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We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new enlarged subdifferential, we provide an algorithmic pattern of descent for weakly convex minimization. Under minimal assumptions, we show subsequential convergence to a critical point. Links with difference-of-convex functions algorithms and criticality conditions are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study optimality conditions as well as sensitivity analysis of parametric convex minimax programming problems. The main tools used here are the formulas for computing the subdifferentials of maximum functions under some suitable regularity conditions. More precisely, we use these tools to study optimality conditions for the problem under consideration. These results are then applied to obtain optimality conditions for multiobjective optimization problems, which constitute the first part of the paper. In the second part, we derive formulas for computing the subdifferential in the sense of convex analysis and singular subdifferential of the optimal value function for the problem in question. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study optimality conditions and duality analysis for the ϵ -quasi optimal solutions of nondifferentiable minimax programming problem (NMPP) via the Aubin–Frankowska’s generalized subdifferentials, where ϵ is a non-negative real number. Employing some advanced tools of generalized subdifferentials and regularity conditions, Karush–Kuhn–Tucker-type necessary optimality conditions for ϵ -quasi optimal solutions of problem (NMPP) are established. Under suitable assumptions on the ϵ -pseudoconvexity and ϵ -quasiconvexity, necessary optimality conditions become sufficient optimality conditions. Besides, these results are applied direct to a constrained vector optimization problem. In addition, a Wolfe-type and Mond–Weir-type dual problem to the primal problem is proposed and weak, strong, and converse duality relations among them are explored. Some illustrative examples are provided for our findings. [ABSTRACT FROM AUTHOR]

In this paper, a subgradient projection method for quasiconvex minimization problems is provided. By employing strong subdifferentials, it is proved that the generated sequence of the proposed algorithm converges to the solution of the minimization problem of a proper, lower semicontinuous, and strongly quasiconvex function (in the sense of Polyak in Soviet Math 7:72–75, 1966), under the same assumptions as those required for convex functions with the convex subdifferentials. Furthermore, a quasi-linear convergence rate of the iterates, extending similar results for the general quasiconvex case, is also provided. [ABSTRACT FROM AUTHOR]

This paper is intended to provide an updated survey of recent optimality theory for infinite-dimensional convex programming. It aims at establishing theoretical support for algorithmic developments. Two alternative strategies inspire the approaches presented in the paper. The first one consists of replacing the family of constraints by a single one, appealing to the supremum function, and is based on various characterizations of the subdifferential of the pointwise supremum of convex functions. The second one uses appropriate characterizations of affine consequent inequalities of the constraint system exploiting ad hoc constraint qualifications. [ABSTRACT FROM AUTHOR]

This paper considers a chief interface between mathematical programming and economics, namely: money-based trade of perfectly divisible and transferable goods. Three important and related features are singled out here: first, convexity enters via acceptable payments, second, convolution of monetary criteria secures Pareto efficiency, and third, competitive equilibrium obtains when agents' subdifferentials intersect. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize generalized convex functions introduced by Linh and Penot Optimization (62: 943–959, 2013) by using generalized monotonicity of the generalized subdifferentials. We use vector variational inequalities in terms of generalized subdifferentials to identify efficient solutions of a multiobjective optimization problem involving quasiconvex functions. We also establish the Minty variational principle by utilizing the mean value theorem established by Kabgani and Soleimani-damaneh (Numer. Funct. Anal. Optim 38: 1548–1563, 2017) for quasiconvex functions in terms of Greenberg–Pierskalla subdifferentials. [ABSTRACT FROM AUTHOR]

In this paper, we first present necessary and sufficient conditions for the existence of global error bounds for a convex function without additional conditions on the function or the solution set. In particular, we obtain characterizations of such global error bounds in Euclidean spaces, which are often simple to check. Second, we prove that under a suitable assumption the subdifferential of the supremum function of an arbitrary family of convex continuous functions coincides with the convex hull of the subdifferentials of functions corresponding to the active indices at given points. As applications, we study the existence of global error bounds for infinite systems of linear and convex inequalities. Several examples are provided as well to explain the advantages of our results with existing ones in the literature. [ABSTRACT FROM AUTHOR]

The problem considered in this paper is a sequence of two optimization problems in which the feasible region of the upper-level optimization problem is determined implicitly by the solution set of the lower-level optimization problem. Using the optimal value reformulation, together with the partial calmness property, we give necessary optimality conditions in terms of tangential subdifferentials. An example that illustrates our finding is also given. [ABSTRACT FROM AUTHOR]

This paper is committed to studying Karush-Kuhn-Tucker (in short, KKT) type necessary and sufficient optimality conditions for non-smooth quasi-convex (geodesic sense) optimization problems on Riemannian manifolds. Recently, Ansari et al. [Ansari QH, Babu F, Zeeshan M. Incremental quasi-subgradient method for minimizing geodesic quasi-convex function on Riemannian manifolds with applications. Numer Funct Anal Optim. 2022;42(13):1492–1521. doi: ] defined the quasi-subdifferential on Riemannian manifolds and established the existence results of the quasi-subdifferential. We provide several auxiliary results for the quasi-subdifferential in the current study. We offer the KKT optimality conditions for the quasi-convex optimization problems on Riemannian manifolds with or without the Slater-constraint qualifications. To verify the suggested outcomes, we formulate numerical examples. In addition, we also provide our results in the Euclidean spaces, which are original and distinct from earlier findings in the Euclidean spaces. [ABSTRACT FROM AUTHOR]

In this paper, we perform optimality conditions and sensitivity analysis for parametric nonconvex minimax programming problems. Our aim is to study the necessary optimality conditions by using the Mordukhovich (limiting) subdifferential and to give upper estimations for the Mordukhovich subdifferential of the optimal value function in the problem under consideration. The optimality conditions and sensitivity analysis are obtained by using upper estimates for Mordukhovich subdifferentials of the maximum function. The results on optimality conditions are then applied to parametric multiobjective optimization problems. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we have focused on a multi-objective fractional semi-infinite programming problems in which the constraints and objective functions are tangentially convex. A result has been established to find the tangential subdifferential of a fractional function, assuming the numerator and the negative of the denominator being tangentially convex functions. With this, optimality conditions have been derived using a non-parametric approach under ϝ -convexity assumption. Further, a Mond–Weir type dual has been considered and weak and strong duality relations have been developed. Moreover, an application in robot trajectory planning has been considered and solved using MATLAB. In addition, considering the same trajectory as in Vaz et al. (Eur J Oper Res 153(3):607–617, 2004), we have compared the results obtained in MATLAB with the results available in Vaz et al. (Eur J Oper Res 153(3):607–617, 2004) and Haaren-Retagne (A semi-infinite programming algorithm for robot trajectory planning, 1992), where the authors have solved using AMPL. It has been observed that our results are more efficient than the previously available results, with the implementation of MATLAB as it substantially reduces the computational time. Throughout the paper, nontrivial examples have also been provided for proper justification of the theorems developed. [ABSTRACT FROM AUTHOR]

This paper aims to study a broad class of multiobjective mathematical problems with switching constraints in which all emerging functions are assumed to be locally Lipschitz. First, we are interested in some Abadie, Guignard, and Cottle types qualification conditions for the problem. Then, these constraint qualifications are applied to obtain several stationarity conditions. The results are based on Clarke's subdifferential. [ABSTRACT FROM AUTHOR]

The main innovation of this article is to establish the existence and approximate controllability of Hilfer fractional differential inclusions of order 1<μ<2$$ 1<\mu <2 $$ using Clarke's subdifferential type in Hilbert spaces. By applying the fixed‐point theorem for multivalued maps, fractional calculus, generalized Clarke's subdifferential, and cosine families, we derive a novel existence result for the mild solutions of the Hilfer fractional differential inclusion system. Furthermore, we develop and demonstrate a new set of sufficient conditions for the approximate controllability of Hilfer fractional differential systems. These criteria are based on the assumption that the associated linear part of the system is approximately controllable. Finally, an example is provided to show the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, by revisiting coderivative calculus rules for convex multifunctions in finite-dimensional spaces, we derive formulae for estimating/computing the basic subdifferential and the coderivative of the efficient point multifunction of parametric convex vector optimization problems. These results are then applied to a broad class of conventional convex vector optimization problems with the presence of operator constraints and equilibrium ones. Examples are also designed to analyze and illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the optimality conditions for a class of nonsmooth multiobjective fractional optimization problems subject to vanishing constraints. In particular, necessary and sufficient conditions for (weak) Pareto solution are presented in terms of the Clark subdifferential. Furthermore, we construct Wolfe and Mond–Weir-type dual models and derive some duality theorems by using generalized quasiconvexity assumptions. Some examples to show the validity of our conclusions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized subdifferentials and the Riemannian gradient subconsistency that are the basis for non-Lipschitz optimization on embedded submanifolds of Rn. We then propose a Riemannian smoothing steepest descent method for non-Lipschitz optimization on complete embedded submanifolds of Rn. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. We also prove that any accumulation point of the sequence generated by our method that satisfies the Riemannian gradient subconsistency is a limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the advantages of Riemannian ℓp (0

We study the combination of proximal gradient descent with multigrid for solving a class of possibly nonsmooth strongly convex optimization problems. We propose a multigrid proximal gradient method called MGProx, which accelerates the proximal gradient method by multigrid, based on using hierarchical information of the optimization problem. MGProx applies a newly introduced adaptive restriction operator to simplify the Minkowski sum of subdifferentials of the nondifferentiable objective function across different levels. We provide a theoretical characterization of MGProx. First we show that the MGProx update operator exhibits a fixed-point property. Next, we show that the coarse correction is a descent direction for the fine variable of the original fine level problem in the general nonsmooth case. Last, under some assumptions we provide the convergence rate for the algorithm. In the numerical tests on the elastic obstacle problem, which is an example of a nonsmooth convex optimization problem where the multigrid method can be applied, we show that MGProx has a faster convergence speed than competing methods. [ABSTRACT FROM AUTHOR]

Some properties of σ-convex functions are studied. Then some relations between the notions of σ-subdifferentials and σ-convex functions are established. Also, we present some results regarding the sum formula for the σ-subdifferential. Moreover, we obtain some particular relationships between the σ-subdifferential and the $ (\sigma,y) $ (σ , y) -conjugate. Finally, by imposing some assumptions on f and σ, the maximal $ 2\sigma $ 2 σ -monotonicity of σ-subdifferential is studied. Indeed, the maximality of the usual Fenchel subdifferentials of lower semicontinuous convex functions follows from our result. [ABSTRACT FROM AUTHOR]

This article is concerned with a class of nonsmooth semi-infinite programming problems on Hadamard manifolds (abbreviated as, (NSIP)). We introduce the Guignard constraint qualification (abbreviated as, (GCQ)) for (NSIP). Subsequently, by employing (GCQ), we establish the Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary optimality conditions for (NSIP). Further, we derive that the Lagrangian function associated with a fixed Lagrange multiplier, corresponding to a known solution, remains constant on the solution set of (NSIP) under geodesic pseudoconvexity assumptions. Moreover, we derive certain characterizations of the solution set of the considered problem (NSIP) in the framework of Hadamard manifolds. We provide illustrative examples that highlight the importance of our established results. To the best of our knowledge, characterizations of the solution set of (NSIP) using Clarke subdifferentials on Hadamard manifolds have not been investigated before. [ABSTRACT FROM AUTHOR]

We explore the possibility to derive basic calculus rules for some subdifferential constructions associated to set-valued maps between normed vector spaces. We use these results in order to write optimality conditions for a special kind of solutions for set optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δpu+aF(u) + ...IK(u) ∈ V* in the whole ℝ N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ N}, where Δp is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D1,p(ℝN) with 1 < p < N, and IK : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...IK. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝN x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- (N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δp + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into Lq(ℝN) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝN. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]

The goal of this article is to establish optimality conditions for the ϵ -quasi efficient solutions of constrained vector optimization problem (CVOP) in terms of the Aubin–Frankowska's generalized subdifferentials, where ϵ ≥ 0. We introduce some new versions of the regularity condition of the (RC3-s) types (s = 1 ... p ¯) and propose some new concepts of ϵ -pseudoconvexity, ϵ -quasiconvexity and ϵ -quasilinearly for calm functions. After that we derive weak and strong KKT-type necessary and sufficient conditions to such a problem. Additionally, a strong KKT-type necessary optimality condition is applied in order to construct a Mond–Weir-type dual vector optimization problem (MWCVOP). Some weak, strong and converse duality theorems for the primal problem (CVOP) and the dual problem (MWCVOP) are explored under some suitable assumptions on the ϵ -pseudoconvexity and pseudoconvexity. Illustrative examples are also provided for our findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ -quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan's lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. [ABSTRACT FROM AUTHOR]

We give exact formulas for the subdifferentials of perturbed distance functions in a normed space. Our method, seemingly novel and different from existing ones, is to turn the involved problem equivalently to a parametric optimization problem and then apply variational analysis technique to the optimal value function. In the convex setting, we obtain new representations for the subdifferential of perturbed distance functions, which do not depend on the relative position of the reference point with respect to the input set, and which are described directly via the input data. Our results complement those of Wang et al. [J. Global Optim. 2010;46:489–501] and of Li and Bounkhel [Nonlinear Anal. 2014;108:173–188], which were established by different methods. [ABSTRACT FROM AUTHOR]

In this paper, we have first formulated semi-infinite equilibrium problems involving data uncertainty. For this class of problems, we have proposed the Linear Independence Constraint Qualification (LICQ) in terms of tangential subdifferentials and obtained robust Karush–Kuhn–Tucker (KKT) necessary optimality conditions. Also, the sufficiency theorem has been derived using pseudo and quasi convexity assumptions. Finally, we have shown the application of our problem in control systems for PID controllers. For that, we have formulated a Model to find the best tuning parameters for the PID controller and solved it using MATLAB. For verification of our results, the Nyquist plot of the loop transfer function is shown, which has come out to be as per Nyquist stability criteria. Moreover, the results for two more PID controller problems with combined sensitivity have been obtained, which are not solved using the concept of H-infinity error anywhere in the literature. Nontrivial examples are shown in appropriate places to justify the theorems derived in this paper. [ABSTRACT FROM AUTHOR]

This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the ε -subdifferential of the optimal value function are derived without qualification conditions. We first calculate the ε -subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the ε -subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions. [ABSTRACT FROM AUTHOR]

In this paper, a general formula concerning the multi-composition rule of convex subdifferential calculus is provided in the setting of Banach spaces under an appropriate regularity condition. As an application, this calculus rule is applied to obtain necessary and sufficient Karush–Kuhn–Tucker type optimality conditions for constrained convex minmax location problems with perturbed minimal time functions and set-up costs. [ABSTRACT FROM AUTHOR]

The goal of this paper is to examine both necessary and sufficient conditions for a specific feasible point to be a global minimizer for reverse quasiconvex programming problems. These results are obtained in terms of adequate approximate subdifferentials and can be viewed as the problem of a convex maximization problem constrained by a convex set. Sufficient conditions for optimality are also established in terms of the Greenberg-Pierskalla subdifferential. Illustrative examples are also given to illustrate the significance of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantitative stability of an optimal control problem with respect to parametric perturbations. We essentially obtain two equivalent conclusions for the stability of this problem by using two independent methods. The first one makes recourse to standard computations based on the famous Gronwall Lemma while our second method employees rather stability of fixed points trough the celebrated Lim's Lemma for which we construct a suitable contracting set-valued mapping over a larger functional space than the one of continuous functions adopted in the close previous works. The second method allows us to introduce a further concept of approximate solutions regarded as approximate values of the optimal control for which we prove similar stability properties as in the case of exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the notion of the ravine of real-valued functions is extended from the finite-dimensional setting to an infinite-dimensional setting. Ravines of quadratic functions are studied in detail. The obtained results solve a problem raised by Professor Joachim Gwinner. In addition, it is proved that a weakly continuous real-valued convex function defined on a reflexive Banach space cannot have any ravine along the null subspace. [ABSTRACT FROM AUTHOR]

In this paper, we study some characterizations of the class of weakly subdifferentiable functions and formulate optimality conditions for nonconvex mathematical programming problems described by the class of weakly subdifferentiable functions in real normed spaces. The necessary and sufficient optimality conditions for a nonconvex scalar function with a global minimum/or a global maximum at a given vector via the weak subdifferentials and augmented normal cones are established. Additionally, the necessary and sufficient optimality conditions for a nonconvex vector function with a weakly efficient solution/or an efficient solution at a given vector via the augmented weak subdifferentials and normal cones are presented too. Finally, our optimality conditions are used to derive the necessary optimality conditions for nonsmooth nonconvex mathematical programming problems with set, inequality, and equality constraints. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of the optimality and scalarization for approximate solutions to a Constrained Vector Equilibrium Problem (CVEP). The optimality conditions are given in terms of Michel-Penot subdifferentials, and the scalarization theorems are proposed via a strongly monotone cone convex function. We firstly establish a necessary condition for an approximate quasi weakly efficient solution to problem (CVEP). Then, a sufficient condition for approximate quasi Benson proper efficient solutions to problem (CVEP) is examined under the newly introduced generalized convexity assumptions. Finally, by using the properties of Bishop-Phelps cone, we present the scalarization theorems for approximate quasi weakly (Benson proper) efficient solutions. [ABSTRACT FROM AUTHOR]

Approximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions. [ABSTRACT FROM AUTHOR]

The level-set subdifferential error bound (LSEB) is weaker than the Kurdyka-Łojasiewicz (KL) property and can replace it to establish linear convergence for various first-order algorithms. In this paper, we mainly study the behaviour of the level-set subdifferential error bound via Moreau envelopes under suitable assumptions. We provide an example that the Moreau envelope does not have the KL property but has the LSEB when the original function does not satisfy the KL property but only the LSEB. [ABSTRACT FROM AUTHOR]

In this article, we employ advanced techniques of convex analysis and $ \mathcal {C} $ C -differentiation to examine KKT-type robust necessary and sufficient optimality conditions and robust duality for an uncertain multiobjective programming problem under uncertainty sets, where $ \mathcal {C} $ C denotes the set of all $ \mathcal {G} $ G -derivatives which are positively homogeneous and convex with respect to the second argument. We first provide the robust constraint qualification of the (RCQ) type via the $ \mathcal {C} $ C -derivatives of uncertain constraint functions. We second establish KKT-type robust necessary conditions for robust (weakly) efficient solutions via the subdifferentials of $ \mathcal {C} $ C -derivatives to such problems. We third propose two new kinds of generalized $ \mathcal {C} $ C -convex functions via the subdifferentials of $ \mathcal {C} $ C -derivatives involving max-functions. Under suitable assumptions on the generalized $ \mathcal {C} $ C -convexity, KKT-type robust necessary optimality conditions become robust sufficient optimality conditions. Furthermore, we formulate as some applications a dual multiobjective programming problem to the underlying programming and examine weak, strong, and converse duality theorems for the same. Some illustrative examples are also provided for our findings. [ABSTRACT FROM AUTHOR]

This article is devoted to studying a nonsmooth semi-infinite multiobjective optimization problem (SIMP) in terms of the Mordukhovich/limiting subdifferentials. We first establish necessary and sufficient conditions for an ε-quasi positively properly efficient solution of a problem (SIMP). We also investigate Mond–Weir type dual problems under assumptions of ε-quasi pseudo-generalized convexity. Next, we provide an application to a nonsmooth fractional semi-infinite multiobjective optimization problem. Finally, some examples are given to illustrate the obtained results. The obtained results improve or include some recent known ones. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the idea of gH-weak subdifferential for interval-valued functions (IVFs) and show how to calculate gH-weak subgradients. It is observed that a nonempty gH-weak subdifferential set is convex and closed. In characterizing the class of functions for which the gH-weak subdifferential set is nonempty, it is identified that this class is the collection of gH-lower Lipschitz IVFs. In checking the validity of the sum rule of gH-weak subdifferential for a pair of IVFs, a counterexample is obtained, which reflects that the sum rule does not hold. However, under a mild restriction on one of the IVFs, one-sided inclusion for the sum rule holds. As applications, we employ gH-weak subdifferential to provide a few optimality conditions for nonsmooth IVFs. Further, a necessary optimality condition for interval optimization problems with a difference of two nonsmooth IVFs as the objective is established. Next, a necessary and sufficient condition via augmented normal cone and gH-weak subdifferential of IVFs for finding weak efficient points is presented. Lastly, in investigating a 'sup-relation' between gH-direction derivative and gH-weak subgradients, we approximately compute gH-weak subgradient at each iterative step. In the sequel, we propose W -gH-weak subgradient method to identify a weak efficient solution of an unconstrained nonsmooth IOP. We apply the proposed method to solve an interval optimization problem by taking a test example. We present a convergence analysis of the proposed method for constant and diminishing step sizes. [ABSTRACT FROM AUTHOR]

In this paper, we consider an infinite inequality system defined by a family of proper lower semicontinuous quasiconvex functions in real normed linear spaces. By using the interior-point condition and approximate continuity assumption of the functions, we establish some sufficient conditions for ensuring the basic constraint qualification for quasiconvex programming. [ABSTRACT FROM AUTHOR]

Outer approximation (OA) for solving convex mixed-integer nonlinear programming (MINLP) problems is heavily dependent on the convexity of functions and a natural issue is to relax the convexity assumption. This paper is devoted to OA for dealing with a pseudo-convex MINLP problem. By solving a sequence of nonlinear subproblems, we use Lagrange multiplier rules via Clarke subdifferentials of subproblems to introduce a parameter and then equivalently reformulate such MINLP as the mixed-integer linear program (MILP) master problem. Then, an OA algorithm is constructed to find the optimal solution to the MNILP by solving a sequence of MILP relaxations. The OA algorithm is proved to terminate after a finite number of steps. Numerical examples are illustrated to test the constructed OA algorithm. [ABSTRACT FROM AUTHOR]

This article deals with a class of constrained nonsmooth multiobjective programming problems (NMOPP) in the setting of Hadamard manifolds. The generalized Guignard constraint qualification (GGCQ), Abadie constraint qualification (ACQ), and the generalized ACQ (GACQ) are introduced in the framework of Hadamard manifolds for NMOPP using the notion of Clarke subdifferentials. Subsequently, by employing GGCQ and geodesic quasiconvexity assumptions, we establish Karush–Kuhn–Tucker (abbreviated as, KKT)-type necessary criteria of Pareto efficiency for NMOPP. Moreover, we establish that ACQ and GACQ are sufficient criteria for satisfaction of GGCQ. Several nontrivial numerical examples are furnished in manifold settings to demonstrate the validity of the derived results. To the best of our knowledge, this is the first time that ACQ, GACQ, GGCQ, and KKT-type necessary criteria of Pareto efficiency for NMOPP have been studied in manifold setting using Clarke subdifferentials. [ABSTRACT FROM AUTHOR]

In a Hilbert framework, we introduce a new class of second-order dynamical systems that combine viscous and geometric damping but also a time rescaling process for nonsmooth convex minimization. A main feature of these systems is to produce trajectories that lie in the graph of the Fenchel subdifferential of the objective. Moreover, they do not incorporate any regularization or smoothing processes. This new class originates from some combination of a continuous Nesterov-like dynamic and the Minty representation of subdifferentials. These models are investigated through first-order reformulations that amount to dynamics involving three variables: two solution trajectories (including an auxiliary one) and another one associated with subgradients. We prove the weak convergence towards equilibria for the solution trajectories, as well as properties of fast convergence to zero for their velocities. Remarkable convergence rates (possibly of exponential-type) are also established for the function values. We additionally state notable properties of fast convergence to zero for the subgradients trajectory and for its velocity. Some numerical experiments are performed so as to illustrate the efficiency of our approach. The proposed models offer a new and well-adapted framework for discrete counterparts, especially for structured minimization problems. Inertial algorithms with a correction term are then suggested relative to this latter context. [ABSTRACT FROM AUTHOR]

In this paper, we perform a sensitivity analysis for the maximal value function, which is the optimal value function for a parametric maximization problem. Our aim is to study various subdifferentials for the maximal value function. We obtain upper estimates of Fréchet, limiting, and horizon subdifferentials of the maximal value function by using some sensitivity analysis techniques sophisticatedly. The derived upper estimates depend only on the union of all solutions and not on its convex hull or only one solution from the solution set. Finally, we apply the derived results to develop some new necessary optimality conditions for nonconvex minimax problems. In the nonconvex-concave setting, our Wolfe duality approach compares favorably with the first-order approach in that the necessary condition is sharper and the constraint qualification is weaker. Funding: L. Guo was supported by the National Natural Science Foundation of China [Grants 72131007, 72140006, and 12271161] and the Natural Science Foundation of Shanghai [Grant 22ZR1415900]. J. J. Ye was partially supported by the Natural Sciences and Engineering Research Council of Canada. J. Zhang was supported by the National Natural Science Foundation of China [Grant 12222106], the Shenzhen Science and Technology Program [Grant RCYX20200714114700072], and the Guangdong Basic and Applied Basic Research Foundation [Grant 2022B1515020082]. [ABSTRACT FROM AUTHOR]

A notion of pseudo-differential on Riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented and then by definition of pseudo-Hessian, a version of Taylor's expansion theorem is obtained. Moreover, characterizations of generalized convexity of continuous functions are obtained in terms of the generalized monotonicity of pseudo-differentials. [ABSTRACT FROM AUTHOR]

In this study, our examination centers around the numerical resolution of non-coercive issues using a multi-grid approach. Our particular emphasis is directed towards employing multi-grid methodologies to tackle non-linear variational inequalities. Our primary goal involves confirming the consistent convergence of the multi-grid algorithm. To attain this objective, we make use of fundamental sub-differential calculus and glean insights from the convergence principles of non-linear multi-grid techniques. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a conservative evolution inclusion on a real Hilbert space with quasi-variational structures for not only time-dependent subdifferentials of convex functions but also inner products. Actually, this paper has three main purposes. The first one is to make it clear how to construct a real Hilbert space which enables us to handle the evolution equation with a mass-conservative property for the consideration. The second one is to show the existence of strong solutions to the Cauchy problem of the evolution inclusion on the Hilbert space, which is constructed in the first main purpose to handle the mass-conservative evolution inclusion. The third one is to investigate the relation between two evolution inclusions. [ABSTRACT FROM AUTHOR]

In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed. [ABSTRACT FROM AUTHOR]

Algorithmic differentiation is a tool used in several branches of computational science, both in conjunction with the interval calculus, and apart of it. This paper presents the ADHC library [14] developed by the author, and identified as package "na60" by Numerical Algorithms journal. This library makes intensive use of the C++ template meta-programming, and it has several unique features. ADHC seems particularly useful for interval-related applications. The library has been used by some solvers, also developed by the author, including HIBA_USNE. The paper describes the library, presenting its features, focusing on the new ones, added in version 2.0; in particular, we describe bounding subdifferentials of non-smooth functions and computing derivatives over various datatypes. Efficiency comparison with respect to other packages is also presented. Then some examples of ADHC applications are given, involving the use of HIBA_USNE and standard benchmark problems for solving nonlinear systems. A particular emphasis is put on examples related to modern machine learning, but not limited to them. Planned extensions and possible directions for future development of ADHC are also outlined and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of approximate convexity to setvalued maps and obtain some relations between approximate convexity and approximate monotonicity of their normal subdifferential. [ABSTRACT FROM AUTHOR]