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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]

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, 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]

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]

In this paper, the notions of fully stable well-posedness and fully stable minimum of optimization problems with an extended real-valued objective function in an Asplund space setting are considered, where the objective function undergoes both tilt perturbations and general parameter perturbations. Special cases of the first notion reduce to the stable Hölder minimizers by Zheng and Ng [SIAM J Optim. 2015;25:416–438] and the fully stable Hölder minimizers introduced by Zheng, Zhu and Ng [SIAM J Optim. 2018;28:2601–2624], respectively. By using techniques of variational analysis and generalized differentiation, the results presented in this paper provide insight into necessary or sufficient conditions for tilt-stable minimizers of nonlinear programmes in Asplund spaces and therefore generalize some existing ones in the recent literature. [ABSTRACT FROM AUTHOR]

Shale rock, being a common caprock for carbon dioxide reservoirs, is subject to extensive research. One of the topics is breakthrough phenomena during injection of supercritical carbon dioxide in shale, the nature of which is still to be fully understood. In the present paper, a two-phase flow model, which may possibly be used to explain the breakthrough behaviour is examined. Capillary effects and geomechanics are included in the model, and influential model parameters are determined. [ABSTRACT FROM AUTHOR]

This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the L 1 -norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called deep quench approximation in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls. [ABSTRACT FROM AUTHOR]

This paper aims at investigating the Painlevé–Kuratowski convergence of the solution sets for set optimization problems with respect to the perturbations of the feasible set and the objective mapping. We introduce the concepts of cone-quasiconnectedness and strictly cone-quasiconnectedness for set-valued mappings and a new convergence for the sequence of set-valued mappings, and then we obtain the Painlevé–Kuratowski convergence of the sets of l-minimal solutions and weak l-minimal solutions for perturbed set optimization problems by using cone-quasiconnectedness and strictly cone-quasiconnectedness. As an application of the main results, we derive Painlevé–Kuratowski convergence of the solution sets for vector optimization problems. [ABSTRACT FROM AUTHOR]

This paper aims to study the stability of a class of semi-infinite vector optimization problems (SVOP) under functional perturbations. By using an important hypothesis H h (p 0) , a necessary and sufficient condition of Hausdorff continuity for weak efficient solution mappings and certain sufficient conditions for Painlevé-Kuratowski convergence of weak efficient solution sets for SVOP are established under the perturbations of both constraint sets and objective functions. [ABSTRACT FROM AUTHOR]

In this paper, in contrast to the literature on the tilt-stability only dealing with local minima, we introduce and study the ψ -tilt-stable global minimum and stable global φ -well-posedness with ψ and φ being the so-called admissible functions. We adopt global strong metric regularity of the subdifferential mapping ∂ ^ f of the objective function f with respect to an admissible function ψ and prove that the global strong metric regularity of ∂ ^ f at 0 with respect to ψ implies the stable global φ -well-posedness of f with φ (t) = ∫ 0 t ψ (s) d s and that if f is convex then the converse implication also holds. Moreover, we establish the relationships between ψ -tilt-stable global minimum and stable global φ -well-posedness. Our results are new even in the convexity case. [ABSTRACT FROM AUTHOR]

In this paper, we use a geometrical approach to sharpen a lower bound given in [5] for the Lipschitz modulus of the optimal value of (finite) linear programs under tilt perturbations of the objective function. The key geometrical idea comes from orthogonally projecting general balls on linear subspaces. Our new lower bound provides a computable expression for the exact modulus (as far as it only depends on the nominal data) in two important cases: when the feasible set has extreme points and when we deal with the Euclidean norm. In these two cases, we are able to compute or estimate the global Lipschitz modulus of the optimal value function in different perturbations frameworks. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and perturbations of all coefficients. Our starting point is an upper bound on this modulus given in Cánovas et al. (4). In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to explore some strategies that can improve the referred upper and lower bounds. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special implicit set-valued map representing solutions to a parametric quasi-optimization problem, (Q O p t) for short. This model finds its motivation in quasi-convex programming and generalized Nash equilibria modelled by the supremum of the so-called Nikaido–Isoda functions. We exploit a new recent variant of the celebrated Lim's Lemma considered in the context of metric regularity and approximate fixed points to establish quantitative stability for ε-approximate solutions to (Q O p t) under parametric perturbations in the spirit of the result presented for convex programming in the seminal contribution by Attouch and Wets [Quantitative stability of variational systems: III. ε-approximatesolutions. Math Program. 1993;61:197–214, Theorem 4.3]. Sharp estimates are then extended to parametric exact solutions to (Q O p t) by means of a qualitative stability analysis stressing the role of Painlevé-Kuratowski and Pompeiu-Hausdorff convergence for sets of approximate minima to a set of exact ones under usual compactness and/or completeness conditions. Finally, we apply our main result to a non-smooth mathematical program under polyhedral convex mappings and situate our contribution in the close recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with parametric vector equilibrium problems with equilibrium constraints. We first introduce concepts of generalized semicontinuity and convexity of vector-valued maps in ordered spaces and discuss their properties, as well as their relationships with well-known concepts. Next, employing mentioned concepts of generalized semicontinuity of vector-valued maps, we study sufficient conditions of upper semicontinuity for solution maps to such problems. Then, combining these generalized semicontinuity along with generalized convexity assumptions, sufficient conditions for lower semicontinuity of solution maps to the reference problems are established. Several examples are provided to illustrate the relaxation and essentialness of assumptions. Our results are new, even for special cases of the underlying problems. [ABSTRACT FROM AUTHOR]

In this paper, we first provide some results on first- and second-order analysis to a circular cone complementary set Γ. In this way, we obtain a parabolical regularity of Γ as well as an exact computation for second epi-derivative of the indicator function to Γ. We then establish first- and second-order optimality conditions for a mathematical programming problem involving a circular cone complementary constraint under the validation of metric subregularity and second-order tangent constraint qualification. Our results are sharpened in comparison with some existing ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the detection of the shape and location of a discontinuous source term from the knowledge of boundary measurements. We propose a non-iterative reconstruction algorithm based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from an energy-like cost function. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. The efficiency of our algorithm is illustrated by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we present a new mathematical model generalizing the model of a single unit repairable device system, with preventive maintenance under warranty period, and the model of a repairable multi-state device system. To study the time dependance solution of the resulting system, we present in this paper a new analytic approach. So, by using the C 0 -semigroup theory of linear operators and some spectral properties, we proof that the dynamic solution of the system converges to steady state solution when the time go to infinity. [ABSTRACT FROM AUTHOR]

In this paper, the approximate efficient solution of generalized semi-infinite vector optimization problems is considered. Utilizing the density of approximate efficient points, the convexity and upper semicontinuity of constraint set mapping, we establish the connectedness of the set of approximate efficient points and the set of approximate efficient solutions to generalized semi-infinite vector optimization problems with set-valued objective maps in normed spaces. As applications, the connectedness of the solution set for semi-infinite vector optimization problems and generalized nonlinear programming problems are also obtained, respectively. Our results are new and extend the results by Gong and the results by Li. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three types of well-posedness for a set optimization problem (u-SOP). Some necessary and sufficient conditions for these well-posedness have been established. Two different scalar optimization problems involving a generalized oriented distance function have been considered. Characterization of u-minimal solutions of (u-SOP) in terms of solutions of these scalar optimization problems have been obtained. Finally, equivalence of well-posedness of (u-SOP) with well-posedness of these scalar optimization problems have been established. [ABSTRACT FROM AUTHOR]

In Part 1 of this paper, we have estimated the Fréchet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to investigate the Lipschitzian-like stability property (namely Aubin property) of the solution map of possibly nonmonotone variational systems with composite superpotentials. Using Mordukhovich coderivative criterion and a second-order subdifferential analysis, we provide simple and verifiable characterizations of this property in terms of the data involved in the problem. Applications are given in nonsmooth mechanics. [ABSTRACT FROM AUTHOR]

Epi-convergence plays an important role when studying the behaviour of a set optimization problem under variations of its data. In this paper, we obtain properties for this notion and compare it with other convergence notions for set-valued maps that have been used recently to study the stability of set optimization problems. [ABSTRACT FROM AUTHOR]

This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. [ABSTRACT FROM AUTHOR]

This paper provides operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the framework of canonical perturbations (i.e., perturbations of the objective function and the right-hand-side of the constraints), the paper provides a computationally tractable formula for the calmness modulus, which goes beyond some preliminary results of the literature. Second, in the framework of full perturbations (perturbations of all coefficients), after characterizing the calmness property for the optimal set mapping, the paper provides an operative upper bound for the corresponding calmness modulus, as well as some illustrative examples. We provide two applications related to algorithms traced out from the literature: the first one to a descent method in LP, and the second to a regularization method for linear programs with complementarity constraints. [ABSTRACT FROM AUTHOR]

In a recent paper, Izmailov (Math Program Ser A 147:581-590, ) derived an extension of Robinson's implicit function theorem for nonsmooth generalized equations in finite dimensions, which reduces to Clarke's inverse function theorem when the generalized equation is just an equation. Páles (J Math Anal Appl 209:202-220, ) gave earlier a generalization of Clarke's inverse function theorem to Banach spaces by employing Ioffe's strict pre-derivative. In this paper we generalize both theorems of Izmailov and Páles to nonsmooth generalized equations in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stability of three kinds of minimal point sets and three kinds of minimizer sets of naturally quasi-functional set-valued optimization problems when the data of the approximate problems converges to the data of the original problems in the sense of Painlevé-Kuratowski. Our main results improve and extend the results of the recent papers. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the asymptotic stability of two wave equations coupled by velocities of anti-symmetric type via only one damping. They adopt the frequency domain method to prove that the system with smooth initial data is logarithmically stable, provided that the coupling domain and the damping domain intersect each other. Moreover, they show, by an example, that this geometric assumption of the intersection is necessary for 1-D case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the difference gap (for brevity, D-gap) function and global error bounds for a class of elliptic variational inequalities (for brevity, EVIs). Firstly, we establish the regularized gap function introduced by Yamashita and Fukushima for EVIs under some suitable conditions. Then the D-gap function for EVIs is proposed by employing these regularized gap functions. Furthermore, we also develop the global error bounds for EVIs in terms of the regularized gap function of the Fukushima type and the D-gap function. Finally, an application to frictional contact problem is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper we study existence of solutions to the generalized equation 0 ∈ f(p,x) + F(x), where f is a function, F is a set-valued mapping, and p is a parameter. Conditions are given, in terms of metric regularity of F, local convex-valuedness of F− 1, and partial calmness of f with respect to x uniformly in p, for the property that, for any p near the reference value, the generalized equation has a solution at a certain distance from the reference solution. Some corollaries and applications of this result are also presented. [ABSTRACT FROM AUTHOR]

In this paper, lexicographic vector equilibrium problems are investigated. By using the idea of sequential process, the upper semicontinuity and closedness of the solution set map are established for a parametric lexicographic strong vector equilibrium problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish Lipschitz continuity of strongly efficient approximate solution mapping to parametric generalized vector equilibrium problems without using monotonicity and any information of the solution mappings. Moreover, we make a new attempt to establish Lipschitz continuity of weakly efficient approximate solution mapping and efficient approximate solution mapping to parametric generalized vector equilibrium problems by using a scalarization method and a density result, respectively. As an application of the main results, we obtain Lipschitz continuity of strongly efficient approximate solution mapping, weakly efficient approximate solution mapping and efficient approximate solution mapping to parametric vector optimization problems. [ABSTRACT FROM AUTHOR]

This paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results. [ABSTRACT FROM AUTHOR]

This paper establishes a bridge between set optimization problems and vector Ky Fan inequality problems. We introduce a general model, called the bifunction-set optimization problem, that provides a unifying framework for the above-mentioned problems. An existence result in our model is obtained, with the help of KKM-Fan's lemma. As applications, we derive some new or sharper existence results for set optimization problems and generalized vector Ky Fan inequalities with efficient solutions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the sensitivity analysis in vector equilibrium problems by using the S-derivative of a set-valued mapping. We first investigate the S-derivative on a kind of set-valued gap function for the vector equilibrium problems. Based on these results, S-derivative estimations on a perturbed mapping for the parametric vector equilibrium problem are given. Moreover, we provide some examples to illustrate the obtained results. Finally, we derive the S-derivative estimations of a solutions mapping of the parametric vector equilibrium problems via S-derivative estimations of a kind of the parametric variational system. [ABSTRACT FROM AUTHOR]

This paper is about the well-posedness and stability in set optimization with applications to vector-valued games involving uncertainty. Some characterizations for several types of well-posedness with their relations in set optimization are given under mild conditions. Some sufficient and necessary conditions for these types of well-posedness in set optimization are also given by using the local C-Lipschitz continuity. Moreover, the upper semi-continuity, lower semi-continuity and compactness of minimal solution mappings are studied for parametric set optimization involving the cone Lipschitz continuous set-valued mapping. Finally, some obtained results are applied to derive the well-posedness for vector-valued games with uncertainty. [ABSTRACT FROM AUTHOR]

The preconditioned iterative solution of large-scale saddle-point systems is of great importance in numerous application areas, many of them involving partial differential equations. Robustness with respect to certain problem parameters is often a concern, and it can be addressed by identifying proper scalings of preconditioner building blocks. In this paper, we consider a new perspective to finding effective and robust preconditioners. Our approach is based on the consideration of the natural physical units underlying the respective saddle-point problem. This point of view, which we refer to as dimensional consistency, suggests a natural combination of the parameters intrinsic to the problem. It turns out that the scaling obtained in this way leads to robustness with respect to problem parameters in many relevant cases. As a consequence, we advertise dimensional consistency based preconditioning as a new and systematic way to designing parameter robust preconditoners for saddle-point systems arising from models for physical phenomena. [ABSTRACT FROM AUTHOR]

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the value function associated with optimal control problems with state constraints. [ABSTRACT FROM AUTHOR]

This paper studies solution stability of a parametric optimal control problem governed by semilinear elliptic equations and nonconvex objective function with mixed pointwise constrains in which the controls act both in the domain and on the boundary. We give sufficient conditions under which the solution map is upper semicontinuous and continuous in parameters. [ABSTRACT FROM AUTHOR]

This paper discusses differential stability of convex discrete optimal control problems with mixed constraints. Among the obtained results, we give formulas for exact computing the subdifferential and the singular subdifferential of the optimal value function via suitable multiplier sets. [ABSTRACT FROM AUTHOR]