Phan Thanh An

Stable generalization of convex functions

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.

Method of orienting curves for determining the convex hull of a finite set of points in the plane

In this article, we present an efficient algorithm to determine the convex hull of a finite planar set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Losung einer regularen Aufgabenklasse der optimalen Steuerung in Grosen mittels Orientierungskurven, Optimization, 18 (1987), pp. 65–81, for solving optimal control problems with state constraints). The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Grahams convex hull algorithm are presented.In this article, we present an efficient algorithm to determine the convex hull of a finite planar set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung in Großen mittels Orientierungskurven, Optimization, 18 (1987), pp. 65–81, for solving optimal control problems with state constraints). The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Grahams convex hull algorithm are presented.

Stability of generalized monotone maps with respect to their characterizations

We show that the known types of generalized monotone maps are not stable with respect to their characterizations (i.e. the characterizations are not maintained if an arbitrary map of this type is disturbed by an element with sufficiently small norm) and introduce s-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, s-quasimonotonicity is related to s-quasiconvexity (introduced by Phu in Optimization, 38, 1996) of the underlying function. A necessary and sufficient condition for a univariate polynomial to be s-quasimonotone is given. Furthermore, some stability properties of s-quasiconvex functions are presented.We show that the known types of generalized monotone maps are not stable with respect to their characterizations (i.e. the characterizations are not maintained if an arbitrary map of this type is disturbed by an element with sufficiently small norm) and introduce s-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, s-quasimonotonicity is related to s-quasiconvexity (introduced by Phu in Optimization, 38, 1996) of the underlying function. A necessary and sufficient condition for a univariate polynomial to be s-quasimonotone is given. Furthermore, some stability properties of s-quasiconvex functions are presented.

An efficient convex hull algorithm for finite point sets in 3D based on the Method of Orienting Curves

This article describes an efficient convex hull algorithm for finite point sets in 3D based on the idea of the Method of Orienting Curves (introduced by Phu in Optimization, 18 (1987) pp. 65–81, for solving optimal control problems with state constraints). The actual run times of our algorithm and known gift-wrapping algorithm on the set of random points (in uniform distribution) show that our algorithm runs significantly faster than the gift-wrapping one.

Direct multiple shooting method for solving approximate shortest path problems

We use the idea of the direct multiple shooting method (presented by Bock in Proceedings of the 9th IFAC World Congress Budapest, Pergamon Press, 1984, for solving optimal control problems) to introduce an algorithm for solving some approximate shortest path problems in motion planning. The algorithm is based on a direct multiple shooting discretization that includes a collinear condition (a continuity condition type in the direct multiple shooting method), multiple shooting structure, and approximation conditions. In the case of monotone polygons, it is implemented by a C code, and a numerical example shows that our algorithm significantly reduces the running time and memory usage of the system.

Incremental Convex Hull as an Orientation to Solving the Shortest Path Problem

The following problem is very classical in motion planning: Let a and b be two vertices of a polygon and P (Q, respectively) be the polyline formed by vertices of the polygon from a to b (from b to a, respectively) in counterclockwise order. We find the Euclidean shortest path in the polygon between a and b. In this paper, an efficient algorithm based on incremental convex hulls is presented. Under some assumption, the shortest path consists of some extreme vertices of the convex hulls of subpolylines of P (Q, respectively), first to start from a, advancing by vertices of P, then by vertices of Q, alternating until the vertex b is reached. Each such convex hull is delivered from the incremental convex hull algorithm for a subpolyline of P (Q, respectively) just before reaching Q (P, respectively). Unlike known algorithms, our algorithm does not rely upon triangulation and graph theory. The algorithm is implemented by a C code then is illustrated by some numerical examples. Therefore, incremental convex hull is an orientation to determine the shortest path. This approach provides a contribution to the solution of the open question raised by J. S. B. Mitchell in J. R. Sack and J. Urrutia, eds, Handbook of Computational Geometry, Elsevier Science B. V., 2000, p. 642. determining convex ropes in robotics (for determining convex hulls, respectively) were introduced in (6) and (7) ((8), respectively). These problems are variations of the shortest path problem and thus can be solved without resorting to a linear-time triangulation algorithm and without resorting to graph theory. Geometrically, we determine the shortest path connecting two points a and b that avoids the obstacles - polylines P and Q. Assume without loss of generality that a and b are the first and the final vertices of P and Q, respectively. In this paper, an O(|P||Q|) time algorithm for determining the shortest path, without resorting to a linear-time triangulation algorithm and without resorting to graph theory, is presented, using the method of incremental convex hull, where |P| (|Q|, respectively) is the number of vertices of P (Q, respectively). Under an assumption on links to P and Q, the shortest path consists of the extreme vertices of the convex hulls downward, first advancing on one convex hull formed by vertices of P including a, then on the other formed by vertices of Q, alternating until the vertex b is reached. Each such convex hull is delivered from the incremental convex hull algorithm for a subpolyline of P (Q, respectively) just before reaching Q (P, respectively). Therefore, incremental convex hull is an orientation to determine the shortest path. The algorithm is implemented by a C code and is illustrated by some numerical examples. This paper also provides a contribution to the solution of the Mitchells open question above.

Multiple shooting approach for computing approximately shortest paths on convex polytopes

Abstract In this paper, we use a multiple shooting approach in solving boundary value problems for ODE to introduce a novel iterative algorithm for computing an approximate shortest path between two points on the surface of a convex polytope in 3D. Namely, the polytope is partitioned into subpolytopes, shooting points and a Straightness condition are established. The algorithm specifies how to combine shortest paths between shooting points in subpolytopes to become the required shortest path by the Straightness condition. In particular, the algorithm does not rely on Steiner points and graph tools on the entire polytope. It is implemented in C++ and a comparison with Agarwal, Har-Peled, and Karia’s algorithm, on the accurate construction of the shortest path, is presented.

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.

Blaschke-Type Theorem and Separation of Disjoint Closed Geodesic Convex Sets

In this paper, we deal with analytic and geometrical properties of geodesic convex sets and geodesic paths. We show that Blaschke’s Theorem for convex sets is also true for geodesic convex sets and geodesic paths in a simple polygon. Some geometrical properties of geodesic triangles are presented. Furthermore, separation of geodesic convex sets is shown.

Piecewise Constant Roughly Convex Functions

AbstractThis paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: ρ-convexity (in the sense of Klötzler and Hartwig), δ-convexity and midpoint δ-convexity (in the sense of Hu, Klee, and Larman), γ-convexity and midpoint γ-convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions