Union Of Convex Sets Research Articles

A NOTE ON VISIBLE SETS

In this paper, we investigate some properties of visible sets and its characterization. Also we obtained relation between visible set and union of convex sets with nonempty intersection.

Per-RMAP: Feasibility-Seeking and Superiorization Methods for Floorplanning with I/O Assignment.

The feasibility-seeking approach provides a systematic scheme to manage and solve complex constraints for continuous problems, and we explore it for the floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be formulated as the union of convex sets. However, the convergence of conventional projection-based algorithms is not guaranteed when the constraints sets are non-convex, which is the case with unions of convex sets. In this work, we propose a resetting strategy to greatly eliminate the divergence issue of the projection-based algorithm for the feasibility-seeking formulation. Furthermore, the superiorization methodology (SM), which lies between feasibility-seeking and constrained optimization, is firstly applied to floorplanning. The SM uses perturbations to steer the iterates of a feasibility-seeking algorithm to a feasible solution with shorter total wirelength. The proposed algorithmic flow is extendable to tackle various constraints and variants of floorplanning problems, e.g., floorplanning with I/O assignment problems. We have evaluated the proposed algorithm on the MCNC benchmarks. We can obtain legal floorplans only two times slower than the branch-and-bound method in its current prototype using MATLAB, with only 3% wirelength inferior to the optimal results. We evaluate the effectiveness of the algorithmic flow by considering the constraints of I/O assignment, and our algorithm achieves 8% improvement on wirelength.

A Technique for Constrained and Quantized Control of Nonlinear Systems Using Second-Order Cone Programming

Abstract This paper presents a novel technique for control of systems with bounded nonlinearity, convex state constraints, and control constraints. The technique is particularly useful for problems whose control constraints may be written as convex sets or the union of convex sets. The problem is reduced to finding bounding solutions associated with linear systems, and it is shown that this can be done with efficient second-order cone program (SOCP) solvers. The nonlinear control may then be interpolated from the bounding solutions. Three engineering problems are solved. These are the Van der Pol oscillator with bounded control and with quantized control, a pendulum driven by a DC motor with bounded voltage control, and a lane change maneuver with bounded rotational control acceleration. For each problem, the resulting second-order cone program solves in approximately 0.1 s or less. It is concluded that the technique provides an efficient means of solving certain control problems with control constraints.

Optimization-Based Collision Avoidance

This article presents a novel method for exactly reformulating nondifferentiable collision avoidance constraints into smooth, differentiable constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid obstacles while moving in an n-dimensional space. The proposed reformulation is exact, does not introduce any approximations, and applies to general obstacles and controlled objects that can be represented as the union of convex sets. We connect our results with the notion of signed distance, which is widely used in traditional trajectory generation algorithms. Our method can be applied to generic navigation and trajectory planning tasks, and the smoothness property allows the use of general-purpose gradient- and Hessian-based optimization algorithms. Finally, in case a collision cannot be avoided, our framework allows us to find “least-intrusive” trajectories, measured in terms of penetration. We demonstrate the efficacy of our framework on an automated parking problem, where our numerical experiments suggest that the proposed method is robust and enables real-time optimization-based trajectory planning in tight environments. Sample code of our example is provided at https://github.com/XiaojingGeorgeZhang/OBCA.

Near-optimal recovery of linear and N-convex functions on unions of convex sets

Abstract In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, ‘$N$-convex functionals’ (an example being the maximum of several fractional-linear functions) of unknown ‘signal’ from indirect noisy observations, the signal assumed to belong to the union of finitely many given convex compact sets. Our main assumption is that the observation scheme in question is good in the sense of Goldenshluger et al. (2015, Electron. J. Stat., 9, 1645–1712), the simplest example being the Gaussian scheme, where the observation is the sum of linear image of the signal and the standard Gaussian noise. The proposed estimates, same as upper bounds on their worst-case risks, stem from solutions to explicit convex optimization problems, making the estimates ‘computation-friendly’.

Marking Observer in Labeled Petri Nets With Application to Supervisory Control

In this paper, we consider the problem of marking estimation in labeled Petri nets whose initial marking is known to belong to a given convex set, in the presence of silent transitions (i.e., transitions labeled with the empty word) and indistinguishable transitions (i.e., transitions sharing the same label with other transitions). First, we demonstrate that all sets of markings consistent with a given sequence of observations can be described in linear algebraic terms (as a union of convex sets); subsequently, this observation is used to construct (offline) a marking observer under appropriate boundedness assumptions. Using the marking observer we show how to derive, at design time, a state feedback control law under the assumption that all transitions sharing a label can be enabled or disabled simultaneously as a group; this way, the most burdensome part of the computations is performed offline.

Carathéodory-type results for the sums and unions of convex sets

Carathéodory-type results for the sums and unions of convex sets

Restricted Normal Cones and the Method of Alternating Projections: Applications

The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving normal cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted normal cone, which is a generalization of the classical Mordukhovich normal cone. Numerous examples are provided to illustrate the theory.

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.

Bias expansion of spatial statistics and approximation of differenced lattice point counts

Investigations of spatial statistics, computed from lattice data in the plane, can lead to a special lattice point counting problem. The statistical goal is to expand the asymptotic expectation or large-sample bias of certain spatial covariance estimators, where this bias typically depends on the shape of a spatial sampling region. In particular, such bias expansions often require approximating a difference between two lattice point counts, where the counts correspond to a set of increasing domain (i.e., the sampling region) and an intersection of this set with a vector translate of itself. Non-trivially, the approximation error needs to be of smaller order than the spatial region’s perimeter length. For all convex regions in 2-dimensional Euclidean space and certain unions of convex sets, we show that a difference in areas can approximate a difference in lattice point counts to this required accuracy, even though area can poorly measure the lattice point count of any single set involved in the difference. When investigating large-sample properties of spatial estimators, this approximation result facilitates direct calculation of limiting bias, because, unlike counts, differences in areas are often tractable to compute even with non-rectangular regions. We illustrate the counting approximations with two statistical examples.

Reachability analysis of linear systems using support functions

This work is concerned with the algorithmic reachability analysis of continuous-time linear systems with constrained initial states and inputs. We propose an approach for computing an over-approximation of the set of states reachable on a bounded time interval. The main contribution over previous works is that it allows us to consider systems whose sets of initial states and inputs are given by arbitrary compact convex sets represented by their support functions. We actually compute two over-approximations of the reachable set. The first one is given by the union of convex sets with computable support functions. As the representation of convex sets by their support function is not suitable for some tasks, we derive from this first over-approximation a second one given by the union of polyhedrons. The overall computational complexity of our approach is comparable to the complexity of the most competitive available specialized algorithms for reachability analysis of linear systems using zonotopes or ellipsoids. The effectiveness of our approach is demonstrated on several examples.

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.

Convex rank 1 subsets of Euclidean buildings (of type A 2)

For a Euclidean building X of type A2, we classify the 0-dimensional subbuildings A of ∂T X that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of A) is (essentially) sufficient. To prove this, we construct new convex subsets as the union of convex sets.

Martin boundary points of a John domain and unions of convex sets

We show that a John domain has finitely many minimal Martin boundary points at each Euclidean boundary point. The number of minimal Martin boundary points is estimated in terms of the John constant. In particular, if the John constant is bigger than $\sqrt3/2$ , then there are at most two minimal Martin boundary points at each Euclidean boundary point. For a class of John domains represented as the union of convex sets we give a sufficient condition for the Martin boundary and the Euclidean boundary to coincide.

Optimal Inequalities in Probability Theory: A Convex Optimization Approach

We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for $P(X\in S)$, for a set S defined by polynomial inequalities and a random vector X defined on $\Omega\subseteq {\cal R}^n$ that has a given collection of up to kth-order moments. In the univariate case, we provide optimal bounds on $P(X\in S)$, when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and $\Omega$ are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NP-hard to find tight bounds for $k\geq 4$ and $\Omega={\cal R}^n$ and for $k\geq 2$ and $\Omega={\cal R}^n_+,$ when the data in the problem is rational. For k = 1 and $\Omega={\cal R}^n_+$ we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and $\Omega$ are unions of convex sets, over which linear functions can be optimized efficiently. For the case k = 2 and $\Omega={\cal R}^n$, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.

Synthesis of H∞ PID controllers: A parametric approach

This paper considers the problem of synthesizing proportional-integral-derivative (PID) controllers for which the closed-loop system is internally stable and the H ∞-norm of a related transfer function is less than a prescribed level for a given single-input single-output plant. It is shown that the problem to be solved can be translated into simultaneous stabilization of the closed-loop characteristic polynomial and a family of complex polynomials. It calls for a generalization of the Hermite–Biehler theorem applicable to complex polynomials. It is shown that the earlier PID stabilization results are a special case of the results developed here. Then a linear programming characterization of all admissible H ∞ PID controllers for a given plant is obtained. This characterization besides being computationally efficient reveals important structural properties of H ∞ PID controllers. For example, it is shown that for a fixed proportional gain, the set of admissible integral and derivative gains lie in a union of convex sets.

Convex decompositions in the plane and continuous pair colorings of the irrationals

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ2 which is not presentable as a countable union of convex sets satisfies the following dichotomy: (1) There is a perfect nonemptyP⊆S so that |C∩P|<3 for every convexC⊆S. In this case coveringS by convex subsets ofS is equivalent to coveringP by finite subsets, hence no nontrivial convex covers ofS can exist. (2) There exists a continuous pair coloringf: [N]2→{0, 1} of the spaceN of irrational numbers so that coveringS by convex subsets is equivalent to coveringN byf-monochromatic sets. In this case it is consistent thatS has a convex cover of cardinality strictly smaller than the continuumc in some forcing extension of the universe.

Extension of smooth functions in infinite dimensions, I: unions of convex sets

Extension of smooth functions in infinite dimensions, I: unions of convex sets

Estimates of the solution set of nonlinear equations with uncertain parameters

We consider the problem of determining the entire solution set of a system of nonlinear equations with interval coefficients. A decomposition method is proposed which produces an upper estimate of the sought solution as the union of convex sets. A numerical simulation example is described.

Sets in a euclidean space which are not a countable union of convex subsets

We prove that if a closed planar setS is not a countable union of convex subsets, then exactly one of the following holds: We show that an analogous theorem is impossible for dimension greater than 2. We give an example of a compact planar set with countable degree of visual independence which is not a countable union of convex subsets, and give a combinatorial criterion for a closed set inR d not to be a countable union of convex sets. We also prove a conjecture of G. Kalai, namely, that a closed planar set with the property that each of its visually independent subsets has at most one accumulation point, is a countable union of convex sets. We also give examples of sets which possess a (small) finite degree of visual independence which are not a countable union of convex subsets.