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In this paper, we consider classes of approximate Minty and Stampacchia type vector variational inequalities using Clarke subdifferential on Hadamard manifolds and a class of nonsmooth multiobjective optimization problems. We investigate the relationship between the solution of these approximate vector variational inequalities and the solution of nonsmooth multiobjective optimization problems involving geodesic approximately convex functions. The results presented in this paper extend and generalize some existing results in the literature. [ABSTRACT FROM AUTHOR]

Given a set H of functions defined on a set X, á function f : X ↦ R ¯ is called abstract H -convex if it is the upper envelope of its H -minorants, i.e. such its minorants which belong to the set H ; and f is called regularly abstract H -convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) H -minorants. In the paper we first present the basic notions of (regular) H -convexity for the case when H is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a H -convex function to be regularly H -convex is formulated. The goal of the paper is to study the particular class of regularly H -convex functions, when H is the set L C ˆ (X , R) of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function f : X ↦ R ¯ to be L C ˆ -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each L C ˆ -convex function is regularly L C ˆ -convex as well. We focus on L C ˆ -subdifferentiability of functions at a given point. We prove that the set of points at which an L C ˆ -convex function is L C ˆ -subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset L C ˆ θ of the set L C ˆ consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of L C ˆ θ -subgradient and L C ˆ θ -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for L C ˆ θ -subdifferentials as well as L C ˆ θ -subdifferential conditions for global extremum points are established. Symmetric notions of abstract L C ˇ -concavity and L C ˇ -superdifferentiability of functions where L C ˇ := L C ˇ (X , R) is the set of Lipschitz continuous convex functions are also considered. [ABSTRACT FROM AUTHOR]

Metric regularity is widely concerned since its important applications in optimization and control theory. For promoting the application of metric regularity, it is valuable to study the estimation of the neighborhood which makes the regularity hold. However, it seems that no result has been established about this issue. This paper investigates the estimation of the neighborhood of metric regularity for quadratic functions. The main result gives the expression of the neighborhood of metric regularity for a kind of convex quadratic functions. [ABSTRACT FROM AUTHOR]

In this paper we present two Fenchel-type dual problems for a DC (difference of convex functions) optimization primal one. They have been built by means of the c-conjugation scheme, a pattern of conjugation which has been shown to be suitable for evenly convex functions. We study characterizations of weak, strong and stable strong duality for both pairs of primal–dual problems. We also give conditions which relate the existence of strong and stable strong duality for both pairs. [ABSTRACT FROM AUTHOR]

In this paper, we study the Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function on Hadamard manifolds. The gH-directional differentiability for interval-valued function is defined by using the generalized Hukuhara difference. The concepts of interval-valued convexity and pseudoconvexity are introduced on Hadamard manifolds, and several properties involving such functions are also given. Under these settings, we derive the KKT optimality conditions and give a numerical example to show that the results obtained in this paper are more general than the corresponding conclusions of Wu [The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res. 2007;176:46–59] in solving the optimization problem with interval-valued objective function. [ABSTRACT FROM AUTHOR]

This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature and also covers those situations where the problem at hand involves additional box-constraints on the control. As a byproduct, our approach shows in particular that Tikhonov regularized optimal control problems for the obstacle problem can be reformulated as state-constrained optimal control problems for the Poisson equation and that problems involving a subharmonic obstacle and a convex objective function are uniquely solvable. The paper concludes with three counterexamples which illustrate that rather peculiar effects can occur in the analysis of second-order optimality conditions for optimal control problems governed by the obstacle problem and that necessary second-order conditions for such problems may be hard to derive. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the split common null point problem in two Banach spaces. Then, using the Bregman generalized resolvents of maximal monotone operators, we prove strong convergence theorems of Halpern type iteration for finding a solution of the split common null point problem in two Banach spaces. As an application of our result, we study the split equilibrium problem in general Banach spaces and approximate a solution of the problem for the first time. Our new technique is based on basic properties of a Bregman distance induced by a Bregman function without using Bregman projection or the requirement of Mosco convergence of the sequences produced by the method. It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances. So, the results of the paper improve and extend many recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-convex min–max fractional problem of quadratic functions subject to convex and non-convex quadratic constraints. First, by using the Dinkelbach-type method, we transform the fractional problem into a univariate nonlinear equation. To evaluate this equation, we need to solve a non-convex quadratically constrained quadratic programming (QCQP) problem. To solve this problem, we propose a new method. In the proposed method, first, by using relaxation and convexification of non-convex constraints of non-convex QCQP problem, an upper bound and a lower bound of the optimal value is obtained. By using these bounds, we construct a parametric QCQP problem with two constraints. Then, by solution of the new problem, the parameters of this problem are updated for the next iteration. We show that the sequence of solutions of new problems is convergent to a global optimal solution of the non-convex QCQP problem. Numerical results are given to show the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose new numerical algorithms in the setting of unconstrained optimization problems and we prove the discrete rate of convergence of order O 1 / n 2 in the iterates of the convex objective function. Our optimization algorithms are obtained via discretizations from dynamical systems with Hessian-driven damping. Finally, some numerical experiments are presented in order to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Based on a coupling conjugation scheme and the perturbational approach, we build Fenchel–Lagrange dual problem of a composed optimization model with infinite constraints in separated locally convex spaces. This paper has mainly two targets. One is to establish strong duality under a new regularity condition (RCA) and an extension closed-type condition (ECRCA). The e-convex counterpart of Fenchel–Moreau theorem plays a key role in analysing the relation between them. The other aim is to achieve the sufficient and necessary characterizations for total duality in terms of c-subdifferentials. For this purpose, a formula for ε-c-subdifferentials of a proper function composed with a linear continuous operator is proved and applied. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonsmooth/nonconvex multiobjective optimization problem with uncertain constraints in arbitrary Asplund spaces. We first provide necessary optimality condition in a fuzzy form for approximate weakly robust efficient solutions and then establish necessary optimality theorem for approximate weakly robust quasi-efficient solutions of the problem in the sense of the limiting subdifferential by exploiting a fuzzy optimality condition in terms of the Fréchet subdifferential. Sufficient conditions for approximate (weakly) robust quasi-efficient solutions to such a problem are also driven under the new concept of generalized pseudo convex functions. Finally, we address an approximate MondWeir-type dual robust problem to the reference problem and explore weak, strong, and converse duality properties under assumptions of pseudo convexity. [ABSTRACT FROM AUTHOR]

In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Douglas–Rachford method (DR) for two closed (possibly nonconvex) sets in Euclidean spaces. We show that under certain regularity conditions, the DR converges locally with-linear rate. In convex settings, we prove that the linear convergence is global. Our study recovers recent results on the same topic. [ABSTRACT FROM AUTHOR]

Differential stability properties of convex optimization problems in Hausdorff locally convex topological vector spaces are considered in this paper. We obtain new formulas for the subdifferential and the singular subdifferential of the optimal value function of convex optimization problems. Namely, instead of using the traditional Moreau–Rockafellar Theorem, we employ a sum rule for subdifferentials of two convex functions from the work of Correa, Hantoute, and López [Weaker conditions for subdifferential calculus of convex functions. J Funct Anal. 2016;271:1177–1212]. Detailed comparisons with some known results are also given in this paper. [ABSTRACT FROM AUTHOR]

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered. [ABSTRACT FROM AUTHOR]

Let g be a strictly convex function on an evenly convex set X ⊂ R n with nonempty interior. Assuming that g is differentiable on i n t X , we consider the Bregman distance D g associated with g. Given a set T ⊂ R n , whose elements are called sites, and a particular site s, the farthest g-Bregman Voronoi cell of s, denoted by F T g (s) , consists of all points that are farther from s than from any other site with respect to D g . In this paper we study farthest g-Bregman Voronoi cells; in particular, we characterize those sets that can be written as F T g (s) for some T ⊂ R n and some s ∈ T. [ABSTRACT FROM AUTHOR]

In this paper, we propose a robust projection twin support vector machine (RPTSVM), where a new truncated L 2 -norm distance measure is applied to the with-class scatter to boost the robustness of the classifier when encountering a large number of outliers. In order to further improve the robustness of the model, chance constraints are employed to specify the lower bound of the probability that the distance from the projected samples to the projection of the other class centre is at least one. RPTSVM considers a pair of non-convex non-smooth problems with chance constraints. To solve these difficult problems, a newly designed method based on the difference of convex functions (DC) programs approach is presented. Extensive experiments on artificial datasets and benchmark datasets demonstrate the robustness and feasibility of our proposed RPTSVM. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, we prove that any accumulation point of the sequence generated by the new method is a stationary point of the problem under consideration. In order to illustrate the practical advantages of the new approach, we report some numerical experiments. In particular, we present an application where our concept of the inexact solutions is quite appealing. [ABSTRACT FROM AUTHOR]

In this paper, a proximal gradient splitting method for solving nondifferentiable vector optimization problems is proposed. The convergence analysis is carried out when the objective function is the sum of two convex functions where one of them is assumed to be continuously differentiable. The proposed splitting method exhibits full convergence to a weakly efficient solution without assuming the Lipschitz continuity of the Jacobian of the differentiable component. To carry this analysis, the popular Beck–Teboulle's line-search procedure is extended to the vectorial setting under mild assumptions. It is also shown that the proposed scheme obtains an ϵ-approximate solution to the vector optimization problem in at most O (1 / ε) iterations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of mixed integer programming problems (MIPs) whose objective functions are DC functions, that is, functions representable in terms of a difference of two convex functions. These MIPs contain a very wide class of computationally difficult non-convex MIPs since the DC functions have powerful expressability. Recently, Maehara, Marumo, and Mutota provided a continuous reformulation without integrality gaps for discrete DC programs having only integral variables. They also presented a new algorithm to solve the reformulated problem. Our aim is to extend their results to MIPs and give two specific algorithms to solve them. First, we propose an algorithm based on DCA originally proposed by Pham Dinh and Le Thi, where convex MIPs are solved iteratively. Next, to handle non-smooth functions efficiently, we incorporate a smoothing technique into the first method. We show that sequences generated by the two methods converge to stationary points under some mild assumptions. [ABSTRACT FROM AUTHOR]

Hemivariational inequalities arise in nonsmooth mechanics of solid involving nonmonotone and multi-valued mechanical relations. Typically, after the finite-element discretization, they lead to constrained nonsmooth nonconvex optimization problems with objective functions being the sum of quadratic functions and nonsmooth terms. In this paper, smoothing approximations are employed to solve the constrained nonsmooth nonconvex optimization problems. After properties of the smoothing functions are analysed, a smoothing quadratic regularization algorithm is presented and studied. The proposed algorithm can be implemented efficiently since the closed form solution is available at each iteration. Convergence of the algorithm is shown, and the worst-case complexity is investigated for reaching an ε-Clarke stationary point. A numerical example is reported to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov's smoothing techniques in infinite dimensions. As an application, we provide a simple approach to image reconstructions based on Nesterov's smoothing and algorithms for minimizing differences of convex (DC) functions that involve the ℓ 1 − ℓ 2 regularization. [ABSTRACT FROM AUTHOR]

The Douglas–Rachford splitting method is a classical and powerful method for minimizing the sum of two convex functions. In this paper, we introduce two dynamical systems based on this method for solving the minimum value problem of the sum of a strongly convex function and a weakly convex function. Under mild conditions, it is shown that the proposed dynamical systems are globally convergent to fixed point sets of the corresponding Douglas–Rachford operators, respectively, and are globally asymptotically stable if the corresponding fixed point sets are singleton. Furthermore, the globally exponential convergence of the proposed dynamical systems is established under some regularity conditions. A numerical example is reported to illustrate the effectiveness of the dynamical splitting method. [ABSTRACT FROM AUTHOR]

In a Hilbert space setting, the authors recently introduced a general class of relaxed inertial proximal algorithms that aim to solve monotone inclusions. In this paper, we specialize this study in the case of non-smooth convex minimization problems. We obtain convergence rates for values which have similarities with the results based on the Nesterov accelerated gradient method. The joint adjustment of inertia, relaxation and proximal terms plays a central role. In doing so, we highlight inertial proximal algorithms that converge for general monotone inclusions, and which, in the case of convex minimization, give fast convergence rates of values in the worst case. [ABSTRACT FROM AUTHOR]

In 2012, Censor et al. proposed the so-called two-subgradient extragradient method (TSEGM) in which both projections have an explicit-formula because the projections are just executed onto half-spaces. However, the convergence of the TSEGM was posted as an open question. In this paper, we answer this open question affirmatively for the case where the closed convex set can be represented as the lower level set of a continuously differentiable convex function. Censor's open question is thus partially answered. More precisely, we combine the inertial technique with the TSEGM and establish the weak convergence of the proposed method under mild conditions. [ABSTRACT FROM AUTHOR]

The paper concerns the study of a new class of conic programming with objectives given as the difference of a composite function and a convex function. We first introduce two new notions of regularity conditions in terms of the subdifferential of the involving functions. Under the new regularity conditions, we provide some necessary and/or sufficient conditions for KKT type optimality conditions to hold. Similarly, saddle point theorems and total Lagrange dualities for conic programming are also given. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The algorithm has low computational complexity since it consists in only one primal step and two dual steps at each iteration and allows one to solve the subproblem of each component inexactly and in parallel. Moreover, the algorithmic parameters are updated automatically without any tuning strategy as it happens in augmented Lagrangian approaches. We analyse the convergence of the algorithm and estimate itsanalytical worst-case complexity for both the primal–dual suboptimality and the primal feasibility violation, whereis a given accuracy. Extensive numerical tests confirm that our method is numerically more efficient than the classical decomposition methods from the literature. [ABSTRACT FROM AUTHOR]

We study semi-convex frontier (SCF) optimization problems where objective functions can be semi-convex and constraint sets can be non-polyhedron, which stem from a growing range of optimization applications such as frontier analysis, multi-objective programming in economics. The new findings of this paper can be summarized as follows: (1) We characterize non-dominated points of a non-polyhedron optimal solution set of a semi-convex frontier program. (2) We obtain optimality conditions of a constant modulus SCF program, of which the objective function is semi-convex with a constant semiconvexity modulus. (3) We obtain a non-smooth Hölder stability of the optimal solutions of a semiconvex frontier program. (4) We use generalized differentiability to establish sensitivity analysis of the optimal value function of a semi-convex frontier program. [ABSTRACT FROM AUTHOR]

In this paper, we first study in a Hilbertian framework the weak convergence of a general Gradient Projection Algorithm for minimizing a convex function of classover a convex constraint set. The way of selecting the stepsizes corresponds to the one used by López et al. for the particular case of the Split Feasibility Problem. This choice allows us to avoid the computation of operator norms. Afterwards, a relaxed version of the Gradient Projection Algorithm is considered where the feasible set is approximated by half-spaces making the projections explicit. Finally, to get the strong convergence, each step of the general Gradient Projection Method is combined with a viscosity step. This is done by adapting Halpern’s algorithm to our problem. The general scheme is then applied to the Split Equality Problem, and also to the Split Feasibility Problem. [ABSTRACT FROM AUTHOR]

In this paper we study the classical Variational Inequality (VI) over the intersection of sub-level sets of finite convex functions in real Hilbert spaces. The Subgradient Extragradient method of Censor et al. extend Korpelevich’s extragradient method by introducing an additional constructible set and then there is the need to calculate only one orthogonal projection onto the feasible set per each iteration instead of two as in. Motivated by this result, we propose a new extension, called the Totally Relaxed and Self-adaptive Subgradient Extragradient Method, which does not require the calculation of any exact projections onto the VI’s feasible set. Hence, any general convex feasible sets can be involved in the VI, such as the finite intersection of sub-level sets of convex functions. In our new scheme we also introduce an adaptive step-size rule which avoids the need to know a priori the Lipschitz constant of the VI associated mapping. Under mild and standard assumptions, we prove weak convergence of the proposed method at rate in the ergodic sense. Two numerical examples are presented to illustrate the behaviour and performances of out proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose two proximal-gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either concave or convex. In the iterative schemes, we perform a proximal step with respect to the nonsmooth numerator and a gradient step with respect to the smooth denominator. The algorithm in case of a concave denominator has the particularity that it generates sequences which approach both the (global) optimal solutions set and the optimal objective value of the underlying fractional programming problem. In case of a convex denominator the numerical scheme approaches the set of critical points of the objective function, provided the latter satisfies the Kurdyka-ᴌojasiewicz property. [ABSTRACT FROM AUTHOR]

This paper considers how an order market might evolve over a fairly short period – say, during a day. It uses elementary convex analysis to model agents’ choice of prudent orders, and it explores whether equilibrium is attainable. [ABSTRACT FROM AUTHOR]

In this paper, we use techniques which originate from proof mining to give rates of asymptotic regularity and metastability for a sequence associated to the composition of two firmly nonexpansive mappings. [ABSTRACT FROM AUTHOR]

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the convergence rate of the proximal algorithm proposed by us in the article [A proximal multiplier method for separable convex minimization. Optimization. 2016; 65:501–537], which has been proposed to solve a separable convex minimization problem. We prove that, under mild assumptions, the primal-dual sequences of the algorithm converge linearly to the optimal solution for a class of proximal distances. [ABSTRACT FROM AUTHOR]

Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A significant progress to go beyond convexity was made by considering the class of functions representable as differences of convex functions. In this paper, we introduce a generalized proximal point algorithm to minimize the difference of a nonconvex function and a convex function. We also study convergence results of this algorithm under the main assumption that the objective function satisfies the Kurdyka–ᴌojasiewicz property. [ABSTRACT FROM AUTHOR]

In this paper, we consider a type of the celebrated convex feasibility problem, named as split quasi-convex feasibility problem (SQFP). The SQFP is to find a point in a sublevel set of a quasi-convex function in one space and its image under a bounded linear operator is contained in a sublevel set of another quasi-convex function in the image space. We propose a new adaptive subgradient algorithm for solving SQFP problem. We also discuss the convergence analyses for two cases: the first case where the functions are upper semicontinuous in the setting of finite dimensional, and the second case where the functions are weakly continuous in the infinite-dimensional settings. Finally some numerical examples in order to support the convergence results are given. [ABSTRACT FROM AUTHOR]

This paper solves a utility maximization problem under utility-based shortfall risk constraint, by proposing an approach using Lagrange multiplier and convex duality. Under mild conditions on the asymptotic elasticity of the utility function and the loss function, we find an optimal wealth process for the constrained problem and characterize the bi-dual relation between the respective value functions of the constrained problem and its dual. This approach applies to both complete and incomplete markets. Moreover, the extension to more complicated cases is illustrated by solving the problem with a consumption process added. Finally, we give an example of utility and loss functions in the Black–Scholes market where the solutions have explicit forms. [ABSTRACT FROM PUBLISHER]

In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case. [ABSTRACT FROM PUBLISHER]

In this paper, we propose an inexact proximal multiplier method using proximal distances for solving convex minimization problems with a separable structure. The proposed method unifies the works of Chen and Teboulle (PCPM method), Kyono and Fukushima (NPCPMM) and Auslender and Teboulle (EPDM), and extends the convergence properties for the class of-divergence distances. We prove, under standard assumptions, that the iterations generated by the method are well defined and the sequence converges to an optimal solution of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an iterative scheme for approximating a common element of the set of fixed points of left Bregman strongly nonexpansive mapping and the set of solutions of split feasibility problem. We further obtain a strong convergence result for finding a common solution of fixed point problem for left Bregman strongly nonexpansive mappings and split feasibility problem in the framework of-uniformly convex Banach spaces which are also uniformly smooth. We give an application of our result to approximating a solution of convexly constrained linear inverse problem which is also a fixed point of a left Bregman strongly nonexpansive mapping in-uniformly convex Banach spaces, which are also uniformly smooth. Finally, we give some numerical example of our result to study its efficiency and implementation. Our results complement many known related results in the literature. [ABSTRACT FROM AUTHOR]

Starting from the innovative ideas of Chicco et al. (Vector optimization problems via improvement sets. J. Optim. Theory Appl. 2011;150:516–529), in this paper, the concepts of improvement set and-efficiency are introduced in a real locally convex Hausdorff topological vector space. Furthermore, some properties of the improvement sets are given and a kind of proper efficiency, named as-Benson proper efficiency, which unifies some proper efficiency and approximate proper efficiency, is proposed via the improvement sets in vector optimization. Moreover, the concept of-subconvexlikeness of set-valued maps is introduced via the improvement sets and an alternative theorem is proved. In the end, some scalarization theorems and Lagrange multiplier theorems of-Benson proper efficiency are established for a vector optimization problem with set-valued maps. [ABSTRACT FROM PUBLISHER]

This paper aims at studying, in the image space, an approximation of a vector optimization problem obtained by substituting the involved functions with their-derivatives. It is shown that, under the hypothesis of-differentiability, the existence of a lower semistationary point is equivalent to the linear separation between the image of the approximated problem and a suitable convex subset of the image space. Applications to optimality conditions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the convergence rate of the sequence of objective function values of a primal-dual proximal-point algorithm recently introduced in the literature for solving a primal convex optimization problem having as objective the sum of linearly composed infimal convolutions, nonsmooth and smooth convex functions and its Fenchel-type dual one. The theoretical part is illustrated by numerical experiments in image processing. [ABSTRACT FROM AUTHOR]

We investigate the long time behaviour of the solutions to the first order differential inclusion where is the subgradient of a given convex and continuous function defined on a real Hilbert space , the operator is the orthogonal projection onto a closed, nonempty and convex subset Q of , and is an absolutely continuous function. We establish that if the objective function Φ has at least one minimizer over Q and behaviours, for t large enough, like for some constant then any solution to (the above equation) converges weakly to a minimizer of Φ over Q and satisfies the following fast decay property: where. Moreover, we prove the strong convergence of the solutions under some simple geometrical assumptions on the function Φ. [ABSTRACT FROM AUTHOR]

We present an interior proximal method for solving constrained nonconvex optimization problems where the objective function is given by the difference of two convex function (DC function). To this end, we consider a linearized proximal method with a proximal distance as regularization. Convergence analysis of particular choices of the proximal distance as second-order homogeneous proximal distances and Bregman distances are considered. Finally, some academic numerical results are presented for a constrained DC problem and generalized Fermat–Weber location problems. [ABSTRACT FROM AUTHOR]

We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories. [ABSTRACT FROM AUTHOR]

The subgradient algorithms for convex optimizations on Riemannian Manifolds of sectional curvatures bounded from below are studied; and the convergence results of the algorithms (employing diminishing step sizes and dynamic step sizes) are established. Some numerical experiments are provided to illustrate the convergence performances of the algorithms. [ABSTRACT FROM AUTHOR]

We introduce and study the notion of e-subdifferential for an e-convex function in locally convex spaces. Some relationships between the e-subdifferential and the Clarke–Rockafellar subdifferential are established. Besides, the e-convex functions are characterized. Moreover, some results regarding the locally Lipschitz properties of e-convex functions are obtained. [ABSTRACT FROM AUTHOR]