Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set {1,… ,n} . Here we are interested in asymptotics, as n→∞ , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.

The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.

Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.

Some finite and symmetric two-player games have no (pure or mixed) symmetric Nash equilibrium when played by partly morally motivated players.The reason is that the “right thing to do” may be not to randomize. We analyze this issue both under complete information between equally moral players and under incomplete information between players of arbitrary degrees of morality. We provide necessary and sufficient conditions for the existence of equilibrium and illustrate the results with examples and counter examples.

In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent (MD), and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a non-degeneracy condition, the algorithm converges to the problem's set of critical points; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is $\mathcal{O}(1/k^\rho)$ for some $\rho\in(0,1]$ that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard non-convex test functions and large-scale traffic assignment problems.

As is well known, equilibrium analysis of evolutionary partnership games can be done by studying a so-called standard quadratic optimization problem, where a possibly indefinite quadratic form is maximized over the standard (probability) simplex. Despite the mathematical simplicity of this model, the nonconvex instances in this problem class allow for remarkably rich patterns of coexisting (strict) local solutions, which correspond to evolutionarily stable states (ESSs) in the game; seen from a dynamic perspective, ESSs form the asymptotically stable fixed points under the continuous-time replicator dynamics. In this study, we develop perturbation methods to enrich existing ESS patterns by a new technique, continuing the research strategy started by Chris Cannings and coworkers in the last quarter of the past century.

In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which is closely related to practical difficulties in solving StQPs globally. In this study, we improve on existing lower bounds for the number of strict local solutions by a new technique to construct instances with a rich solution structure. Furthermore, we provide extensive case studies where the system of supports (the so-called pattern) of solutions are analyzed in detail. Note that by naive simulation, in accordance to theory, most of the interesting patterns would not be encountered, since random instances have, with a high probability, quite sparse solutions (with singleton or doubleton supports), and likewise their expected numbers are considerably lower than in the worst case. Hence, instances with a rich solution pattern are rather rare. On the other hand, by concentrating on (thin) subsets of promising instances, we are able to give an empirical answer on the size distribution of supports of strict local solutions to the StQP and their patterns, complementing average-case analysis of this NP-hard problem class.

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order, and a further improvement for such matrices of order six.

Let $p_n$ denote the largest possible cp-rank of an $n\times n$ completely positive matrix. This matrix parameterhas itssignificance both in theory and applications, as it sheds light on the geometry and structure of the solution set ofhardoptimization problems in their completely positive formulation.Known bounds for $p_n$ are $s_n=\binom{n+1}2-4$, the current best upper bound, and the Drew--Johnson--Loewy (DJL)lower bound$d_n=\lfloor\frac{n^2}4\rfloor$. The famous DJL conjecture (1994) states that $p_n=d_n$. Here we show$p_n=\frac {n^2}2 +{\mathcal O}(n^{3/2}) = 2d_n+{\mathcal O}(n^{3/2}){{p_n=\frac {n^2}2 +{\mathcal O}(n^{3/2}) = 2d_n+{\mathcal O}(n^{3/2})}}$,and construct counterexamples to the DJL conjecture for all $n\ge {12}$ (for orders seven through elevencounterexamples were already given in[I. M. Bomze, W. Schachinger, and R. Ullrich, Linear Algebra Appl. , 459 (2014), pp. 208--221].

We study n×n completely positive matrices M on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix S which generates a quadratic form with finitely many zeroes in the standard simplex. Constructing particular instances of S, we are able to construct counterexamples to the famous Drew–Johnson–Loewy conjecture (1994) for matrices of order seven through eleven.

We show that the maximal cp-rank of $n\times n$ completely positive matrices is attained at a positive-definite matrix onthe boundary of the cone of $n\times n$ completely positive matrices, thus answering a long-standing question.We also show that the maximal cp-rank of $5\times 5$ matrices equals six, which proves the famousDrew--Johnson--Loewy conjecture [ Linear Multilinear Algebra , 37 (1994), pp. 303--310] for matrices of this order. In addition we present asimple scheme for generating completely positive matrices of high cp-rank and investigate the structure of a minimal cp factorization.

Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone \({\mathcal{C}}\) of all completely positive symmetric n × n matrices (which can be factorized into \({FF^\top}\) , where F is a rectangular matrix with no negative entry), and its dual cone \({\mathcal{C}^*}\) , which coincides with the cone of all copositive matrices (those which generate a quadratic form taking no negative value over the positive orthant). We provide structural algebraic properties of these cones, and numerous (counter-)examples which demonstrate that many relations familiar from semidefinite optimization may fail in the copositive context, illustrating the transition from polynomial-time to NP-hard worst-case behaviour. In course of this development we also present a systematic construction principle for non-attainability phenomena, which apparently has not been noted before in an explicit way. Last but not least, also seemingly for the first time, a somehow systematic clustering of the vast and scattered literature is attempted in this paper.

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic program, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained problems with 1000 variables is reported.

A Standard Quadratic Optimization Problem (StQP) consists of maximizing a (possibly indefinite) quadratic form over the standard simplex. Likewise, in a multi-StQP we have to maximize a (possibly indefinite) quadratic form over the Cartesian product of several standard simplices (of possibly different dimensions). Among many other applications, multi-StQPs occur in Machine Learning Problems. Several converging monotone interior point methods are established, which differ from the usual ones used in cone programming. Further, we prove an exact cone programming reformulation for establishing rigorous yet affordable bounds and finding improving directions.

The famous Frank--Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive semidefiniteness or copositivity conditions, despite the fact that the objective can be even linear. This paper studies exact penalizations of (classical) quadratic programs, i.e., optimization of quadratic functions over a polyhedron, and applies the results to establish a Frank--Wolfe-type theorem for the primal-dual pair of a class of conic programs that frequently arises in applications. One result is that uniqueness of the solution of the primal ensures dual attainability, i.e., existence of the solution of the dual.

We improve the best known lower bounds on the distance between two points of an optimal Morse cluster, with rho is an element of [4.967, 15]. We develop a generalization of a method previously applied to the Lennard-Jones potential, that also leads to improvements of lower bounds for the Morse potential. (C) 2009 Elsevier B.V. All rights reserved.

We improve lower bounds on the minimal distance between two points of a minimum energy configuration w.r.t. the Morse potential. This is achieved by generalizing a method that was already applied to the Lennard-Jones potential in Schachinger et al. (Comput. Optim. Appl. 38:329–349, 2007), resulting in improvements of the currently best bounds known for ρ∈[4.967,15] both for minimal distance and for energy of optimal configurations.

We establish new lower bounds on the distance between two points of a minimum energy configuration of N points in ℝ3 interacting according to a pairwise potential function. For the Lennard-Jones case, this bound is 0.67985 (and 0.7633 in the planar case). A similar argument yields an estimate for the minimal distance in Morse clusters, which improves previously known lower bounds. Moreover, we prove that the optimal configuration cannot be two-dimensional, and establish an upper bound for the distance to the nearest neighbour of every particle, which depends on the position of this particle. On the boundary of the optimal configuration polytope, this is unity while in the interior, this bound depends on the potential function. In the Lennard-Jones case, we get the value \(\sqrt[6]{\frac {11}{5}}\approx1.1404\). Also, denoting by V N the global minimum in an N point minimum energy configuration, we prove in Lennard-Jones clusters \(\frac{V_{N}}{N}\ge-41.66\) for all N≥2, while asymptotically \(\lim_{N\to\infty}\frac{V_{N}}{N}\le-8.611\) holds (as opposed to \(\frac{V_{N}}{N}\ge-8.22\) in the planar case, confirming non-planarity for large N).

We show that, for a certain class of probabilistic models, the number of internal nodes $S_n$ of a trie built from $n$ independent and identically distributed keys is concentrated around its mean, in the sense that $\Var S_n=\Oh(\EE S_n)$. Keys are sequences of symbols which may be taken from varying alphabets, and the choice of the alphabet from which the $k$th symbol is taken, as well as its distribution, may depend on all the preceding symbols. In the construction of the trie we also allow for bucket sizes greater than 1. The property that characterizes our models is the following: there is a constant $C$ such that for any word $v$ that may occur as a prefix of a key the size $S^v_n$ of a trie built from the suffixes of $n$ independent keys conditioned to have the common prefix $v$ has the property $\EE S^v_n\leq Cn$. This class of models contains memoryless and Markovian source models as well as the probabilistic dynamical source models that were recently introduced and thoroughly developed by Vallée [Algorithmica29 (2001) 262–306], in particular the continued fraction source. Furthermore we study the external path length $L_n$, which obeys $\EE L_n=\Oh(n\ln n)$ and $\Var L_n=\Oh(n\ln^2 n)$.

In this paper we study the costs CN of partial match retrievals in K-dimensional tries (K-d tries), constructed from N records. The probabilistic model that we assume is the asymmetric Bernoulli model: keys are sequences of independently and identically distributed random variables, which assume the values 0 and 1 with probability $p \neq \frac{1}{2}$ and 1-p, and are pairwise independent. We determine the extremal asymptotic orders that the sequence of expectations $(\mathbb{E}\,C_N)_{N \ge 0}$ may have for different fixed queries, as well as the narrow region that contains $(\mathbb{E}\,C_N)_{N \geq 0}$ for almost every query. Furthermore we show that $(\mathbb{E}\,C_N)_{N \geq 0}$ and $({\rm Var}\,C_N)_{N \ge 0}$ have the same asymptotics up to a logarithmic factor and, employing a central limit theorem for martingale difference arrays, we prove asymptotic normality of $\frac{C_N - \mathbb{E}\,C_N}{\sqrt{{\rm Var}\,C_N}}$. For random queries, assumed to be independent of the keys and having their specified components distributed according to the same Bernoulli model, no limiting distribution for CN exists, but we can prove asymptotic normality of $\ln C_N$, when appropriately normalized, and determine ${\rm Var}\,C_N$ up to a logarithmic factor, where now $(\mathbb{E}\,C_N)^2 = o({\rm Var}\,C_N)$.