On the Expected Probability of Constraint Violation in Sampled Convex Programs

In this note, we derive an exact expression for the expected probability V of constraint violation in a sampled convex program (see Calafiore and Campi in Math. Program. 102(1):25–46, 2005; IEEE Trans. Autom. Control 51(5):742–753, 2006 for definitions and an introduction to this topic):

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective

Journal of Optimization Theory and Applications - We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible...     

On Shallit’s Minimization Problem

Mathematical Notes - In Shallit’s problem (SIAM Review, 1994), it was proposed to justify a two-term asymptotics of the minimum of a rational function of $$n$$ variables defined as the sum of...     

On a Problem of Cameron’s on Inexhaustible Graphs

A graph G is inexhaustible if whenever a vertex of G is deleted the remaining graph is isomorphic to G. We address a question of Cameron [6], who asked which countable graphs are inexhaustible. In particular, we prove that there are continuum many countable inexhaustible graphs with properties in common with the infinite random graph, including adjacency properties and universality. Locally finite inexhaustible graphs and forests are investigated, as is a semigroup structure on the class of inexhaustible graphs. We extend a result of [7] on homogeneous inexhaustible graphs to pseudo-homogeneous inexhaustible graphs.The authors gratefully acknowledge support from the Natural Science and Engineering Research Council of Canada (NSERC).     

A Bilevel Stochastic Programming Problem with Random Parameters in the Follower’s Objective Function

Under study is a bilevel stochastic linear programming problem with quantile criterion. Bilevel programming problems can be considered as formalization of the process of interaction between two parties. The first party is a Leader making a decision first; the second is a Follower making a decision knowing the Leader’s strategy and the realization of the random parameters. It is assumed that the Follower’s problem is linear if the realization of the random parameters and the Leader’s strategy are given. The aim of the Leader is the minimization of the quantile function of a loss function that depends on his own strategy and the optimal Follower’s strategy. It is shown that the Follower’s problem has a unique solution with probability 1 if the distribution of the random parameters is absolutely continuous. The lower-semicontinuity of the loss function is proved and some conditions are obtained of the solvability of the problem under consideration. Some example shows that the continuity of the quantile function cannot be provided. The sample average approximation of the problem is formulated. The conditions are given to provide that, as the sample size increases, the sample average approximation converges to the original problem with respect to the strategy and the objective value. It is shown that the convergence conditions hold for almost all values of the reliability level. A model example is given of determining the tax rate, and the numerical experiments are executed for this example.     

On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

On a Functional Equation Associated with Simpson’s Rule

In this paper, we determine the general solution of the functional equation $$f(x)-g(y)=(x-y)\lbrack h(x+y)+\psi (x)+\phi (y)\rbrack$$ for all real numbers x and y. This equation arises in connection with Simpson’s Rule for the numerical evaluation of definite integrals. The solution of this functional equation is achieved through the functional equation $$g(x)-g(y)=(x-y)f(x+y)+(x+y)f(x-y).$$      

A Fenchel–Lagrange Duality Approach for a Bilevel Programming Problem with Extremal-Value Function

In this paper, for a bilevel programming problem (S) with an extremal-value function, we first give its Fenchel–Lagrange dual problem. Under appropriate assumptions, we show that a strong duality holds between them. Then, we provide optimality conditions for (S) and its dual. Finally, we show that the resolution of the dual problem is equivalent to the resolution of a one-level convex minimization problem.     

On a Problem of Hayman

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On Basic Constraint Qualifications for Infinite System of Convex Inequalities in Banach Spaces

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi-continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.     

On Some Limit Laws in the Coupon Collector’s Problem

Three limit distributions are obtained for waiting times in the collectors problem with group drawings. Bibliography: 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 126–133.     

On Kolmogorov’s ε-Entropy for a Compact Set of Infinitely Differentiable Aperiodic Functions (Babenko’s Problem)

The asymptotics of Kolmogorov’s ε-entropy for a compact set of infinitely differentiable aperiodic functions that are boundedly embedded in the space of continuous functions on a finite interval is calculated.     

On the Geometrical and Physical Meaning of Newton’s Solution to Kepler’s Problem

In the short treatise De Motu (1684),which serves as a precursor to the Principia Mathematica (1687),Newton essentially deals with the following two problems.     

On a generalization of Gross’s formula

We prove a relation between Whittaker functionals of cusp forms on ${{\widetilde{SL}}_2}$ and the toric periods of forms on the quaternion algebra. As an application we prove a generalization of a formula of Gross.     

On a generalization of Hua’s theorem with five squares of primes

We sharpen Hua’s theorem with five squares of primes by proving that every sufficiently large integer N congruent to 5 modulo 24 can be written in the form

On a Problem on Chebyshev Centers

Let X be a normed linear space, A, G be subsets of X. Define r_A(G) = inf sup ||α - g||then r_A(G) is called the relative Chebyshev radius of A with respect to G. If     

On a Generalization of Voronin’s Theorem

Mathematical Notes - Voronin’s theorem states that the Riemann zeta-function ζ(s) is universal in the sense that all analytic functions that are defined and have no zeros on the right...     

On Plateau’s problem for soap films with a bound on energy

We prove existence and almost everywhere regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in Euclidean space R ³.The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched set which is a union of Lipschitz Jordan curves of finite total length.