This paper focuses on maximal marginal functions, also known as optimal value functions, which frequently arise in variational analysis, parametric optimization, and various other fields. These functions represent the value of the objective function at the optimal solution of a maximization problem, with respect to the problem’s parameters. Generally, they are inherently nonsmooth, necessitating the use of generalized differentiation techniques for their analysis. The main results of this paper provide sufficient conditions for the tangential convexity of the marginal function and offer an upper estimate of its tangential subdifferential when this property holds. The structure of the upper estimate closely resembles results already established in the literature. Examples are included to illustrate our findings.
The Variation Evolving Method (VEM), established through the variational dynamic analysis, facilitates the computation of Optimal Control Problems (OCPs) by transforming them to Initial-value Problems (IVPs) with respect to the variational time τ . In this paper, it is reformulated in the primal variable space for better performance. The Evolution Partial Differential Equation (EPDE), which theoretically guarantees the variational motion of control u towards the optimal solution, is derived in the first-order and the second-order forms. The costate-free optimality conditions are established, and the explicit analytic expressions for the costates and the Lagrange multipliers adjoining the terminal constraints are presented, in terms of the state and the control variables. Using the semi-discrete method, the EPDE is solved as a finite-dimensional IVP on Ordinary Differential Equations (ODEs), and it is proved that with a reasonable step-size, the ODE numerical solution for the transformed nonlinear stable dynamical system is precise and has an exponential convergence rate to the optimal solution.
A system of equilibrium problems is a problem consisting in finding a common solution to several equilibrium problems. In this paper, we develop algorithms based on the proximal point algorithm and the bundle concept to solve such problems. To deal with, we firstly reformulate the system of equilibrium problems as a single unconstrained equilibrium problems or with simpler constraints. Afterwards, we show that the latter is equivalent to an unconstrained optimization problem or with simpler constraints, for which we propose a general framework to construct various proximal bundle methods, where sequences of simpler optimization problems, typically quadratic programs, are to be solved. We show that our algorithms converge for every starting point, feasible or infeasible, to a solution of the system of equilibrium problems under different conditions, and for problems involving cyclically antimonotone bifunctions.
For solution mappings of parameterized models (such as optimization problems, variational inequalities, and generalized equations), standard stability (such as Aubin property) fails to hold as the parameter approaches the boundary of the feasible domain. A remedy is relative stability restricted to a constraint set (e.g., the feasible domain), which is our focus in this paper. We establish generalized differentiation criteria that characterize stability and strong stability of a solution mapping relative to a broad class of nonconvex constraint sets. Beyond this class, we give a counterexample that invalidates all known generalized differentiation criteria. Applied to generalized equations, our results yield characterizations of relative stability and relative strong stability of their solution mappings, which are further explicitly specified for affine variational inequalities. Finally, we prove a global relative stability criterion, which provides a different perspective on stability analysis and also generalizes the mean value theorem to set-valued, non-smooth mappings.
In this paper, we investigate, from a variational analysis standpoint, the so-called Set of Sustainable Thresholds associated with a given control system. This set corresponds to a certain collection of parameters that ensure, for a given initial state, that the pathwise constraints of the control systems are satisfied within a prescribed period of time. Our goal is to study properties of the Set of Sustainable Thresholds when it is seen as a set-valued map that depends on the initial state of the control system. The novelty of this work is that our analysis is carried out in the context of continuous-time control systems, extending recent developments reported for the discrete-time framework. Using tools from differential inclusions, we establish conditions for ensuring that the Set of Sustainable Thresholds defines a closed set-valued map. Furthermore, we investigate convexity properties of this set-valued map by using some recent developments in monotonicity analysis for nonlinear dynamical systems. Finally, we study continuity properties of this set-valued mapping, with emphasis on lower semicontinuity and Lipchitz continuity.
We develop refined Karush-Kuhn-Tucker (KKT) and Fritz-John (FJ)-type optimality conditions for nonsmooth, nonconvex mathematical programming problems. We pay special attention in the case that the functional constraint belongs to a specific class of generalized convex functions known as strongly quasiconvex functions. After analyzing a specialized subdifferential, named the strong subdifferential, we compute the normal cone of the supremum function in terms of such subdifferentials, and apply this result to the mathematical programming problem. We illustrate our important results by examples.
In this paper, we will extend the Fibre Contraction Principle (FCP) to the case of multivalued operators. Starting with an extended variant of the Multivalued Contraction Principle, a Multivalued Fibre Contraction Principle will be proved. The special case of multivalued operators having strict fixed points and some open questions for the case of nonself operators are also presented. Some applications to a triangular system of integral inclusions and to multivalued fibre fractals are also given.
We study stability and robustness of global attractors for parabolic inclusions subjected to external disturbances entering through the boundary conditions. To this end we develop an approach consisting in investigation of a suitable family of non-autonomous semiprocesses possessing a uniform attractor, which depends on disturbances. Based on the property of its upper semicontinuity, we establish conditions guaranteeing the asymptotic gain property for the global attractor of the unperturbed system w.r.t. disturbances.
Iteration mappings provide a unifying language for analyzing how numerical minimization methods cluster toward candidate minimizers, and basin geometry is increasingly used to compare robustness and reliability across methods. Recent work developed strong basins, boundary notions induced by pre-distance functions, and inverse-iteration representations of tolerance basins for exact iteration mappings, including single-valued and multiset-valued cases. Motivated by the practical reality that implementations are inexact (finite precision, approximate derivatives, truncated subproblem solves, nonunique internal decisions, and stochastic perturbations), we develop a robust and parametric theory of basins for inexact iteration mappings modeled as set-valued perturbations. We introduce robust tolerance basins defined by eventual capture into a prescribed neighborhood uniformly over all admissible errors. Under a local contraction hypothesis with contraction factor strictly less than one, we derive explicit disturbance-to-neighborhood bounds showing that every inexact orbit starting sufficiently close to a strong attractor remains in a forward-invariant neighborhood and stays within a computable asymptotic radius determined by the error level and the contraction strength; the bound is sharp in general. We also treat parametric families of iteration mappings and establish Lipschitz-type sensitivity of fixed points and certified robust basin neighborhoods with respect to parameter changes. Finally, we develop a robust inverse-iteration viewpoint: robust basin membership implies a necessary weak-preimage inclusion, while a strong-preimage construction yields a necessary-and-sufficient characterization consistent with the universal quantifier inherent in robustness. Examples and computational recipes, including boundary localization via amplification diagnostics, illustrate how to certify robust interiors and numerically detect thin basin boundaries.
We generalize an abstract variational principle in Banach spaces, introduced by Topalova Zlateva [3], by showing that the set ℙ_0 of perturbations for which a perturbed lower semi-continuous function f is WPMC (Well Posed Modulus Compact) (equivalently, well posed in generalized sense in [1] and [2]) not only contains a dense G_δ subset, but is also a complement to a σ -porous subset in a specifically defined positive cone. Moreover, if the space is a Musielak-Orlicz sequence space satisfying ℓ _Φ≅ h_Φ , then the notion WPMC is replaced by the stronger notion of Tikhonov well posedness, which is proved to be equivalent to the single-valuedness and upper semi-continuity of the multivalued mapping assigning a parameter to the solution set. We give several applications. The first one is that the Musielak-Orlicz sequence spaces have the Radon-Nikodym property and, therefore, are dentable by proving the validity of Stegall’s variational principle. As a consequence we obtain that the duals of Musielak-Orlicz sequence spaces are w^* -Asplund. We establish also a sufficient condition for Musielak-Orlicz and Nakano sequence spaces to be Asplund spaces. The next applications are for determining the type of the smoothness of certain Musielak-Orlicz, Nakano, and weighted Orlicz sequence spaces. We illustrate by an example that it is possible to consider an Orlicz function without the Δ _2 condition, by a particular choice of the weighted sequence {w_n}_n=1^∞ to get ℓ _M(w)≅ h_M(w) and to be able to apply the main result.
In this paper, we study a broad class of constrained best approximation problems in the face of data uncertainty. Adopting the deterministic robust optimisation framework, we model uncertainty using spectrahedral sets, which unify many convex uncertainty models commonly employed in practice. In this setting, the robust best approximation problem requires the solution to satisfy all constraints for every parameter in the prescribed spectrahedral sets, giving rise to a computationally difficult infinite-dimensional problem that may also involve infinitely many constraints. To address this difficulty, we employ convex optimisation duality and semi-definite optimisation techniques, and a variable transformation to reformulate the Lagrangian dual of the robust best approximation problem into a finite-dimensional convex semi-definite program. This enables the best approximation to be found from the solution of the dual problem via a solution recovery formula. By further reformulation of the dual to a convex composite unconstrained optimisation problem using convex conjugation techniques, we present a readily implementable first-order primal-dual proximal splitting method to compute the solution to the robust best approximation problem. Finally, we present the results of numerical experiments to illustrate our proposed approach.
An “attractor” for a numerical minimization method was defined in (Levy, in Springer Briefs in Optimization, Springer Nature Switzerland AG, Cham, 2018) in terms of “iteration mappings” that assign the output of an iteration of the method to its input, and the related concept of “basin of attraction” was defined as the collection of initial inputs from which the method clusters toward a given attractor. This idea was broadened in (Levy, in Set-Valued Var. Anal. 29:1–28, 2021) to ϵ -basins of attraction that collect the inputs from which repeated application of the iteration mapping eventually generates outputs within ϵ >0 of the attractor. In the present paper, we introduce a third type of basin of attraction and refine and unify the understanding of all three types of basins by analyzing basin intersections and boundaries. We introduce several new notions of continuity for iteration mappings including a type of Lipschitz continuity whose absence identifies basin boundaries. We also study inverse-iteration and provide a characterization of ϵ -basins as a union of inverse-images. We use illustrative examples and simulations, and we demonstrate how our results allow us to address limitations of simulation.
In this paper we investigate the weak solvability of differential inclusions with mixed boundary conditions of the type (P): {[ -div(ϕ '(|∇ u|)/|∇ u|∇ u)+ ϕ '(|u|)/|u|u∈∂ _C f(x,u), in Ω ,; u=0, on Γ _1; ϕ '(|∇ u|)/|∇ u|∂ u/∂ν∈∂ _C g(x,u), on Γ _2, ]. where Ω⊂ℝ^N is a bounded domain with Lipschitz boundary ∂Ω =Γ̅_1∪Γ̅_2 , Γ _1∩Γ _2=∅ , ϕ :[0,∞ )→ [0,∞ ) is an N -function of C^1 class, f(x,t) and g(x,t) are measurable w.r.t. the first variable and locally Lipschitz w.r.t. the second variable such that their corresponding Clarke subdifferential satisfy an Orlicz-type growth condition. Using nonsmooth critical point theory we are able to prove existence and multiplicity results provided the function ϕ either completely dominates or it is completely dominated by both functions controlling the growth of ∂ _C f(x,· ) and ∂ _C g(x,· ) , respectively. An important feature of the paper is that we allow meas(Γ _i)=0 , i∈{1,2} , thus obtaining in the limiting case either a pure Dirichlet or Neumann-type boundary condition.
This article introduces a fully relaxed variable metric variant of Tseng’s splitting method for finding zeros of the sum of two operators in a real Hilbert space, one maximal monotone and the other Lipschitz or even uniformly continuous and monotone. In the case of the variable metric operator being the identity operator, it allows the relaxation factor to vary freely in any closed subinterval of (0,2) . To ensure its weak convergence, and with the hope of improving performance, two new strategies for choosing the step length are developed. One is based on a novel equality to evaluate when the backtracking can be accepted. The other is based on its corresponding strict equality, but avoids the backtracking and adopts an innovative parabolic update to determine the step length once per iteration. In addition, if the forward operator is Lipschitz continuous, then we can give special strategies for choosing the step length. On top of convergence guarantees, we show that, under a mild growth condition on the sum operator, the method enjoys a locally linear rate of convergence. Numerical results demonstrate the necessity of introducing both the relaxation factor, with values close to 2, and new strategies beyond the backtracking.
We study, from a view of variational convergence, approximations of weak and strong set-valued quasi-variational inequalities and traffic network problems with arc capacity constraints, set-valued travel costs, and elastic demands depending on equilibrium flows. We define a bifunction associated to the considered problem and show that when such bifunctions of problems approximating this problem converge in appropriate types of variational convergence to the associated bifunction of the original problem, their approximate solutions metrically converge to solutions of the original problem. Here, we propose some new concepts of approximate solutions, saturatedness of arcs and paths, and equilibrium flows. The paper is the first attempt to consider global approximations of the aforementioned optimization models in terms of variational convergence. Hence, the novelty of results is high and they suggest further developments of the topic.
This paper investigates the notion of compact R-continuity and its specifications for set-valued mappings between Banach spaces. We establish several important properties of compact R-continuity in general settings and show that in finite dimensions, this notion is supported by the classical Łojasiewicz inequality for analytic functions. Applications of compact R-continuity and the results obtained herein are provided through the convergence analysis for a broad class of descent algorithms in nonsmooth optimization. Moreover, we demonstrate that compact R-continuity plays a key role in the design and theoretical justification of a novel R-class of algorithms for solving general inclusion problems.
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for any target level of smoothness. Our constructed smoothings, by virtue of their optimality, establish a fundamental limit on potential applications of smoothing techniques. Optimal smoothings restricting to either inner or outer approximations also follow from our theory. Finally, we apply our theory to provide insights into smoothing amenable functions given by compositions with sublinear functions and generic convex sets by expressing them as conic sections.
This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To overcome these challenges, we introduce a regularized system based on a fixed Lipschitz approximation of the convexified mapping. From this approximation, we extract a single-valued Lipschitz selection that preserves the essential geometric features of the original system. This framework enables the application of nonsmooth Lyapunov methods and Hamiltonian-based stability criteria. Instead of approximating trajectories, we focus on analyzing a simplified system that faithfully reflects the structure of the original dynamics. Several examples are provided to illustrate the method’s practicality and scope. The analysis is carried out under a uniform boundedness assumption on the convexified limit mapping.
This paper investigates a recently introduced notion of strong variational sufficiency in optimization problems whose importance has been highly recognized in optimization theory, numerical methods, and applications. We address a general class of composite optimization problems and establish complete characterizations of strong variational sufficiency for their local minimizers in terms of a generalized version of the strong second-order sufficient condition (SSOSC) and the positive-definiteness of an appropriate generalized Hessian of the augmented Lagrangian calculated at the point in question. The generalized SSOSC is expressed via a novel second-order variational function, which reflects specific features of nonconvex composite models. The imposed assumptions describe the spectrum of composite optimization problems covered by our approach while being constructively implemented for nonpolyhedral problems that involve the nuclear norm function and the indicator function of the positive-semidefinite cone without any constraint qualifications.
We introduce radial variants of the Wijsman and Attouch-Wets topologies for the family 𝒮_rc^d of star-shaped sets (with respect to the origin) in ℝ^d that are radially closed. These topologies give rise to new types of convergence for star-shaped sets, even when such sets are not closed or bounded. Our approach relies on a new family of functionals, called radial distance functionals, which measure “radial distances” between points and star-shaped sets in 𝒮_rc^d . These are natural radial analogues of the distance functionals for closed sets. However, unlike the radial functions of star-shaped sets, our radial distance functionals are real-valued maps and thus admit a natural treatment within the framework of classical function spaces. We prove that our radial Wijsman type topology τ _W^r is not metrizable on 𝒮_rc^d , while our radial Attouch-Wets type topology τ _AW^r is completely metrizable. A corresponding radial Attouch-Wets distance d_AW^r is introduced, and we prove that d_AW(A,K) ≤ d_AW^r(A,K) for all closed A,K ∈𝒮_rc^d , where d_AW denotes the Attouch-Wets distance.
