Tverberg's theorem with constraints

The topological Tverberg theorem claims that for any continuous map of the (q−1)(d+1)-simplex σ(d+1)(q−1) to Rd there are q disjoint faces of σ(d+1)(q−1) such that their images have a non-empty intersection. This has been proved for affine maps, and if q is a prime power, but not in general.We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem.The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for d=2 and q=3.     

A lagrange multiplier theorem for control problems with state constraints

We prove an existence theorem of Lagrange multipliers for an abstract control problem in Banach spaces. This theorem may be applied to obtain optimality conditions for control problems governed by partial differential equations in the presence of pointwise state constraints.     

Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect

We consider the problem of minimization of an integral functional with nonconvex with respect to the control integrand. We minimize our functional over the solution set of a control system described by two ordinary differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values. The coefficients of the equations and the constraint depend on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input–output relation of hysteresis type.     

Shape evolutions under state constraints: A viability theorem

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite-dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space).In this morphological framework, the evolution of compact subsets of is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis.Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order)—correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space.We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in , but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set .     

Regularity theorem for a one-sided problem with convex constraints on the gradient of the solution

One considers a one-sided problem for a second-order linear elliptic operator, according to the conditions of which the gradient of the solution at each point x must belong to a given strictly convex set K(x). Under certain conditions ensuring the solvability of the problem in the class one proves that the first-order derivatives of the solution are locally Lipschitz continuous.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 166–171, 1984.     

Mixed constraints in optimal control: an implicit function theorem approach

^** Email: mrpinho{at}fe.up.pt^*** Corresponding author. Email: jfrl{at}servidor.unam.mx This paper concerns a derivation of second-order necessary conditionsfor a fixed-endpoint control problem of Lagrange involving mixedequality and/or inequality constraints, posed over piecewisecontinuous controls. These conditions are obtained in a clearand transparent way by reducing the original problem, throughan implicit function theorem approach, to an unconstrained controlproblem.     

Subtour elimination constraints imply a matrix-tree theorem SDP constraint for the TSP

De Klerk et al., (2008) give a semidefinite programming constraint for the Traveling Salesman Problem (TSP) based on the matrix-tree theorem. This constraint says that the aggregate weight of all spanning trees in a solution to a TSP relaxation is at least that of a cycle graph. In this note, we show that the semidefinite constraint holds for any weighted 2-edge-connected graph and, in particular, is implied by the subtour elimination constraints.     

Denjoy's theorem with exponents

If is the (unique) minimal set for a diffeomorphism of the circle without periodic orbits, , then the upper box dimension of is at least . The method of proof is to introduce the exponent into the proof of Denjoy's theorem.

     

Decidable fan theorem and uniform continuity theorem with continuous moduli

The uniform continuity theorem $(\mathsf{UCT})$ states that every pointwise continuous real-valued function on the unit interval is uniformly continuous. In constructive mathematics, $\mathsf{UCT}$ is strictly stronger than the decidable fan theorem $(\mathsf{DFT})$, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real-valued functions as type-one functions. However, the precise relation between such type-one functions and continuous real-valued functions (usually described as type-two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real-valued function on [0, 1], and show that real-valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that $\mathsf{DFT}$ is equivalent to the statement that every pointwise continuous real-valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that $\mathsf{DFT}$ is equivalent to a similar principle for real-valued functions on the Cantor space ${\{0,1\}}^{\mathbb{N}}$. These results extend Berger's [2] characterisation of $\mathsf{DFT}$ for integer-valued functions on ${\{0,1\}}^{\mathbb{N}}$ and unify some characterisations of $\mathsf{DFT}$ in terms of functions having continuous moduli.     

Least squares problems with inequality constraints as quadratic constraints

Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and one from a hydrological application. Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ² regularization method. The χ² regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.     

Lagrange interpolation with constraints

Uniform convergence of Lagrange interpolation at the zeros of Jacobi polynomials in the presence of constraints is investigated. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial respecting the given constraints well approximates a given function and its derivatives.     

Approximation with interpolatory constraints

Best approximation with interpolatory constraints is considered. A sufficient condition for an approximating function to be a unique best approximation is presented. A necessary condition is deduced if uniqueness holds.     

Variational problems with constraints

Summary The problem considered is that of maximizing dt subject to x=G(x, y), x(0)=c, and0 ≤y≤x. An essentially new feature is determining in what regions y=x,0<y<x, and y=0.     

Variational equations with constraints

In this work we deal with a constrained variational equation associated with the usual weak formulation of an elliptic boundary value problem in the context of Banach spaces, which generalizes the classical results of existence and uniqueness. Furthermore, we give a precise estimation of the norm of the solution.     

Decision-support with preference constraints

One approach to Human Centered Processing is to take into account preferences of users within the context of multiple criteria optimization. The preference model of a problem encloses all the information needed to evaluate the quality of solutions.     

Approximation with interpolatory constraints

Given a triangular array of points on satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a near-optimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.

     

Record values with constraints

Record values are very popular in probability and mathematical statistics. There are many books and papers concerned with classical record values and record times, i.e., records in sequences of independent equally distributed random variables. In recent times, new types of record values (records in the F ^α-scheme, record values in sequences of unequally distributed random variables, records with confirmations, exceedance record values) have been proposed and examined. The present paper proposes another record scheme (so-called “records with constraint”). Various cases are studied in which these records may be useful. For these record values, we give their joint density functions and discover some of their properties. For particular cases of utmost importance, when the initial random variables are independent and have equal exponential distribution, we obtain fairly simple representations of records with constraints as sums of independent equally distributed random terms.