Separating Hyperplane Theorem Research Articles

Trajectory Generation for Drones in Confined Spaces Using an Ellipsoid Model of the Body

We address the problem of motion planning for aerial drones to allow them to fly through confined spaces and narrow gaps between obstacles. Finding safe and feasible trajectories for steering multiple drones towards some desired positions in tight spaces or guiding them through gaps smaller in width than their diameters is impossible without taking the drones' orientations into account. To incorporate the orientation into the motion planning problem we employ an ellipsoid model of the drone body, and utilize the separating hyperplane theorem for convex sets to derive constraints that guarantee collision avoidance between the ellipsoid-shaped drone and obstacles modeled as ellipsoids, spheres, or polygons. The resulting set of constraints is seamlessly integrated into the motion planning method based on Bézier curve whose properties are exploited for efficient evaluation of the constraints. The efficacy of the proposed method in generating feasible and collision-free trajectories is demonstrated via different simulations involving single and multiple drones.

Afriat and arbitrage

This paper presents a proof of Afriat’s (Int Econ Rev 8:67–77) theorem on revealed preference by using the idea that a rational consumer should not be vulnerable to arbitrage. The main mathematical tool is the separating hyperplane theorem.

An algorithmic separating hyperplane theorem and its applications

We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets K,K′ of the Euclidean space intersect, and when they are disjoint. The theorem, referred as distance duality, is distinct from classical separation theorems. Next, utilizing the theorem we develop a substantially generalized and stronger version of the Triangle Algorithm, originally designed for the convex hull membership problem. If δ∗=d(K,K′), the Euclidean distance between the sets, ρ∗ the maximum of their diameters, and ε a prescribed tolerance, the Triangle Algorithm approximates δ∗, or induces a separating hyperplane, or approximates optimal supporting hyperplanes. Specifically, it computes (p,p′)∈K×K′ satisfying any of the following conditions desired:(1) d(p,p′)≤εd(p,v), v∈K, or d(p,p′)≤εd(p′,v′), v′∈K′ (when δ∗=0);(2) the orthogonal bisector of pp′ separates K from K′;(3) d(p,p′)−δ∗≤εd(p,p′) (when δ∗>0);(4) the supporting hyperplanes orthogonal to pp′ satisfy δ∗−d(H,H′)≤εd(p,p′).The corresponding number of iterations to solve these are, O(1∕ε2), O(ρ∗2∕δ∗2), and O(ρ∗2∕δ∗2ε) for the last two, all independent of K,K′. The complexity in each iteration of the first two tasks is computing for a given pair of iterates (p,p′)∈K×K′ a pivot, i.e. v∈K with d(p,v)≥d(p′,v), or v′∈K′ with d(p′,v′)≥d(p,v′). For the last two the complexity of each iteration is either computing a pivot, or supporting hyperplanes (H,H′) orthogonal to pp′. In the worst-case the complexity of each iteration is solving a linear program over K or K′.Special cases include when K and K′ are convex hulls of finite sets, or polytopes described as the intersection of halfspaces. The corresponding problems include, linear and quadratic programming, SVM and more. In separate work we describe computational comparison between Triangle Algorithm, Frank–Wolfe, Gilbert’s Algorithm, SMO, and more. Future work includes extensions to unbounded convex sets, non-Euclidean norms, also combinatorial and NP-complete problems.

A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle

We prove constructively that every uniformly continuous convex function f:X→R+ has positive infimum, where X is the convex hull of finitely many vectors. Using this result, we prove that a separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle are constructively equivalent. This is the first time that important theorems are classified into Markov's principle within constructive reverse mathematics.

Bounded real lemmas for positive descriptor systems

A well known result in the theory of linear positive systems is the existence of positive definite diagonal matrix (PDDM) solutions to some well known linear matrix inequalities (LMIs). In this paper, based on the positivity characterization, a novel bounded real lemma for continuous positive descriptor systems in terms of strict LMI is first established by the separating hyperplane theorem. The result developed here provides a necessary and sufficient condition for systems to possess H∞ norm less than γ and shows the existence of PDDM solution. Moreover, under certain condition, a simple model reduction method is introduced, which can preserve positivity, stability and H∞ norm of the original systems. An advantage of such method is that systems׳ matrices of the reduced order systems do not involve solving of LMIs conditions. Then, the obtained results are extended to discrete case. Finally, a numerical example is given to illustrate the effectiveness of the obtained results.

Prices and Efficient Assignments Without Transfers

We study the assignment of indivisible objects in environments, such as school choice, in which transfer payments are not used. Our main result shows that every efficient assignments can be decentralized through prices. We thus establish the Second Welfare Theorem for the no-transfer environments with possibly satiated agents. The result builds on a new Separating Hyperplane Theorem, which is of independent interest.

H∞Control for Linear Positive Discrete-Time Systems

This paper is concerned withH∞control for linear positive discrete-time systems. Positive systems are characterized by nonnegative restriction on systems’ variables. This restriction results in some remarkable results which are available only for linear positive systems. One of them is the celebrated diagonal positive definite matrix solutions to some existed well-known results for linear systems without nonnegative restriction. We provide an alternative proof for criterion ofH∞norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also considerH∞control problem for linear positive discrete-time systems via state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions.

A constructive version of Ky Fan's coincidence theorem

In this paper we examine Ky Fan's coincidence theorem from the point of view of Bishop style constructive mathematics using constructive versions of Kakutani's fixed point theorem, Berge's maximum theorem and the separating hyperplane theorem.

Generalized Probabilism: Dutch Books and Accuracy Domination

This paper explores De Finetti’s generalized versions of Dutch Book and Accuracy Domination theorems. Following proposals due to Jeff Paris, we construe these as underpinning a generalized probabilism appropriate to belief states against either a classical or a non-classical background. Both results are straightforward corollaries of the separating hyperplane theorem; their geometrical relationship is examined. It is shown that each point of Accuracy Domination for b induces a Dutch Book on b; but Dutch Books may need to be ‘scaled’ in order to find a point of Accuracy-Domination. Finally, diachronic Dutch Book defences of conditionalization are examined in the general setting. The formulation and limitations of the generalized conditionalization this delivers are examined.

On economic efficiency under non-convexity

This paper investigates economic efficiency under non-convexity. The analysis relies on a generalization of the separating hyperplane theorem under non-convexity. The concept of zero-maximality is used to characterize Pareto efficiency under non-convexity. We show the existence of a separating hypersurface that can be used to provide a dual characterization of efficient allocations. When the separating hypersurface is non-linear, this implies that non-linear pricing is an integral part of economic efficiency. Implications for the decentralization of economic decisions under non-convexity are discussed.

Ordinal Efficiency and the Polyhedral Separating Hyperplane Theorem

We study problems in which each of finitely many agents must be allocated a single object, based on the agents' rankings of pure outcomes. A random allocation is ordinally efficient if it is not ordinally dominated in the sense of there being another random assignment that gives each agent a first order stochastically dominant distribution of objects. We show that any ordinally efficient random assignment maximizes the sum of expected utilities for some vector of vNM utility functions that are consistent with the given ordinal preferences. One method of proof uses a new version of the separating hyperplane theorem for polyhedra. Journal of Economic Literature Classification Numbers: C78, D61.

GPM ANALYSIS IN COMBINING INTERIOR POINTS AND THE SIMPLEX METHOD

ABSTRACT By geometric viewpoint in search directions of linear programming problems (LP), there are two major approaches: the simplex method and the interior-point algorithm originated from Karmarkar's approach. Many of their variants developed both in theory and applications are still in progress. Roughly speaking, the main difference among them is that the simplex method is devoted for each the exact optimal solution while Karmarkar-based method is computationally fast in approaching to the neighborhood of the optimal solution, but it becomes slow near the optimal point. By the separating hyperplane theorem, we know that the optimal solution of a linear programming problem would locate at the boundary point. However, Karmarkar's algorithm is claimed as an interior-point approach which takes a solution trajectory path through the interior of the feasible region. On the other hand, these algorithms will coincide the zig-zag situation following the trajectory path before attaining the optimal point meanwhile it cannot achieve the optimal point. Therefore, the purpose of this paper is to study a possible approach by combining the interior point and the simplex method. By means of QR factorization, we give a general analysis on Gradient Projection Method(GPM) in solving linear programming problems. Moreover, we prove that a general assumption-technique constraint matrix A of full rank can be relaxed by using our approach. Finally, we analyze the complexity of this approach to ensure that its upper bound is O(n3).

Some topological minimax theorems

Two topological variants of the minimax theorem are proved with no restrictions on one of the spaces except for those related to the function under consideration. The conditions concerning the behavior of the function deal only with the interval between the maximin and minimax. As corollaries, we obtain the well-known theorems of Sion and Hoang-Tui on quasiconvex-quasiconcave semicontinuous functions. The scheme of arguments goes back to the Hahn-Banach theorem and the separating hyperplane theorem. It is shown how this scheme can be explicitly realized in the proof of the Hahn-Banach theorem.

The analysis of finite security markets using martingales

The theory of finite security markets developed by Harrison and Pliska [1] used the separating hyperplane theorem to establish the relationship between the lack of arbitrage opportunities and the existence of a certain martingale measure. In this paper we treat this theory by examining certain geometric properties of the sample paths of the price process, that is, we focus on the price increments of the stocks between one time period to the next and convert them to martingale differences through an equivalent change of measure. Thus, in contrast to Harrison and Pliska&s functional analytic derivation, our approach is based on probabilistic methods and allows a geometric interpretation which not only provides a connection to linear programming but also yields an algorithm for analyzing finite security markets. Moreover, we can make precise the connection between diverse expressions of economic equilibrium such as ‘absence of arbitrage’, ‘martingale property, and ‘complementary slackness property’.

The analysis of finite security markets using martingales

The theory of finite security markets developed by Harrison and Pliska [1] used the separating hyperplane theorem to establish the relationship between the lack of arbitrage opportunities and the existence of a certain martingale measure. In this paper we treat this theory by examining certain geometric properties of the sample paths of the price process, that is, we focus on the price increments of the stocks between one time period to the next and convert them to martingale differences through an equivalent change of measure. Thus, in contrast to Harrison and Pliska&s functional analytic derivation, our approach is based on probabilistic methods and allows a geometric interpretation which not only provides a connection to linear programming but also yields an algorithm for analyzing finite security markets. Moreover, we can make precise the connection between diverse expressions of economic equilibrium such as ‘absence of arbitrage’, ‘martingale property, and ‘complementary slackness property’.

An axiomatic derivation of subjective probability, utility, and evaluation functions

Subjective expected utility maximization is derived from four axioms, using an argument based on the separating hyperplane theorem. It is also shown that the first three of these axioms imply a more general maximization formula, involving an evaluation function, which can still serve as a basis for decision analysis.

Necessity and Existence Results on Constrained Optimization of Separable Functionals by a Multiplier Rule

Functionals E (extended real-valued) and C (vector-valued) are defined by $E(q) = \smallint _X e(x,q(x))d\mu x$ and $C(q) = \smallint _X c(x,q(x))d\mu x$, in a rather abstract setting, without differentiability assumptions. If, among other assumptions, either $\mu $ is a nonatomic measure or convexity conditions hold, then from Blackwell’s generalization of Lyapunov’s convexity theorem and a separating hyperplane theorem, it follows that for $q^ * $ to maximize E subject to an equality (inequality) constraint on C, it is necessary that there exist a vector $\lambda $ (nonnegative vector $\lambda $) such that $q^ * $ maximizes $E - \lambda \cdot C$. For the latter to hold, by von Neumann’s selection theorem under the principal condition that e and c are Borel functions, it is necessary that $q^ * (x)$ maximize $e(x, \cdot ) - \lambda \cdot c(x, \cdot )$ for a.e. $x \in X$. This much extends methods and results of Aumann and Perles. Existence results are derived from “upper closure” of the range of $(C,E)$, under boundedness assumptions, as the range of a vector integral over selections from a set-valued function. This development utilizes and extends results and methods of Olech and Blackwell. The main result asserts existence of optimal functions under the principal conditions that the set-valued function is upper closed, e and c are Borel functions, C is bounded, and either $\mu $ is nonatomic or E is bounded above. Examples show that the results cannot be strengthened in various ways.