusion coefficient does not depend on the central bank interest rate, we derive a semi-closed valuation formula for contingent derivatives, in particular for Zero Coupon Bonds (ZCBs). By using ZCB yields as observations, we implement the Kalman filter and obtain a dynamical estimate of the short-term interest rate. In turn, by this estimate, at each time step, we calibrate the model parameters under the risk-neutral measure and the coefficient of the risk premium. We compare the market values of German interest rate yields for several maturities with the corresponding values predicted by our model, from 2007 to 2015. The numerical results validate both our model and our numerical procedure.

Characterizations for Strong Abadie Constraint Qualification and Applications to Calmness

In this paper, we mainly study the Abadie constraint qualification (ACQ) and the strong ACQ of a convex multifunction. To characterize the general difference between strong ACQ and ACQ, we prove that the strong ACQ is essentially equivalent to the ACQ plus the finite distance of two image sets of the tangent derivative multifunction on the sphere and the origin, respectively. This characterization for the strong ACQ is used to provide the exact calmness modulus of a convex multifunction. Finally, we apply these results to local and global error bounds for a convex inequality defined by a proper convex function. The characterization of the strong ACQ enables us to give primal equivalent criteria for local and global error bounds in terms of contingent cones and directional derivatives.

Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

Higher-order variational set of the benson proper perturbation map in set-valued optimization

In the paper, we study sensitivity analysis in set-valued optimization, a research direction has been attracting much attention of many mathematicians in the world recently. The main derivative used in the paper is higher-order variational set (introduced by Khanh and Tuan in 2008) which is considered as a generalization of the contingent derivative (known as the first and the most popular derivative in set-valued optimization). Firstly, we establish relationships between higher-order variational sets of a given set-valued map and those of its profile (extended by a ordering cone). Then, we give results on higher-order variational set of the Benson proper perturbation map for a kind of set-valued optimization problem, the perturbation map is defined in the objective space. Finally, we apply the obtained results to sensitivity analysis for optimal-value map of a parametrized constrained set-valued optimization problem whose the objective map and constrained maps depends on some parameter. More precisely, some results on sensitivity analysis for parametrized constrained set-valued optimization problem are obtained. The content of the paper gives us more applications of higher-order variational set in set-valued optimization.

Sensitivity analysis for set-valued equilibrium problems

In the paper, we mention a parametrized vector equilibrium problem via sum of two given set-valued maps, considered as a unified model of some problems in variational analysis and optimization. Then, we study sensitivity analysis for this problem in terms of the second-order contingent derivatives. Finally, corresponding results for special cases are implied.

Second-Order Composed Contingent Derivative of the Perturbation Map in Multiobjective Optimization

In view of the structural advantage of second-order composed derivatives, the purpose of this paper is to analyze quantitatively the behavior of perturbation maps for the first time by using this concept. First, new concepts of the second-order composed adjacent derivative and the second-order composed lower Dini derivative are introduced. Some relationships among the second-order composed contingent derivative, the second-order composed adjacent derivative and the second-order composed lower Dini derivative are discussed. Second, the relationships between second-order composed lower Dini derivable and Aubin property are provided. Third, by virtue of second-order composed contingent derivatives and the above relationships, some results concerning second-order sensitivity analysis are established without the assumption of the locally Lipschitz property or the locally Hölder continuity. Finally, we give some complete characterizations of second-order composed contingent derivatives of the perturbation maps.

Sensitivity analysis in parametric vector optimization in Banach spaces viaτw-contingent derivatives

This paper is concerned with sensitivity analysis in parametric vector optimization problems via ${\tau}^w$-contingent derivatives. Firstly, relationships between the ${\tau}^w$-contingent derivative of the Borwein proper perturbation map and the ${\tau}^w$-contingent derivative of feasible map in objective space are considered. Then, the formulas for estimating the ${\tau}^w$-contingent derivative of the Borwein proper perturbation map via the ${\tau}^w$-contingent of the constraint map and the Hadamard derivative of the objective map are obtained.

Contingent derivatives of the set-valued solution map of a noncoercive saddle point problem. A cross-fertilization between variational analysis and inverse problems

Contingent derivatives of the set-valued solution map of a noncoercive saddle point problem. A cross-fertilization between variational analysis and inverse problems

Second-order composed contingent derivatives of perturbation maps in set-valued optimization

In the paper, we study calculus rules of second-order composed contingent derivatives. More precisely, chain rule and sum rule are established and their applications to some particular mathematical models are obtained. Then sensitivity analysis in set-valued optimization using second-order composed contingent derivatives are proposed. Our results are new and many examples are given to illustrate them.

Second-order sensitivity analysis for parametric equilibrium problems in set-valued optimization

In the paper, we first establish relationships between second-order contingent derivatives of a given set-valued map and that of the weak perturbation map. Then, these results are applied to sensitivity analysis for parametric equilibrium problems in set-valued optimization.

On Higher-Order Sensitivity Analysis of Parametric Henig Set-Valued Equilibrium Problems

In the article, we discuss sensitivity analysis of a parametric Henig set-valued equilibrium problem. In detail, relationships between the higher-order contingent derivative of the solution map of this problem and those of objective and constraint maps are established. Finally, applications to parametric set-valued optimization problems are given.

New Second-Order Optimality Conditions for Vector Equilibrium Problems with Constraints in Terms of Contingent Derivatives

In this paper, we establish the primal and dual second-order necessary and sufficient optimality conditions for (local) weakly efficient solutions of vector equilibrium problems with constraints in terms of contingent derivatives with 2-steady functions in real finite spaces. Using the 2-steadiness of objective and constraint functions at a given optimal point, the second-order necessary conditions for weakly efficient solutions of constrained vector equilibrium problems are derived. Under the suitable assumptions on 2-steadiness of objective and constraint functions, the dual second-order necessary optimality conditions will become the dual second-order sufficient optimality conditions. We also give several examples that illustrate our results.

Calculus rules of the generalized contingent derivative and applications to set-valued optimization

In the paper, we develop sum and chain rules of the generalized contingent derivative for set-valued mappings. Then, their applications to sensitivity analysis and optimality conditions for some particular optimization problems are given. Our results extend some recent existing ones in the literature.

Characterizing the Lagrange multiplier rule in nonconvex set-valued optimization

In this article, by using the notions of contingent derivative, contingent epiderivative and generalized contingent epiderivative, we obtain some characterizations of the Lagrange multiplier rule at points which are not necessarily local minima.

Second-order efficiency conditions for $C^{1,1}$-vector equilibrium problems in terms of contingent derivatives and applications

Second-order efficiency conditions for $C^{1,1}$-vector equilibrium problems in terms of contingent derivatives and applications

The second-order contingent derivative of generalized perturbation maps

In the paper, we give some remarks on [1]. Then, we modify main results concerning the sum rule of second-order contingent derivatives for set-valued maps and its application to the sensitivity analysis of generalized perturbation maps. The obtained results are new and better than those in [1]. Some examples are proposed to illustrate our results.