Hoang Xuan Phu

Six kinds of roughly convex functions

This paper considers six kinds of roughly convex functions, namely: δ-convex, midpoint δ-convex, ρ-convex, γ-convex, lightly γ-convex, and midpoint γ-convex functions. The relations between these concepts are presented. It is pointed out that these roughly convex functions have two optimization properties: each r-local minimizer is a global minimizer, and if they assume their maximum on a bounded convex domain D (in a Hilbert space), then they do so at least at one r-extreme point of D, where r denotes the roughness degree of these functions. Furthermore, analytical properties are investigated, such as boundedness, continuity, and conservation properties.

ROUGH CONVERGENCE IN NORMED LINEAR SPACES

x * ∈ X is said to be an r-limit point of a sequence (xi ) in some normed linear space (X,∥ · ∥) if (r ≥ 0). The set of all r-limit points of (xi , denoted by LIM r x i , is bounded closed and convex. Further properties, in particular the relation between this rough convergence and other convergence notions, and the dependence of LIM r x i on the roughness degree r, are investigated. For instance, the set-valued mapping r ↦ LIM r x i is strictly increasing and continuous on ( ), where . For a so-called ρ-Cauchy sequence (xi ) satisfying it is shown in case X = R n that r = (n/(n + 1))ρ (or for Euclidean space) is the best convergence degree such that LIM r x i ≠ Ø.

γ-Subdifferential and γ-convexity of functions on the real line

In this paper the γ-subdifferential and γ-convexity of real-valued functions on the real line are introduced. By means of the γ-subdifferential, a new necessary condition for global minima (or maxima) is formulated which many local minima (or maxima) cannot satisfy. The γ-convexity is used to state sufficient conditions for global minima. The class of γ-convex functions is relatively large. For example, there are γ-convex functions which are not continuous anywhere. Nevertheless, a γ-local minimum of a γ-convex function is always a global minimum. Furthermore, if a γ-convex function attains its global minimum, then it does so near the boundary of its domain.

On the stability of solutions to quadratic programming problems

Abstract.We consider the parametric programming problem (Qp) of minimizing the quadratic function f(x,p):=xTAx+bTx subject to the constraint Cx≤d, where x∈ℝn, A∈ℝn×n, b∈ℝn, C∈ℝm×n, d∈ℝm, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Qp) are denoted by M(p) and Mloc(p), respectively. It is proved that if the point-to-set mapping Mloc(·) is lower semicontinuous at p then Mloc(p) is a nonempty set which consists of at most ?m,n points, where ?m,n=

Stable generalization of convex functions

\binom{m}{{\text{min}}\{[m/2],n\}}

Essential supremum and supremum of summable functions

is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.

Rough Convergence in Infinite Dimensional Normed Spaces

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.

Some properties of globally δ-convex functions ∗

Let D ≌ RN , 0 0}, where ess sup ƒ denotes the essential supremum of ƒ. These properties can be used for computing ess sup ƒ. As example, two algorithms are stated. If the function ƒ is dense, or lower semicontinuous, or if −ƒ is robust, then sup ƒ = ess sup ƒ. In this case, the algorithms mentioned can be applied for determining the supremum of ƒ, i.e., also the global maximum of ƒ if it exists.

ROUGH CONTINUITY OF LINEAR OPERATORS

Abstract For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, and it is called a ρ-Cauchy sequence if This article investigates different aspects of this rough convergence, especially in infinite dimensional spaces, such as analytical properties of r-limit set, relation to other convergence notions, and the dependence of the r-limit set on the roughness degree r. Moreover, by using the Jung constant we find the minimal value of r such that an arbitrary ρ-Cauchy sequence in X is certainly r-convergent.Abstract For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, and it is called a ρ-Cauchy sequence if This article investigates different aspects of this rough convergence, especially in infinite dimensional spaces, such as analytical properties of r-limit set, relation to other convergence notions, and the dependence of the r-limit set on the roughness degree r. Moreover, by using the Jung constant we find the minimal value of r such that an arbitrary ρ-Cauchy sequence in X is certainly r-convergent.

The method of orienting curves and its application to manipulator trajectory planning

The notions of δ-convex and midpoint δ-convex functions were introduced by Hu, Klee, and Larman (SIAM Journal on Control and Optimization, Vol. 27 1989). It is known that such functions have some important optimization properties: each r-local minimum is a global minimum, and if they assume their global maximum on a bounded convex domain of a Hilbert space then they do so at least at some r-extreme points of this domain. In this paper some analytical properties of δ-convex and midpoint δ-convex functions are investigated. Concretely, it is shown when they are bounded (from above or from below). For instance, δ-convex functions defined on the entire real line is always locally bounded, and midpoint δ-convex function on the real line is either locally bounded or totally unbounded. Further on, it is proved that there are totally discontinuous (i.e., nowhere differentiable) δ-convex and midpoint δ-convex functions on the real line