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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 paper, we prove that a radially lower semicontinuous function of one variable, defined on some interval, is pseudoconvex, if and only if its domain of definition can be split into three parts such that the function is strictly monotone decreasing without stationary points over the first subinterval, it is constant over the second one, and it is strictly monotone increasing without stationary points over the third subinterval. Each one or two of these parts may be empty or degenerate into a single point. The proof of this property is easy, when the function is differentiable. We consider functions, which are pseudoconvex with respect to the lower Dini directional derivative. This result follows from some known claims, but our theorem is a shot, directed to the target. We apply this characterization to obtain a complete characterization of strictly pseudoconvex functions. We also derive the respective results, when the function is radially lower semicontinuous in a real linear space. Several applications of the characterization are provided. A result due to Diewert, Avriel and Zang is extended to radially continuous functions. [ABSTRACT FROM AUTHOR]

In this paper, we study some problems with a continuously differentiable and quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over the solution set, and the normalized gradient is constant over it; (II) the gradient of the objective function is equal to zero over the solution set. As a consequence, we obtain characterizations of the solution set of a program with a continuously differentiable and quasiconvex objective function, provided that one of the solutions is known. We also derive Lagrange multiplier characterizations of the solutions set of an inequality constrained problem with continuously differentiable objective function and differentiable constraints, which are all quasiconvex on some convex set, not necessarily open. We compare our results with the previous ones. Several examples are provided. [ABSTRACT FROM AUTHOR]

In this article, a generalization of theECP algorithm to cover a class of nondifferentiable Mixed-Integer NonLinear Programming problems is studied. In the generalization constraint functions are required to be-pseudoconvex instead of pseudoconvex functions. This enables the functions to be nonsmooth. The objective function is first assumed to be linear but also-pseudoconvex case is considered. Furthermore, the gradients used in theECP algorithm are replaced by the subgradients of Clarke subdifferential. With some additional assumptions, the resulting algorithm shall be proven to converge to a global minimum. [ABSTRACT FROM AUTHOR]

Some properties of pseudoconvex functions w.r.t. an equilibrium function, defined by means of limiting subdifferential, are obtained. Furthermore, the equivalence between generalized Minty vector equilibrium problems (in short, GMVEPs) (resp., generalized Stampacchia vector equilibrium problems (in short, GMVEPs)) involving limiting subdifferential and nonsmooth vector optimization problems (in short, VOPs) are studied under pseudoconvexity condition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a one-layer recurrent projection neural network is proposed for solving pseudoconvex optimization problems with general convex constraints. The proposed network in this paper deals with the constraints into two parts, which brings the network simpler structure and better properties. By the Tikhonov-like regularization method, the proposed network need not estimate the exact penalty parameter in advance. Moreover, comparing with some existing neural networks, the proposed network can solve more general constrained pseudoconvex optimization problems. When the solution of the proposed network is bounded, it converges to the optimal solution set of considered optimization problem, which may be nonsmooth and nonconvex. Meantime, some sufficient conditions are presented to guarantee the boundedness of the solution of the proposed network. Numerical examples with simulation results are given to illustrate the effectiveness and good characteristics of the proposed network for solving constrained pseudoconvex optimization. [Copyright &y& Elsevier]

We study some classes of generalized convex functions, using a generalized differential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdifferential or a pseudo-differential in the sense of Jeyakumar and Luc. Such a general framework allows us to avoid technical assumptions related to specific constructions. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced. We devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange–Kuhn–Tucker multipliers for mathematical programming problems. [ABSTRACT FROM PUBLISHER]

We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known. [ABSTRACT FROM AUTHOR]

We study some classes of generalized affine functions, using a generalized differential. We study some properties and characterizations of these classes and we devise some characterizations of solution sets of optimization problems involving such functions or functions of related classes. [ABSTRACT FROM AUTHOR]

A quasiconvex function is a function which has convex sublevel sets. This paper studies robustly quasiconvex functions, that is, quasiconvex functions which remain quasiconvex under small linear perturbations. Relations to pseudoconvexity and other generalized convexity concepts and necessary and sufficient first-order conditions for robust quasiconvexity of smooth functions are presented. Convex-analytic properties and convexification of robustly quasiconvex functions are studied. Supporting robustly quasiconvex functions by simpler functions is discussed, with duality theory as motivation. Through the use of a second-order condition for robust quasiconvexity of nonsmooth functions, nontrivial examples of such functions are given. [ABSTRACT FROM AUTHOR]

The notions of continuous (strictly) pseudoconvex functions and set-valued (strictly) pseudomonotone maps are introduced. Relations between these notions are given using an asymptotic mean value theorem. Characterizations of a continuous quasiconvex function are derived. [ABSTRACT FROM AUTHOR]