Compact Convex Set Research Articles

On a colorful problem by Dol'nikov concerning translates of convex bodies

In this note we study a conjecture by Jerónimo-Castro, Magazinov and Soberón which generalized a question posed by Dol'nikov. Let F1,F2,…,Fn be families of translates of a convex compact set K in the plane so that each two sets from distinct families intersect. We show that, for some j, ⋃i≠jFi can be pierced by at most 4 points. To do so, we use previous ideas from Gomez-Navarro and Roldán-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square.

On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets

On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets

Computation of the Hausdorff Distance between Two Compact Convex Sets.

The Hausdorff distance between two closed sets has important theoretical and practical applications. Yet apart from finite point clouds, there appear to be no generic algorithms for computing this quantity. Because many infinite sets are defined by algebraic equalities and inequalities, this a huge gap. The current paper constructs Frank-Wolfe and projected gradient ascent algorithms for computing the Hausdorff distance between two compact convex sets. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, we investigate a homotopy method that gradually deforms two balls into the two target sets. The Frank-Wolfe and projected gradient algorithms are tested on two pairs of compact convex sets, where: (1) is the box translated by and is the intersection of the unit ball and the non-negative orthant; and (2) is the probability simplex and is the unit ball translated by . For problem (2), we find the Hausdorff distance analytically. Projected gradient ascent is more reliable than the Frank-Wolfe algorithm and finds the exact solution of problem (2). Homotopy improves the performance of both algorithms when the exact solution is unknown or unattained.

Adaptive controller design and disturbance attenuation for minimum phase multiple‐input multiple‐output linear systems with noisy output measurements and with measured disturbances

SummaryIn this paper, we present a systematic procedure for robust adaptive control design for minimum phase uncertain multiple‐input multiple‐output linear systems that are right invertible and can be dynamically extended to a linear system with vector relative degree using a dynamic compensator that is known. For this class of systems, it is always possible to dynamically extend them, and/or integrate a select set of output channels, and/or padding dummy state variables to arrive at a system model that admits uniform vector relative degree and uniform observability indices that is further minimum phase according to Reference 13. We assume that the uniform vector relative degree is known and an upper bound for the uniform observability indices is known. We also assume that the unknown parameter vector lies in a convex compact set such that the high frequency gain matrix remains invertible for any parameter vector value in the set. These are the assumptions that allow for a successful design of a robust adaptive controller. A numerical example is included to fully illustrate the controller design and the effectiveness of the controller.

Optimal Learning for Structured Bandits

We study structured multiarmed bandits, which is the problem of online decision-making under uncertainty in the presence of structural information. In this problem, the decision-maker needs to discover the best course of action despite observing only uncertain rewards over time. The decision-maker is aware of certain convex structural information regarding the reward distributions; that is, the decision-maker knows that the reward distributions of the arms belong to a convex compact set. In the presence of such structural information, the decision-maker then would like to minimize his or her regret by exploiting this information, where the regret is its performance difference against a benchmark policy that knows the best action ahead of time. In the absence of structural information, the classical upper confidence bound (UCB) and Thomson sampling algorithms are well known to suffer minimal regret. However, as recently pointed out by Russo and Van Roy (2018) and Lattimore and Szepesvari (2017) , neither algorithm is capable of exploiting structural information that is commonly available in practice. We propose a novel learning algorithm that we call “DUSA,” whose regret matches the information-theoretic regret lower bound up to a constant factor and can handle a wide range of structural information. Our algorithm DUSA solves a dual counterpart of the regret lower bound at the empirical reward distribution and follows its suggested play. We show that this idea leads to the first computationally viable learning policy with asymptotic minimal regret for various structural information, including well-known structured bandits such as linear, Lipschitz, and convex bandits and novel structured bandits that have not been studied in the literature because of the lack of a unified and flexible framework. Funding: N. Golrezaei was supported in part by the Young Investigator Program (YIP) Award from the Office of Naval Research (ONR) [Grant N00014-21-1-2776] and the MIT Research Support Award. This paper was accepted by Chung Piaw Teo, Optimization

Linear Recurrent Equations in the Space of Convex Compact Sets and the Diameters of Their Solutions

Linear Recurrent Equations in the Space of Convex Compact Sets and the Diameters of Their Solutions

Conditional Gradient Method for Optimization Problems with a Constraint in the Form of the Intersection of a Convex Smooth Surface and a Convex Compact Set

The conditional gradient method is generalized to nonconvex sets of constraints representing the set-theoretic intersection of a convex smooth surface and a convex compact set. Necessary optimality conditions are studied, and the convergence of the method is analyzed.

Conditional Gradient Method for Optimization Problems with a Constraint in the Form of the Intersection of a Convex Smooth Surface and a Convex Compact Set

Conditional Gradient Method for Optimization Problems with a Constraint in the Form of the Intersection of a Convex Smooth Surface and a Convex Compact Set

Compact Hölder retractions and nearest point maps

In this paper, an α-Hölder retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is constructed for every positive α<1. This constitutes a positive solution to a Hölder version of a question raised by Godefroy and Ozawa. In fact, compact convex sets are found to be absolute α-Hölder retracts under certain assumption of flatness.

Sufficient Conditions for the Linear Convergence of an Algorithm for Finding the Metric Projection of a Point onto a Convex Compact Set

Sufficient Conditions for the Linear Convergence of an Algorithm for Finding the Metric Projection of a Point onto a Convex Compact Set

Solving Optimization Problems with Blackwell Approachability

In this paper, we propose a new algorithm for solving convex-concave saddle-point problems using regret minimization in the repeated game framework. To do so, we introduce the Conic Blackwell Algorithm+ ([Formula: see text]), a new parameter- and scale-free regret minimizer for general convex compact sets. [Formula: see text] is based on Blackwell approachability and attains [Formula: see text] regret. We show how to efficiently instantiate [Formula: see text] for many decision sets of interest, including the simplex, [Formula: see text] norm balls, and ellipsoidal confidence regions in the simplex. Based on [Formula: see text], we introduce [Formula: see text], a new parameter-free algorithm for solving convex-concave saddle-point problems achieving a [Formula: see text] ergodic convergence rate. In our simulations, we demonstrate the wide applicability of [Formula: see text] on several standard saddle-point problems from the optimization and operations research literature, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, [Formula: see text] achieves state-of-the-art numerical performance and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters. Funding: J. Grand-Clément is supported by the Agence Nationale de la Recherche [Grant 11-LABX-0047] and by Hi! Paris. C. Kroer is supported by the Office of Naval Research [Grant N00014-22-1-2530] and by the National Science Foundation [Grant IIS-2147361].

Quasi-Solution Method and Global Minimization of the Residual Functional in Conditionally Well-Posed Inverse Problems

A class of conditionally well-posed problems characterized by a Hölder conditional stability estimate on a convex compact set in a Hilbert space is considered. The operator of the direct problem and the right-hand side of the equation are given with errors, and the derivatives of the exact and perturbed operators are not assumed to be close to each other. The convexity and single-extremality of the residual functional of the quasi-solution method are examined. For this functional, each of its stationary points on the set of conditional well-posedness that lies not too far from the sought solution of the original inverse problem is shown to belong to a small neighborhood of the solution. The diameter of this neighborhood is estimated in terms of the errors in the input data. It is shown that this neighborhood is an attractor of the iterations of the gradient projection method, and the convergence rate of the iterations to the attractor is estimated. The necessity of the used conditional stability estimate for the existence of iterative processes with the indicated properties is established.

Optimization of the Reachable Set of a Linear System with Respect to Another Set

Given a linear controlled autonomous system, we consider the problem of including a convex compact set in the reachable set of the system in the minimum time and the problem of determining the maximum time when the reachable set can be included in a convex compact set. Additionally, the initial point and the time at which the extreme time is achieved in each problem are determined. Each problem is discretized on a grid of unit vectors and is then reduced to a linear programming problem to find an approximate solution of the original problem. Additionally, error estimates for the solution are found. The problems are united by a common ideology going back to the problem of finding the Chebyshev center.

New conditions on finite-time stability of linear discrete-time system

This paper is concerned with the problems of set-based finite-time stability (SFTS) and set-based finite-time boundedness (SFTB) for both certain and uncertain linear time-varying systems. The concepts of SFTS and SFTB are defined. Different from existing results, sufficient conditions for SFTS and SFTB are directly derived from the basic definitions of finite-time stability (FTS) and finite-time boundedness (FTB) by using the convex hull technique rather than utilizing the weighted quadratic functions. Thus, more practical constraints on the system states can be dealt with. Furthermore, intervals, zonotopes and polytopes are employed to describe the typical compact convex sets. For linear uncertain systems, the uncertain time-varying state sets are assumed to be represented by interval matrices and matrix zonotopes, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the main results.

On a group pursuit problem on time scales

In a finite-dimensional Euclidean space $\mathbb R^k$, we consider a linear problem of pursuit of one evader by a group of pursuers, which is described on the given time scale $\mathbb{T}$ by equations of the form \begin{gather*} z_i^{\Delta} = a z_i + u_i - v, \end{gather*} where $z_i^{\Delta}$ is the $\Delta$-derivative of the functions $z_i$ on the time scale $\mathbb{T}$, $a$ is an arbitrary number not equal to zero. The set of admissible controls for each participant is a unit ball centered at the origin, the terminal sets are given convex compact sets in $\mathbb R^k$. The pursuers act according to the counter-strategies based on the information about the initial positions and the evader control history. In terms of initial positions and game parameters, a sufficient capture condition has been obtained. For the case of setting the time scale in the form $\mathbb T = \{ \tau k \mid k \in \mathbb Z,\ \tau \in \mathbb R,\ \tau &gt;0\}$ sufficient pursuit and evasion problems solvability conditions have been found. In the study, in both cases, the resolving function method is used as basic one.

Extreme states and boundary representations of operator spaces in ternary rings of operators

This paper explores different notions of extreme states on operator systems and on operator spaces. We prove that boundary points of the generalized state space of an operator system in finite dimensions are precisely the restrictions of boundary representations of the generated [Formula: see text]-algebra. We study weak TRO-extreme states on operator spaces in ternary rings of operators as a non self-adjoint version of [Formula: see text]-extreme states on operator systems in [Formula: see text]-algebras. Rectangular boundary point for a rectangular matrix convex set is introduced and its connection with boundary representations of the associated operator space is established in finite dimensional setting. Also, we show that the set of all rectangular boundary points is the minimal spanning set for a compact rectangular matrix convex set in finite dimensions.

On distal flows and common fixed point theorems in Banach spaces

A semigroup S consisting of self-mappings defined on a set B of a Banach space X is said to be a distal flow on B provided0∉{Sx−Sy:S∈S}‾,for allx≠y∈B. In this paper, we show that if T is a nonexpansive self-mapping defined on a nonempty compact convex set C of X so that S≡{Tn:n∈N} forms a distal flow, then T:C→C is a surjective isometry; in particular, if C is of finite dimension, then T is a surjective affine isometry. Consequently, i) every distal S of nonexpansive mappings on a convex compact set C consists of surjective self-isometries on C; ii) all S in the distal S have a common fixed point in the Chebyshev center of C; iii) if, in addition, C is finite dimensional, then every S in S is a surjective affine self-isometry on C. Therefore, it can also be understood as a refinement of a recent common fixed point theorem due to Wiśnicki. We also show that the conclusion i) is not true if C is not compact. Finally, we give a localized setting of Lim-Lin's common fixed point theorem for isometries.

The application of a universal separating vector lemma to optimal sampled-data control problems with nonsmooth Mayer cost function

&lt;p style='text-indent:20px;'&gt;In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called &lt;i&gt;universal separating vector lemma&lt;/i&gt; is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.&lt;/p&gt;

Приведение возмущенного движения точки на базовую траекторию при наличии геометрических ограничений на дополнительные управления

The deriving the perturbed material point motion in a small neighborhood of the base trajectory problem is solved in this paper. Assumed that geometric constraints in the form of convex compact sets are imposed on the additional controls. In particular, the cases of spheres, ellipsoids and rectangular parallelepipeds were considered. As a result of the constructed control, all the requirements for the kinematics of the point flight were satisfied: the point hitting prohibition with given regions and the necessity of its location in the given vertical strip above the ground surface. The numerical experiment confirmed the effectiveness of the proposed algorithm.

Continuous global optimization of multivariable functions based on Sergeev and Kvasov diagonal approach

Abstract. One of modern global optimization algorithms is method of Strongin and Piyavskii modified by Sergeev and Kvasov diagonal approach. In recent paper we propose an extension of this approach to continuous multivariable functions defined on the multidimensional parallelepiped. It is known that Sergeev and Kvasov method applies only to a Lipschitz continuous function though it effectively extends one-dimensional algorithm to multidimensional case. So authors modify We modify mentioned method to a continuous functions using introduced by Vanderbei ε-Lipschitz property that generalizes conventional Lipschitz inequality. Vanderbei proved that a real valued function is uniformly continuous on a convex domain if and only if it is ε-Lipschitz. Because multidimensional parallelepiped is a convex compact set, we demand objective function to be only continuous on a search domain. We describe extended Strongin’s and Piyavskii’s methods in the Sergeev and Kvasov modification and prove the sufficient conditions for the convergence. As an example of proposed method’s application, at the end of this article we show numerical optimization results of different continuous but not Lipschitz functions using three known partition strategies: “partition on 2”, “partition on 2N” and “effective”. For the first two of them we present formulas for computing a new iteration point and for recalculating the ε-Lipschitz constant estimate. We also show algorithm modification that allows to find a new search point on any algorithm’s step.