Finite Union Of Convex Sets Research Articles

Adaptive risk bounds in unimodal regression

We study the statistical properties of the least squares estimator in unimodal sequence estimation. Although closely related to isotonic regression, unimodal regression has not been as extensively studied. We show that the unimodal least squares estimator is adaptive in the sense that the risk scales as a function of the number of values in the true underlying sequence. Such adaptivity properties have been shown for isotonic regression by Chatterjee et al. (Ann. Statist. 43 (2015) 1774–1800) and Bellec (Sharp oracle inequalities for Least Squares estimators in shape restricted regression (2016)). A technical complication in unimodal regression is the non-convexity of the underlying parameter space. We develop a general variational representation of the risk that holds whenever the parameter space can be expressed as a finite union of convex sets, using techniques that may be of interest in other settings.

Matrices commuting with a given normal tropical matrix

Consider the space Mnnor of square normal matrices X=(xij) over R∪{−∞}, i.e., −∞≤xij≤0 and xii=0. Endow Mnnor with the tropical sum ⊕ and multiplication ⊙. Fix a real matrix A∈Mnnor and consider the set Ω(A) of matrices in Mnnor which commute with A. We prove that Ω(A) is a finite union of alcoved polytopes; in particular, Ω(A) is a finite union of convex sets. The set ΩA(A) of X such that A⊙X=X⊙A=A is also a finite union of alcoved polytopes. The same is true for the set Ω′(A) of X such that A⊙X=X⊙A=X.A topology is given to Mnnor. Then, the set ΩA(A) is a neighborhood of the identity matrix I. If A is strictly normal, then Ω′(A) is a neighborhood of the zero matrix. In one case, Ω(A) is a neighborhood of A. We give an upper bound for the dimension of Ω′(A). We explore the relationship between the polyhedral complexes spanA, spanX and span(AX), when A and X commute. Two matrices, denoted A̲ and A¯, arise from A, in connection with Ω(A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.

On the local convergence of the Douglas–Rachford algorithm

We discuss the Douglas-Rachford algorithm to solve the feasibility problem for two closed sets $A,B$ in $\mathbb{R}^d$. We prove its local convergence to a fixed point when $A,B$ are finite unions of convex sets. We also show that for more general nonconvex sets the scheme may fail to converge and start to cycle, and may then even fail to solve the feasibility problem.

Simple Wavelet Sets in $\mathbb{R}^{n}$

Wavelet sets that are finite unions of convex sets are constructed in $\mathbb{R}^{n}$ , n≥2, for dilation by any expansive matrix that has a power equal to a scalar times the identity and also has all singular values greater than $\sqrt{n}$ . In particular, we produce simple wavelet sets in every dimension for dilation by any real scalar greater than 1.

STOCHASTIC DOMINANCE: CONVEXITY AND SOME EFFICIENCY TESTS

This paper points out the importance of Stochastic Dominance (SD) efficient sets being convex. We review classic convexity and efficient set characterization results on SD efficiency of a given portfolio relative to a diversified set of assets and generalize them in the following aspects. First, we propose a linear programming SSD test that is more efficient than that of Post (2003). Secondly, we expand the SSD efficiency criteria developed by Dybvig and Ross (1982) onto the Third Order Stochastic Dominance and further to Decreasing Absolute and Increasing Relative Risk Aversion Stochastic Dominance. The efficient sets for those are finite unions of convex sets.

Simple Wavelet Sets for Matrix Dilations in ℝ2

A partial answer is given to the question of which expansive integer matrix dilations in ℝ2 have wavelet sets that are finite unions of convex sets. New results are given supporting a conjecture that among matrices with determinant greater than 2, it is exactly matrices that have a power equal to a scalar that have such wavelet sets.

Finite Sets as Complements of Finite Unions of Convex Sets

Suppose S⊆ℝ d is a set of (finite) cardinality n, whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem of Björner and Kalai yields a lower bound. Much cruder estimates are then obtained when the openness restriction is dropped. For a given set S the problem of determining the smallest number of convex sets whose union is ℝ d ∖S is shown to be equivalent to the problem of finding the chromatic number of a certain (infinite) hypergraph ℋ S . We consider the graph $\mathcal {G}_{S}$ whose edges are the 2-element edges of ℋ S , and we show that, when d=2, for any sufficiently large set S, the chromatic number of $\mathcal{G}_{S}$ will be large, even though there exist arbitrarily large finite sets S for which $\mathcal{G}_{S}$ does not contain large cliques.

Quasiconvex Minimization on a Locally Finite Union of Convex Sets

Extending the approach initiated in Aussel and Hadjisavvas (SIAM J. Optim. 16:358–367, 2005) and Aussel and Ye (Optimization 55:433–457, 2006), we obtain the existence of a local minimizer of a quasiconvex function on the locally finite union of closed convex subsets of a Banach space. We apply the existence result to some difficult nonconvex optimization problems such as the disjunctive programming problem and the bilevel programming problem.

Dual Characterizations of Set Containments with Strict Convex Inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.

Reasoning About Space: The Modal Way

We investigate the topological interpretation of modal logic in modern terms, using a new notion of bisimulation. We look at modal logics with interesting topological content, presenting, among others, a new proof of McKinsey and Tarski's theorem on completeness of S4 with respect to the real line, and a completeness proof for the logic of finite unions of convex sets of reals. We conclude with a broader picture of extended modal languages of space, for which the main logical questions are still wide open.

Expansion of a model of a weakly o-minimal theory by a family of unary predicates

AbstractA subsetA⊆Mof a totally ordered structureMis said to beconvex, if for anya, b∈A: [a<b→ ∀t(a<tb→t∈A)]. A complete theory of first order isweakly o-minimal(M. Dickmann [D]) if any modelMis totally ordered by some ∅-definable formula and any subset ofMwhich is definable with parameters fromMis a finite union of convex sets. We prove here that for any modelMof a weakly o-minimal theoryT. any expansionM+ofMby a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63). that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.

A-Convexity

Let S be a subset of R n , and let 0⩽ a⩽1. We call S a-convex if x, y∈S imply a x+(1−a) y∈S . It is incorrectly stated in (Rosenfeld, A., Kak, A.C., 1976. Digital Picture Processing, Academic Press, New York, p. 389; Rosenfeld, A., Kak, A.C., 1982. Digital Picture Processing, second ed. Vol. 2, p. 268) that 1/2-convexity implies convexity. We show that this implication is in fact valid for closed sets and for finite unions of convex sets. We also define fuzzy a-convexity, and show that it implies fuzzy convexity for fuzzy sets whose membership functions are continuous or piecewise constant.

Hunting Voronoi vertices

Given three sites in the plane, a Voronoi vertex is a point that is equidistant from each. In this paper, we consider the problem of computing Voronoi vertices for three disjoint planar sites of fixed but possibly unknown shape; we only require the ability to query the closest point on an object from a given point. Each site is assumed to be either convex or a finite union of convex sets. Our technique is simple and iterative in nature: beginning from some initial seed point, it computes a sequence of points based on intermediate closest-point queries. This technique is observed to either converge to a Voronoi vertex or oscillate with some finite period. We study geometric conditions on shape/placement of the objects and choice of the initial point that guarantee convergence or oscillation. We show that our technique is probabilistically complete: selecting seed points at random will guarantee convergence to a finite Voronoi vertex, if one exists. Our motivation for seeking Voronoi vertices comes from robot motion planning: Voronoi vertices are natural havens for mobile robots avoiding obstacles. We conclude by briefly describing the implementation of a retraction-like path planner for a planar robot based on our iterative strategy for seeking Voronoi vertices.

On the Euler characteristic of finite unions of convex sets

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.

A note on non-parametric tests of consumer behaviour

This note extends the Afriat-Diewert-Varian conditions for consistency of demand data with the maximization hypothesis to the case where the choice set is a finite union of convex sets. Applications include models with discrete goods, regressive taxes, decreasing block prices and fixed costs.

A Decomposition for Sets Having a Segment Convexity Property

Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments.

The Princess and Monster Differential Game

In this game of R. Isaacs, the players are a hider and a searcher. The payoff is the time until capture. The hider is assumed to be mobile within a bounded set $\mathcal{D}$. The searcher has an arbitrarily small detection radius. There is incomplete information in that neither player knows the present or past location of the other. For various sets $\mathcal{D}$, the value of the game is demonstrated by the presentation of $\varepsilon $-optimal strategies. Specifically, the game is solved in case $\mathcal{D}$ is convex or is the finite union of convex sets in $\mathbb{R}^n $ with $n \geqq 2$. Also the game is discussed in the case where $\mathcal{D}$ is a network. The results settle two conjectures of S. Gal.

A better decomposition theorem for simply connected m-convex sets

A set S in R d is said to be m-convex, m ⩾ 2, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. For S a closed m-convex set in R 2, various decomposition theorems have been obtained to express S as a finite union of convex sets. However, the previous bounds may be lowered further, and we have the following result: In case S is simply connected, then S is a union of σ( m) or fewer convex sets, where σ(m) = [(m − N)(m − 3 2 ) + 3 2 ] . Moreover, this result induces an improved decomposition in the general case as well.

Points of Local Nonconvexity and Finite Unions of Convex Sets

Let 5 be a subset of Rd. A point x in 5 is a point of local convexity of S if and only if there is some neighborhood U of x such that, if y, z ϵ 5 ⌒ U, then [y, z] ⊆ S. If S fails to be locally convex at some point q in 5, then q is called a point of local nonconvexity (lnc point) of S.Several interesting properties are known about sets whose lnc points Q may be decomposed into n convex sets. For S closed, connected, S ∼ Q connected, and Q having cardinality n, Guay and Kay [2] have proved that S is expressible as a union of n + 1 or fewer closed convex sets (and their result is valid in a locally convex topological vector space).

Finite Unions of Convex Sets

In this paper it is shown that a set is the union of k convex subsets if and only if every finite subset of it is contained in some k convex subsets of it. This is a characterization of a set as the union of a finite number of convex sets by conditions on its finite subsets. Also, a proof of McKinney’s theorem for unions of two convex sets is given using similar methods.