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In this paper, we consider a general Mayer optimal control problem governed by a mono-input affine control system whose optimal solution involves a second-order singular arc (leading to chattering). The objective of the paper is to present a numerical scheme to approach the chattering control by controls with a simpler structure (concatenation of bang-bang controls with a finite number of switching times and first-order singular arcs). Doing so, we consider a sequence of vector fields converging to the drift such that the associated optimal control problems involve only first-order singular arcs (and thus, optimal controls necessarily have a finite number of bang arcs). Up to a subsequence, we prove convergence of the sequence of extremals to an extremal of the original optimal control problem as well as convergence of the value functions. Next, we consider several examples of problems involving chattering. For each of them, we give an explicit family of approximated optimal control problems whose solutions involve bang arcs and first-order singular arcs. This allows us to approximate numerically solutions (with chattering) to these original optimal control problems. [ABSTRACT FROM AUTHOR]

The paper investigates stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may nonlinearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper–the metric subregularity of the mapping associated with the system of first order necessary optimality conditions. This property also enables error estimates for approximation methods. A Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product. [ABSTRACT FROM AUTHOR]

In this paper, we establish some existence results for the solution of the generalized symmetric set-valued quasi-equilibrium problem (GSSQEP). The new forms of GSSQEP via improvement set and scalar generalized symmetric set-valued quasi-equilibrium problem (GSSQEP Δ for short) are also introduced. By using Kakutani–Fan–Glicksberg fixed point method, maximal element principle and nonlinear scalarization technique, we develop two classes of sufficient conditions for the existence of solutions to GSSQEP. The drawback of some existing work for this problem is overcome. Moreover, some applications to the symmetric set-valued equilibrium problem (SSEP), symmetric vector quasi-equilibrium problem (SVQEP) and the set-valued equilibrium problem (SEP) are also given in this paper. Our results improve a few existing ones in Fakhar and Zafarani [Generalized symmetric vector quasiequilibrium problems. J Optim Theory Appl. 2008;136:397–409], Farajzadeh et al. [On the existence of solutions of symmetric vector equilibrium problems via nonlinear scalarization. Bull Iran Math Soc. 2019;45:35–58], Fu [Symmetric vector quasi-equilibrium problems. J Math Anal Appl. 2003;285:708–713], Han et al. [Existence of solutions for symmetric vector set-valued quasi-equilibrium problems with applications. Pacific J Optim. 2018;14:31–49]. [ABSTRACT FROM AUTHOR]

In this paper, we discuss well-posedness for a generalized vector variational inequality problem (GVVIP, in short) in the framework of topological vector spaces. Unlike in the available literature, we have adopted a topological approach using admissibility and convergence of nets, instead of monotonicity and convexity etc of the function involved. We provide necessary and sufficient conditions for a GVVIP to be well-posed in generalized sense. We give a characterization for GVVIP to be well-posed in generalized sense in terms of the upper semi-continuity of the approximate solution set map. Also, we provide some necessary conditions for a GVVIP to be well-posed in generalized sense in terms of Painlevé–Kuratowski convergence. [ABSTRACT FROM AUTHOR]

The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953–986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples. [ABSTRACT FROM AUTHOR]

The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative tools used in the analysis of the robustness of well-posedness of mathematical problems and related regularity properties of mappings involved in the statements. We also expand the selection of classes of perturbations, for which the formula for the radius of strong subregularity is valid. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a newly introduced tool, projectional coderivatives, and the corresponding calculus rules in finite dimensional spaces. We show that when the restricted set has some nice properties, more specifically, it is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to give a complete characterization of the relative Lipschitz-like property under such a setting. Chain rules and sum rules are obtained to facilitate the application of the tool to a wider range of parametric problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to explore the stability of (weak)-minimal solutions for set-valued optimization problems via improvement sets. Firstly, the optimality and closedness of solution sets for the set-valued optimization problem under the upper order relation are discussed. Then, a new convergence concept for set-valued mapping sequences is introduced, and some properties of the set-valued mapping sequences are shown under the new convergence assumption. Moreover, by means of upper level sets, Painlevé-Kuratowski convergences of (weak) E-u-solutions to set-valued optimization problems with respect to the perturbations of feasible sets and objective mappings are established under mild conditions. The order that we use to establish the result depends on the improvement set, which is not necessarily a cone order. Our results can be seen as the extension of the related work established recently in this field. [ABSTRACT FROM AUTHOR]

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution. [ABSTRACT FROM AUTHOR]

In this paper, a family of discrete optimal control problems that depend on parameters is considered. The control problems are reformulated as parametric optimization problems. By establishing/exploiting abstract results on subdifferentials of optimal value functions of parametric optimization problems, we derive formulas for estimating/computing subdifferentials of optimal value functions of parametric discrete optimal control problems in both nonconvex and convex cases. Namely, for control problems with nonconvex costs, upper-evaluations on the regular subdifferential and the limiting (Mordukhovich) subdifferential of the optimal value function are obtained without using the (strict) differentiability of the costs. Meanwhile, for control problems with convex costs, besides results on estimating/computing the subdifferential (in the sense of convex analysis) of the optimal value function, it is worth pointing out that some properties of the optimal value function are first discussed in this paper. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish some new results on the Levitin–Polyak well-posedness to a class of split hemivariational inequality problems on Hadamard manifolds. We first consider a new class of split hemivariational inequality problems (for short, SHIP) on Hadamard manifolds and introduce the regularized gap functions for these problems. Then, we study the notion of Levitin–Polyak well-posedness by perturbations to SHIP and show the equivalence between the Levitin–Polyak well-posedness by perturbations and the existence of solutions for SHIP under suitable conditions. Furthermore, based on the regularized gap functions for the perturbed SHIP, we establish the criterion for the Levitin–Polyak well-posedness by perturbations for SHIP via the split optimization problems on Hadamard manifolds. Our main results presented in paper are new even in the special case of hemivariational inequality problems. [ABSTRACT FROM AUTHOR]

This paper aims to study the well-posedness of semi-infinite set optimization problems. We first introduce a notion of global well-posedness for efficient solutions of the reference problems. This notion is then characterized in terms of qualitative properties of approximately minimal solution maps. Next, we extend this notion for perturbed problems obtained by perturbing both the objective set-valued maps and the constraints. Finally, we establish sufficient conditions for the proposed well-posedness by utilizing the Slater constraint qualification and converse properties of the objective set-valued maps. [ABSTRACT FROM AUTHOR]

The paper extends the radius of metric regularity theorem by Dontchev, Lewis and Rockafellar (2003) by providing an exact formula for the radius with respect to Lipschitz continuous perturbations in general Asplund spaces, thus, answering affirmatively an open question raised twenty years ago by Ioffe. In the non-Asplund case, we give a natural upper bound for the radius complementing the conventional lower bound in the theorem by Dontchev, Lewis and Rockafellar. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a class of multicriteria games. We establish an existence theorem for the weak multicriteria Nash equilibrium of multicriteria games in which a weak form of upper semicontinuity and quasiconvexity on payoff functions is allowed. Moreover, we introduce the concept of Γ C -convergence for the sequence of payoff mappings and obtain the sufficient conditions for generalized Hadamard well-posedness of multicriteria games. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed. [ABSTRACT FROM AUTHOR]

This paper presents sufficient conditions for strong metric subregularity (SMsR) of the optimality mapping associated with the local Pontryagin maximum principle for Mayer-type optimal control problems with pointwise control constraints given by a finite number of inequalities G j (u) ≤ 0 . It is assumed that all data are twice smooth, and that at each feasible point the gradients G j ′ (u) of the active constraints are linearly independent. The main result is that the second-order sufficient optimality condition for a weak local minimum is also sufficient for a version of the SMSR property, which involves two norms in the control space in order to deal with the so-called two-norm-discrepancy. [ABSTRACT FROM AUTHOR]

This paper is concerned with the calmness of a partial perturbation to the composite rank constraint system, an intersection of the rank constraint set and a general closed set, which is shown to be equivalent to a local Lipschitz-type error bound and also a global Lipschitz-type error bound under a certain compactness. Based on its lifted formulation, we derive two criteria for identifying those closed sets such that the associated partial perturbation possesses the calmness, and provide a collection of examples to demonstrate that the criteria are satisfied by common nonnegative and positive semidefinite rank constraint sets. Then, we use the calmness of this perturbation to obtain several global exact penalties for rank constrained optimization problems, and a family of equivalent DC surrogates for rank regularized problems. [ABSTRACT FROM AUTHOR]

In this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed. [ABSTRACT FROM AUTHOR]

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in R n . To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of R n + 1 . In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of a nonconvex regular polynomial vector optimization problem on a nonempty closed set. We introduce regularity conditions for the polynomial vector optimization problem and study their properties and characterizations. Under the regularity conditions, we establish the nonemptiness and boundedness of the solution sets of the problem. As a by-product, we infer two Frank–Wolfe type theorems for the nonconvex polynomial vector optimization problem. Finally, we investigate the solution stability of the problem. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic mathematical model for Dengue Fever (DF) and Zika virus (ZIKV) co-infection transmission dynamics is formulated and analyzed. Two sub-models, namely ZIKV-only and DF-only sub-models, are considered first of all. Rigorous qualitative analysis of the sub-models reveals that each of the diseases undergoes the phenomenon of backward bifurcation when the associated reproduction number of both ZIKV-only and DF-only sub-models (denoted by R 0 z + and R 0 d respectively) is less than unity. It is shown, using the centre manifold theory that the full ZIKV-DF co-infection model undergoes a backward bifurcation phenomenon. In addition, simulation of the full ZIKV-DF model shows the clear picturization of disease transmission and also we provide the solution of model numerically in detail. [ABSTRACT FROM AUTHOR]

Given an arbitrary non-empty set of sites, T in ℝ n , and a specific site s ∈ T, the farthest Voronoi cell of s is the set consisting of all points in ℝ n that are farther from s, with respect to the Euclidean norm, than from any other site in T. In this paper, we propose a new concept of a radius of robust feasibility for farthest Voronoi cells in the ball uncertainty framework, based only on the uncertainty of the sites, and apply it to obtain geometrical and stability properties of these cells. We provide a formula for its computation based on the data. The positiveness of this radius is related to a Slater condition and allows to characterize the interior and the full dimension of the farthest Voronoi cells. We also give an application to full dimensional tessellations of ℝ n obtained from the farthest Voronoi diagrams. Moreover, we address the stability of the nonemptiness of farthest Voronoi cells applying the robust approach to uncertainty about either T \ s or s. The complete case of uncertainty on T is also analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider parametric set-valued equilibrium problems in normed spaces. By virtue of the Gerstewitz nonlinear scalarization function along with relaxed concavity assumptions, we obtain the Lipschitz continuity property of solution maps to such problems. The treatment and obtained results for these problems are new and different from the existing ones in the literature. We apply the main results to the Browder variational inclusion to illustrate for their applicability. [ABSTRACT FROM AUTHOR]

The paper aims to introduce inverse tensor variational inequalities and analyze their application to an economic control equilibrium model. More precisely, some existence and uniqueness results are established and the well-posedness analysis is investigated. Moreover, the Tikhonov regularization method is extended to tensor inverse problems to study them when they are ill-posed. Lastly, the policymaker's point of view for the oligopolistic market equilibrium problem is introduced. The equivalence between the equilibrium conditions and a suitable inverse tensor variational inequality is established. [ABSTRACT FROM AUTHOR]

In this paper, we study an uncertain inequality system, where the input data are uncertain and belong to prescribed uncertainty sets. Using the deterministic approach in robust optimization, we treat this uncertain system by examining the so-called robust system. This approach enables us to compute the second order tangent sets for the solution set of the robust system and then obtain the second order epi-subderivative for the indicator function of its solution set. In this way, we are able to calculate the graphical derivative for the normal cone mapping of solution set of the robust system under certain qualification conditions. As applications, we establish second order necessary and sufficient optimality conditions, and derive necessary and sufficient conditions for stability properties such as the isolated calmness of optimization problems involving uncertain constraints under weak qualification conditions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with vector optimization problems where the images of the objective maps are given in linear spaces not endowed with any topological structure. Firstly, we propose new concepts of semicontinuity of vector-valued maps acting from metric spaces into linear ones and discuss their properties and characterizations. Next, using Zorn's Lemma and these properties, we formulate existence conditions for such problems. Then, we introduce two notions of Levitin-Polyak well-posedness for the reference problems, and study sufficient conditions for their fulfillment, as well as relationships of these notions. Finally, using the Kuratowski measure of noncompactness, several metric characterizations of such well-posedness properties for vector optimization problems are investigated. Many examples are given to analyze our results. [ABSTRACT FROM AUTHOR]

In this paper, we derive formulae for computing the S-derivative of the extremum multifunction in a multi-objective parametric discrete optimal control problem with nonconvex objective functions and control constraints. Particularly, we obtain formulae for upper and lower evaluation on the S-derivative of the extremum multifunction via the solution of state equations, the tangent cone to the constraint sets, and the Fréchet derivative of the objective functions. By establishing an abstract result on the S-derivative of the extremum multifunction in a multi-objective parametric mathematical programming problem, we derive formulae for upper and lower evaluation on the S-derivative of the extremum multifunction in a multi-objective parametric discrete optimal control problem. [ABSTRACT FROM AUTHOR]

The paper investigates the Lipschitz/Hölder stability with respect to perturbations of optimal control problems with linear dynamic and cost functional which is quadratic in the state and linear in the control variable. The optimal control is assumed to be of bang-bang type and the problem to enjoy certain convexity properties. Conditions for bi-metric regularity and (Hölder) metric sub-regularity are established, involving only the order of the zeros of the associated switching function and smoothness of the data. These results provide a basis for the investigation of various approximation methods. They are utilized in this paper for the convergence analysis of a Newton-type method applied to optimal control problems which are affine with respect to the control. [ABSTRACT FROM AUTHOR]

We first introduce notions of strong t-quasiconvexity and strong t-quasiconvex-likeness along with relaxed Hölder continuity with respect to an ordering cone of a set-valued map. Then, these new conditions are utilized to establish sufficient conditions for the non-emptiness of solution sets and the stability in the sense of Hölder continuity of solution maps to parametric set-valued Ky Fan inequalities. Our approach is different from the existing ones. At the end of the paper, we present an application of the main results to set-valued Nash equilibrium in games. [ABSTRACT FROM AUTHOR]

In this paper, by considering some properties associated with scalar functionals of multiple-integral type, we study the well-posedness and generalized well-posedness for a new variational inequality-constrained optimization problems By using the set of approximating solutions, we state some characterization theorems on well-posedness and generalized well-posedness. Also, in order to validate the derived results, some examples are given. [ABSTRACT FROM AUTHOR]

This paper studies generalized differentiation for a class of normal cone operators and sensitivity of solution to a class of parametric discrete optimal control problems. We first establish the Fréchet and Mordukhovich coderivatives of a class of normal cone operators in Asplund spaces. We then use the obtained results to compute and estimate the Fréchet coderivative of the solution map to parametric discrete optimal control problems with nonlinear state equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider parametric optimal control problems with linear state equations and control constraints. Applying the KKM fixed point theorem and the Gronwall Lemma, conditions of the existence of solutions to such problems are gained. Under suitable tools and techniques, we formulate sufficient conditions of a Hölder property of solution maps to reference problems without assuming conditions related to differentiability properties. Also, we suggest and investigate a concept of Hölder well-posedness for these problems. Finally, as applications, employing the main results, we discuss the Hölder calmness property of the solution maps to two practical situations, namely linear quadratic regulator and fuel-optimal frictionless horizontal motion of a mass point problems. [ABSTRACT FROM AUTHOR]

From a computational point of view, this paper provides a significant advance in the study of the calmness property of ordinary (finite) linear programs under canonical perturbations (i.e., perturbations of the objective function coefficient vector and the right-hand side of the constraint system). In the recent literature we find, for both the feasible and the optimal set (argmin) mappings, computable expressions for the corresponding calmness moduli. These expressions are called point-based as far as they only depend on the nominal data. In this paper we show that both calmness moduli are indeed (sharp) calmness constants, and provide point-based expressions for neighborhoods where such constants work. We show that our results cannot be extended to general convex inequality systems under right-hand side perturbations. [ABSTRACT FROM AUTHOR]

In this paper, the Hausdorff quasi-upper and Hausdorff quasi-lower semicontinuity for set-valued mappings are introduced to derive the upper semicontinuity and lower semicontinuity of cone constraint mapping in normed vector spaces. We discuss existence of solutions for generalized vector equilibrium problems with cone constraints. Moreover, we establish the lower semicontinuity of (weakly) efficient approximate solution mappings, upper semicontinuity of weakly efficient approximate solution mappings, and Hausdorff upper semicontinuity of efficient approximate solution mappings to parametric generalized vector equilibrium problems with cone constraints under some suitable conditions. Some examples are also given to illustrate our main results. [ABSTRACT FROM AUTHOR]