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Let μ be a probability measure on R d with barycenter at the origin, which is supported on an open, bounded and convex set K having smooth boundary and positive Gauss curvature. Suppose that d μ x = ϱ x − α d x , with α > d + 1 and ϱ : K → 0 , + ∞ is C ∞ -smooth, satisfies ɛ 0 -convex condition for some ɛ 0 > 0 such that ϱ and its first derivatives are bounded in K. In this paper, we prove that for any convex solution ψ : R d → 0 , + ∞ to the following Monge–Ampère equation (1) [ ϱ ∇ ψ x ] − α det ( ∇ 2 ψ x ) = (ψ x ) − α , x ∈ R d the function x ↦ log ψ x has second derivatives to be bounded by a constant C ɛ 0 depending on ɛ 0 , α , ψ (x 0) and values of first, second and fourth derivatives of ψ at a fixed point x 0 ∈ R d . This estimate is as a variant case of the Caffarelli contraction theorem by replacing target densities of e − W by densities of ϱ − α that Cauchy distributions (i.e., ϱ x − α = 1 + | x | 2 − α ) are typical examples. [ABSTRACT FROM AUTHOR]

Abstract The paper is concerned with the least W 1 , 1 -energy required to produce maps from a domain Ω ⊂ R 2 with values into S 1 having prescribed singularities (a i) 1 ≤ i ≤ k. The value of the infimum has been known for a long time and corresponds to the length of minimal configurations connecting the points (a i) between themselves and/or to the boundary. We tackle here the question whether the infimum of this W 1 , 1 -energy is achieved. This natural topic turns out to be delicate and we have a complete answer only when k = 1. The bottom line for k ≥ 1 is that the infimum is "rarely" achieved. As a "substitute", we give a full description of the asymptotic behavior of all minimizing sequences and show that they "concentrate" along "convex combinations" of minimal configurations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study some non-traditional schemes of proximal point algorithm for nonsmooth convex functionals in a Banach space. The proximal approximations to their minimal points and/or their minimal values are considered separately for unconstrained and constrained minimization problems on convex closed sets. For the latter we use proximal point algorithms with the metric projection operators and first establish the estimates of the convergence rate with respect to functionals. We also investigate the perturbed projection proximal point algorithms and prove their stability. Some results concerning the classical proximal point method for minimization problems in a Banach space is also presented in this paper. [Copyright &y& Elsevier]

This paper will deal with a locally constrained inverse curvature flow which keeps the convexity and preserves the enclosed area of the evolving curve, and the limiting curve converges to a circle in the C ∞ sense. As applications of this flow, an inequality for the integral of the power of support function for convex curves and the classical isoperimetric inequality are obtained. [ABSTRACT FROM AUTHOR]

The paper investigates various properties of Moreau envelopes and proximal mappings of s -lower regular functions. In particular, for such a function on uniformly convex space, the Moreau envelope is shown to be differentiable with its derivative locally Hölder continuous. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we extend some classical results of Convex Analysis to the sub-Riemannian setting of the Heisenberg group. In particular, we provide a horizontal version of Minty’s theorem concerning maximal H-monotone operators defined in the Heisenberg group with values in the first layer of its Lie algebra. [Copyright &y& Elsevier]

Abstract: The paper deals with the similarity solutions of a boundary layer problem, which arises in steady two-dimensional flow for power-law fluids, and which is converted to a boundary value problem including a negative parameter under certain boundary conditions. We establish the existence and uniqueness results for normal convex solutions or generalized convex solutions to the boundary value problem, and give the asymptotic behavior of normal convex solutions at the infinity by applying some new conclusions on a singular integral equation. [Copyright &y& Elsevier]

Abstract: Existence and uniqueness of solutions to a Hamilton–Jacobi equation with convex with respect to the last variable can be proved by associating to either a Calculus of Variations or an optimal control problem. The data of the new problem should be so that its Hamiltonian coincides with and should also inherit appropriate continuity/local Lipschitz continuity properties of . In other words, can be represented by functions describing an optimization problem. In this paper we provide further developments of representation theorems from Rampazzo (2005). In particular, our results imply continuous dependence of representations on the mapping . We apply them to study existence of solutions to the Hamilton–Jacobi equation with possibly discontinuous in . [Copyright &y& Elsevier]

Abstract: In this paper we study the behaviour of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch–Wets convergences. Our analysis is devoted to proximal, Fréchet, and Mordukhovich limiting normal cones and subdifferentials. The results obtained can be seen as extensions of the Attouch theorem to the context of non-convex functions on locally uniformly convex Banach space. They also generalize, to sequences of subsmooth sets or functions, various results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence and some qualitative properties of solutions of variational inequalities of the form: and of quasi-variational inequalities of the form: where is a second-order elliptic operator of Leray–Lions type, is a multivalued lower order term, and are convex functionals, and also depends on . We concentrate here in noncoercive cases and use sub-supersolution methods to study the existence and enclosure of solutions, and also the existence of extremal solutions between sub and supersolutions. [Copyright &y& Elsevier]

Abstract: In this paper, our main purpose is to establish the existence of multiple solutions of a -Laplacian equation with convex and Sobolev critical nonlinearity by some standard variational methods, whose key is to construct homotopies between and levels of the functional , and some analytical techniques. [Copyright &y& Elsevier]

Abstract: In this paper, a new class of nonsmooth nonconvex multiobjective programming problems in Banach spaces is considered. The sufficient optimality conditions for efficiency and proper efficiency are established under nonsmooth -invexity and generalized -invexity notions, after extending these concepts to the functions between Banach spaces. Further, Mond–Weir duality theorems are also established under -invexity for the considered vector optimization problems in Banach spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we establish conditions ensuring Hölder and Lipschitz continuity of minimizers in convex programming. Lipschitz continuity is proved by establishing and applying a generalized version of the implicit function theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we explore some properties of the Moreau envelope function of and the associated proximal mapping in the sense of the Bregman distance induced by a convex function . Precisely, we study the continuity, differentiability, and Clarke regularity of the Moreau envelope function and the upper semicontinuity and single-valuedness of the proximal mapping as well as its relation to the convexity of , where is a positive parameter. [Copyright &y& Elsevier]

Abstract: In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition. [Copyright &y& Elsevier]

Abstract: In this paper, we prove sufficient conditions for controllability and strong controllability in terms of the Mordukhovich subdifferential for two classes of differential inclusions. The first one is the class of sub-Lipschitz multivalued functions introduced by Loewen–Rockafellar (1994) . The second one, introduced recently by Clarke (2005) , is the class of multivalued functions which are pseudo-Lipschitz and satisfy the so-called tempered growth condition. To do this, we establish an error bound result in terms of the Mordukhovich subdifferential outside Asplund spaces. [Copyright &y& Elsevier]

Abstract: The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter. Concavity of the optimal value is established and implies further regularity properties of the optimal value. Boundedness, outer semicontinuity, single-valuedness, continuity, and Lipschitz continuity of the minimizer mapping are concluded under various assumptions. Sharp minimizers are given further attention. Several examples are given to illustrate our results. [Copyright &y& Elsevier]

Abstract: The aim of this note is to give an alternative proof for a recent result due to Dorsch et al., which provides an upper estimate for the Clarke subdifferential of an infimal value function. We show the validity of this result under a weaker condition than the one assumed in the aforementioned paper, while the use of the Mordukhovich subdifferential, as an intermediate step, will considerably shorten its proof. [Copyright &y& Elsevier]

Abstract: This paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms. By exploiting the effect of the coefficient of the critical nonlinearity and a variational method, we establish the main result which is based on the argument of the compactness. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new class of functions called geodesic semi--convex functions and discuss some of their properties. [Copyright &y& Elsevier]

Abstract: The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35–61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, and , that can be used to approximate common zeros of and . In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals. [Copyright &y& Elsevier]

Abstract: In this paper, we consider optimality conditions and a constraint qualification for quasiconvex programming. For this purpose, we introduce a generator and a new subdifferential for quasiconvex functions by using Penot and Volle’s theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with a class of nondifferentiable minimax programming problems and its two types of higher-order dual models. We establish weak, strong and strict converse duality theorems in the framework of generalized convexity in order to relate the optimal solutions of primal and dual problems. Our study improves and extends some of the known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper we show that the positive semi-definiteness (PSD) of the Fréchet and/or Mordukhovich second-order subdifferentials can recognize the convexity of functions. However, the PSD is insufficient for ensuring the convexity of a locally Lipschitz function in general. A complete characterization of strong convexity via the second-order subdifferentials is also given. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave–convex functions are obtained as well. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the -concave–convex operator. Without any compact or continuous assumptions, we prove the existence and uniqueness of fixed points, giving monotone iterative sequences for the unique fixed point for the operator. Finally, we apply the results to an integral equation of polynomial type which possesses items of measurable functions. [Copyright &y& Elsevier]

Abstract: Local convergence analysis of the proximal point method for a special class of nonconvex functions on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of second order differential inclusions attached to which the convex potential varies depending on the time, and the boundary conditions are described by the subdifferentials of convex functions. Applying the theory of lower semicontinuous, convex and proper functionals, and fixed points, we give some results about the representation of the subdifferential operators of some convex functionals and the existence of solutions for the boundary value problems. [Copyright &y& Elsevier]

Abstract: In this paper we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary. We establish an explicit and general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. [Copyright &y& Elsevier]

Abstract: In this paper we prove two characterizations of reflexivity for a Banach space . The first one is based on the existence in of a closed convex cone with a nonempty interior such that all the bases generated by a strictly positive functional are bounded, while the second one is stated in terms of the nonexistence of a cone such that it has bounded and unbounded bases (both generated by strictly positive functionals) simultaneously. We call such a cone mixed based cone. We study the features of this class of cones. In particular, we show that every cone conically isomorphic to the nonnegative orthant of is a mixed based cone and that every mixed based cone contains a conically isomorphic copy of . Moreover we give a detailed description of the structure of a mixed based cone. This approach allows us to prove some results concerning the embeddings of and in a Banach space. [Copyright &y& Elsevier]

Abstract: This paper deals with an eigenvalue problem for hemivariational inequalities on domains of the type ( is a bounded open subset of , ) and it involves concave–convex nonlinearities. Under suitable conditions on the nonlinearities, two nontrivial solutions are obtained which belong to a special closed convex cone of whenever the eigenvalues are of certain range. Our approach is variational based on the theories of non-smooth analysis. Our results are a generalization of the case of Laplacian from A. Kristály, et al. to the case of -Laplacian. [Copyright &y& Elsevier]

Abstract: The principal aim of this paper is to show that weakly cone-convex vector-valued functions, as well as weakly cone-quasiconvex vector-valued functions, can be characterized in terms of usual weakly convexity and weakly quasiconvexity of certain real-valued functions, defined by means of the extreme directions of the polar cone or by Gerstewitz’s scalarization functions. [Copyright &y& Elsevier]

Abstract: Let be a bounded Lipschitz domain, a suitably quasiconvex integrand and consider the energy functional over the space of measure preserving maps In this paper we discuss the question of existence of multiple strong local minimizers for over . Moreover, motivated by their significance in topology and the study of mapping class groups, we consider a class of maps, referred to as twists, and examine them in connection with the corresponding Euler–Lagrange equations and investigate various qualitative properties of the resulting solutions, the stationary twists. Particular attention is paid to the special case of the so-called -Dirichlet energy, i.e., when . [Copyright &y& Elsevier]

Abstract: In this paper, we present a new primal-dual interior-point algorithm for solving a special case of convex quadratic semi-definite optimization based on a parametric kernel function. The proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the -center for the algorithm. These properties enable us to derive the currently best known iteration bounds for the algorithm with large- and small-update methods, namely, and , respectively, which reduce the gap between the practical behavior of the algorithm and its theoretical performance results. [Copyright &y& Elsevier]

Abstract: The paper is organized into two parts. In the first part, we establish some new vectorial set-valued variant of Hahn–Banach theorems with a K-concave continuous (resp. lower semicontinuous, upper semicontinuous) set-valued map dominated by a K-convex set valued map. The existence of the extension of continuity for dominated set-valued maps in order-complete topological vector spaces is also given. The main purpose of the second part is to present some applications to the existence theorems of K-quasigradients of K-convex set-valued maps, a generalized Lagrange multiplier theorem and a generalized sandwich theorem. [Copyright &y& Elsevier]

Abstract: In this paper we generalize a result of Mercer [A.McD. Mercer, A variant of Jensen’s inequality, J. Inequal. Pure & Appl. Math. 4 (4) (2003) Article 73. Online:http://jipam.vu.edu.au/article.php?sid=314] on convex functions. A relationship between this result and majorization ordering is pointed out. An extension of Mercer’s result to pairs of similarly separable m-tuples is given. In addition, applications are discussed for nonincreasing in mean tuples, star-shaped tuples and convex tuples. [Copyright &y& Elsevier]

Abstract: In this paper, we first establish that the Kuhn–Tucker necessary optimality condition is sufficient for global optimality of the class of convexifiable programming problems with bounds on variables for which a local minimizer is global. This result yields easily verifiable Kuhn–Tucker sufficient conditions for non-convex quadratic programs. We also present new conditions for a feasible point which satisfies the Kuhn–Tucker conditions to be a global minimizer of multi-extremal mathematical programming problems which may have local minimizers that are not global. In the multi-extremal case, the convexifiability of an augmented Lagrangian function plays a key role in deriving the result. As an application, we also derive sufficient optimality conditions for multi-extremal bivalent programming problems. Several examples are given to illustrate the significance of the results. [Copyright &y& Elsevier]

Abstract: In this paper some properties of nonsmooth quasiconvex and pseudoconvex functions in weakly compactly generated Asplund spaces are proved. [Copyright &y& Elsevier]

Abstract: The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset of and , the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 or −1 implying the existence of the nearest points to from is equivalent to being compactly locally uniformly convex. Similar results for uniqueness of the nearest point are also established. [Copyright &y& Elsevier]

Abstract: Dini derivatives in Riemannian manifold settings are studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given. [Copyright &y& Elsevier]

Abstract: Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following degenerate parabolic problem: where is a continuous function which could be negative in a bounded set of and a convex increasing function. Under additive conditions, we prove the uniqueness of the positive stationary solution and, then, give the asymptotic behaviour of solutions to when . [Copyright &y& Elsevier]

Abstract: Several kinds of behaviors of extended-real-valued lower semicontinuous functions are known to be equivalent to certain appropriate conditions in terms of the Clarke subdifferential. The paper provides a systematic study showing that any such condition with the Clarke subdifferential is valid if and only if it holds with any operator representing the Clarke subdifferential like in the subdifferential proximal formula. [Copyright &y& Elsevier]

Abstract: In this paper, a definition of a kind of quasi-convex functional was proposed and two properties of the quasi-convex functional were proved. A mini max theorem due to Stepan A. Tersian was generalized by using the properties of the quasi-convex functional. The existence and uniqueness of solution of the boundary value problem for the nonlinear beam equation was probed and an existence and uniqueness theorem was presented. [Copyright &y& Elsevier]

In this paper, we prove H 2 + α regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar C 2 , α regularity result due to Silvestre and Sirakov proved in [17] for solutions to non-convex fully nonlinear elliptic equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present new dual constraint qualifications which completely characterize the zero duality gap property for convex programming problems in general Banach spaces. As an application, we derive constraint qualifications, characterizing zero duality gaps of semidefinite programs. Our approach makes use of convex majorants of support functions together with conjugate analysis and approximate subdifferentials of convex functions. [Copyright &y& Elsevier]