randomized sampling for large zero sum games this paper addresses the solution of large zero sum matrix games using randomized methods we formalize a procedure termed as the sampled security policy ssp algorithm by which a player can compute policies that with a high confidence are security policies against an adversary using randomized methods to explore the possible outcomes of the game the ssp algorithm essentially consists of solving a stochastically sampled subgame that is much smaller than the original game we also propose a randomized algorithm termed as the sampled security value ssv algorithm which computes a high confidence security level i e worst case outcome for a given policy which may or may not have been obtained using the ssp algorithm for both the ssp and the ssv algorithms we provide results to determine how many samples are needed to guarantee a desired level of confidence we start by providing results when the two players sample policies with the same distribution and subsequently extend these results to the case of mismatched distributions we demonstrate the usefulness of these results in a hide and seek game that exhibits exponential complexity

a multiplicity result for a special class of gradient type systems with non differentiable term abstract we prove the existence of multiple solutions of certain systems of hemivariational inequalities using a recent minimax result of b ricceri we apply both our main theorem and the principle of symmetric criticality to obtain multiple solutions of systems of hemivariational inequalities defined on certain sobolev spaces

asynchronous gossip based random projection algorithms over networks we consider a distributed constrained convex optimization problem over a multi agent no central coordinator network we propose a completely decentralized and asynchronous gossip based random projection grp algorithm that solves the distributed problem using only local communications and computations we analyze the convergence properties of the algorithm for a diminishing and a constant stepsize which are uncoordinated among agents for a diminishing stepsize we prove that the iterates of all agents converge to the same optimal point with probability 1 for a constant stepsize we establish an error bound on the expected distance from the iterates of the algorithm to the optimal point we also provide simulation results on a distributed robust model predictive control problem

notes on the scenario design approach the scenario optimization method developed in is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design that would otherwise be hard to tackle via standard deterministic techniques in this note we explore some further aspects of the scenario methodology and present two results pertaining to the tightness of the sample complexity bounds we also state a new theorem that enables the user to make a priori probabilistic claims on the scenario solution with one level of probability only

survey paper research on probabilistic methods for control system design a novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty the main objective of this paper is to provide a broad perspective on this area of research known as probabilistic robust control and to address in a systematic manner recent advances the focal point is on design methods based on the interplay between uncertainty randomization and convex optimization and on the illustration of specific control applications

secant zeta functions abstract we study the series s z n 1 sec n z n s and prove that it converges under mild restrictions on z and s the function possesses a modular transformation property which allows us to evaluate s z explicitly at certain quadratic irrational values of z this supports our conjecture that k k j q whenever k and j are positive integers with k even we conclude with some speculations on the bernoulli numbers

quasilinear elliptic boundary value problems on unbounded cylinders and a related mountain pass lemma sufficient conditions for the existence of a weak solution to quasilinear elliptic boundary value problems on unbounded cylinders are obtained

on the sample size of random convex programs with structured dependence on the uncertainty the scenario approach provides an intuitive method to address chance constrained problems arising in control design for uncertain systems it addresses these problems by replacing the chance constraint with a finite number of sampled constraints scenarios the sample size critically depends on helly s dimension a quantity always upper bounded by the number of decision variables however this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability we derive improved bounds of helly s dimension for problems where the chance constraint has certain structural properties the improved bounds lower the number of scenarios required for these problems leading both to improved objective value and reduced computational complexity our results are generally applicable to randomized model predictive control of chance constrained linear systems with additive uncertainty and affine disturbance feedback

circle and divisor problems and double series of bessel functions abstract in approximately 1915 ramanujan recorded two identities involving doubly infinite series of bessel functions the identities were brought to the mathematical public for the first time when his lost notebook was published in 1988 and are connected with the classical long standing circle and divisor problems respectively we provide a proof of the first identity for the first time by analytically continuing a new kind of dirichlet series delicate estimates of exponential sums are needed and the new methods we introduce may be of independent interest

relations between elliptic multiple zeta values and a special derivation algebra we investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length our method is based on representing elliptic multiple zeta values as iterated integrals over eisenstein series and exploiting the connection with a special derivation algebra its commutator relations give rise to constraints on the iterated integrals over eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives conversely the above connection suggests apparently new relations in the derivation algebra under https tools aei mpg de emzv we provide relations for elliptic multiple zeta values over a wide range of weights and lengths ar x iv 1 50 7 02 25 4v 1 he pth 8 j ul 2 01 5

on the computational complexity and generalization properties of multi stage and stage wise coupled scenario programs abstract we discuss the computational complexity and feasibility properties of scenario sampling techniques for uncertain optimization programs we propose an alternative way of dealing with a special class of stage wise coupled programs and compare it with existing methods in the literature in terms of feasibility and computational complexity we identify trade offs between different methods depending on the problem structure and the desired probability of constraint satisfaction to illustrate our results an example from the area of approximate dynamic programming is considered

varying confidence levels for cvar risk measures and minimax limits conditional value at risk cvar has been widely studied as a risk measure in this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with cvar appearing in the objective and also the constraints we start by considering a problem in which cvar is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one we make use of a consistent tail condition which ensures that the cvar of a random function will converge uniformly to its supremum as the confidence level increases and establish an error bound for the cvar optimal solution under second order growth conditions the results are extended to a minimization problem with a constraint on the cvar value which in the limit as the confidence level approaches one coincides with a problem having semi infinite constraints we study the sample average approximation scheme for the cvar constraints and establish an exponential rate of convergence for the sample averaged optimal solution we propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to cvar constrained problems out of sample our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance

distributed random convex programming via constraints consensus this paper discusses distributed approaches for the solution of random convex programs rcps rcps are convex optimization problems with a usually large number n of randomly extracted constraints they arise in several application areas especially in the context of decision making under uncertainty see g c calafiore siam j optim 20 2010 pp 3427 3464 g c calafiore and m c campi ieee trans automat control 51 2006 pp 742 753 we here consider a setup in which instances of the random constraints the scenario are not held by a single centralized processing unit but are instead distributed among different nodes of a network each node sees only a small subset of the constraints and may communicate with neighbors the objective is to make all nodes converge to the same solution as the centralized rcp problem to this end we develop two distributed algorithms that are variants of the constraints consensus algorithm g notarstefano and f bullo proceedings of the 46th iee

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

an existence result for gradient type systems with a nondifferentiable term on unbounded strips in this paper we study the existence of nontrivial solutions for a class of gradient type systems on strip like domains where the nonlinear term is not necessarily continuously differentiable the proof of the main result is based on a nonsmooth version of the mountain pass theorem which involves the cerami compactness condition and on the principle of symmetric criticality for locally lipschitz functions

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

robust model predictive control via scenario optimization this paper discusses a novel probabilistic approach for the design of robust model predictive control mpc laws for discrete time linear systems affected by parametric uncertainty and additive disturbances the proposed technique is based on the iterated solution at each step of a finite horizon optimal control problem fhocp that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances followed by a specific command selection rule implemented in a receding horizon fashion the scenario fhocp is always convex also when the uncertain parameters and disturbance belong to nonconvex sets and irrespective of how the model uncertainty influences the system s matrices moreover the computational complexity of the proposed approach does not depend on the uncertainty disturbance dimensions and scales quadratically with the control horizon the main result in this work is related to the analysis of the closed loop system under receding horizon implementation of the scenario fhocp and essentially states that the devised control law guarantees constraint satisfaction at each step with some a priori assigned probability p while the system s state reaches the target set either asymptotically or in finite time with probability at least p the proposed method may be a valid alternative when other existing techniques either deterministic or stochastic are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance constrained optimization problem

bifurcation problems for the laplacian in ⁿ in this paper we consider the bifurcation problem div ivulp 2vu ag x julp 2u f a x u in rn with p 1 we show that a continuum of positive solutions bifurcates out from the principal eigenvalue a1 of the problem div ivuip 2vu ag x ulp 2u here both functions f and g may change sign

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

a constraint sampling approach for multi stage robust optimization we propose a tractable approximation scheme for convex not necessarily linear multi stage robust optimization problems we approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization we obtain a priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi infinite problem arising from the decision rule approximation we demonstrate that for a suitable choice of basis functions the approximation converges as the size of the basis and the number of sampled constraints tend to infinity the approach yields an algorithm parameterized in the basis size the probability of constraint violation and the confidence that this probability will not be exceeded these three parameters serve to tune the trade off between optimality and feasibility of the decision rules and the computational cost of the algorithm we assess the convergence and scalability properties of our approach in the context of two inventory management problems

selberg s integral and linear forms in zeta values using selberg s integral we present some new euler type integral representations of certain nearly poised hypergeometric series these integrals are also shown to produce linear forms in odd and or even zeta values that generalize previous work of the author

a real space renormalization approach to the kubo greenwood formula in mirror fibonacci systems an exact real space renormalization method is developed to address the electronic transport in mirror fibonacci chains at a macroscopic scale by means of the kubo greenwood formula the results show that the mirror symmetry induces a large number of transparent states in the dc conductivity spectra contrary to the simple fibonacci case a length scaling analysis over ten orders of magnitude reveals the existence of critically localized states and their ac conduction spectra show a highly oscillating behaviour for multidimensional quasiperiodic systems a novel renormalization plus convolution method is proposed this combined renormalization convolution method has shown an extremely elevated computing efficiency being able to calculate electrical conductance of a three dimensional non crystalline solid with 10 30 atoms finally the dc and ac conductances of mirror fibonacci nanowires are also investigated where a quantized dc conductance variation with the fermi energy is found as observed in gold nanowires

optimization with few violated constraints for linear bounded error parameter estimation in the context of linear constrained optimization we study the problem of finding an optimal solution satisfying all but k of the given n constraints a solution is obtained by means of an algorithm of the complexity min o n spl middot k sup d o n spl middot d sup k 1 where d is the dimension of the problem we then use these results to solve the problem of robust identification in the presence of outliers in the setting of bounded error parameter identification finally we show that the estimate obtained converges to the true but unknown parameter in the presence of outliers

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

technical communique policy set iteration for markov decision processes this communique presents an algorithm called policy set iteration psi for solving infinite horizon discounted markov decision processes with finite state and action spaces as a simple generalization of policy iteration pi psi generates a monotonically improving sequence of stationary markovian policies p k based on a set manipulation as opposed to pi s single policy manipulation at each iteration k when the set involved with psi at k contains n independently generated sample policies from a given distribution d the probability that the expected value of any sampled policy from d with respect to an initial state distribution is greater than that of p k converges to zero with o n k rate moreover psi converges to an optimal policy no slower than pi in terms of the number of iterations for any d

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

reliable approximations of probability constrained stochastic linear quadratic control here we consider a state constrained stochastic linear quadratic control problem this problem has linear dynamics and a quadratic cost and states are required to satisfy a probabilistic constraint in this paper the joint probabilistic constraint in the model is converted to a conservative deterministic constraint using a multi dimensional chebyshev bound a maximum volume inscribed ellipsoid problem is solved to obtain this probability bound using the probability bound we develop a recursive state feedback control algorithm for a special class of state constrained stochastic linear quadratic regulator lqr the performance of this approach is explored in a numerical example

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

weighted divisor sums and bessel function series on page 335 in his lost notebook ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary bessel functions we provide the first published proof of this result the identity yields as corollaries representations of weighted divisor sums in particular the summatory function for r 2 n the number of representations of the positive integer n as a sum of two squares

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

probabilistic optimal estimation with uniformly distributed noise the classical approach to system identification is based on stochastic assumptions about the measurement error and provides estimates that have random nature worst case identification on the other hand only assumes the knowledge of deterministic error bounds and establishes guaranteed estimates thus being in principle better suited for the use in control design however a main limitation of such deterministic bounds lies in their potential conservatism thus leading to estimates of restricted use in this paper we propose a rapprochement between the stochastic and worst case system identification viewpoints which is based on the probabilistic setting of information based complexity the main idea in this line of research is to discard sets of measure at most where is a probabilistic accuracy from the set of deterministic estimates therefore we are decreasing the so called worst case radius of information at the expense of a given probabilistic risk in the case of uniformly distributed noise we derive new computational results and in particular we compute a trade off curve called violation function which shows how the radius of information decreases as a function of the accuracy to this end we construct randomized and deterministic algorithms which provide approximations of this function we report extensive simulations showing numerical comparisons between the stochastic worst case and probabilistic approaches thus demonstrating the efficacy of the methods proposed in this paper

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

singularly perturbed markov decision processes a multiresolution algorithm singular perturbation techniques allow the derivation of an aggregate model whose solution is asymptotically optimal for markov decision processes with strong and weak interactions we develop an algorithm that takes advantage of the asymptotic optimality of the aggregate model in order to compute the solution of the original model we derive conditions for which the proposed algorithm has better worst case complexity than conventional contraction algorithms based on our complexity analysis we show that the major benefit of aggregation is that the reduced order model is no longer ill conditioned the reduction in the number of states due to aggregation is a secondary benefit this is a surprising result since intuition would suggest that the reduced order model can be solved more efficiently because it has fewer states however we show that this is not necessarily the case our theoretical analysis and numerical experiments show that the proposed algorithm can compute the optimal solution with a redu

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

generalized lambert series and arithmetic nature of odd zeta values it is pointed out that the generalized lambert series sum nolimits n 1 infty n n 2h e n nx 1 studied by kanemitsu tanigawa and yoshimoto can be found on page 332 of ramanujan s lost notebook in a slightly more general form we extend an important transformation of this series obtained by kanemitsu tanigawa and yoshimoto by removing restrictions on the parameters n and h that they impose from our extension we deduce a beautiful new generalization of ramanujan s famous formula for odd zeta values which for n odd and m 0 gives a relation between 2 m 1 and 2 nm 1 a result complementary to the aforementioned generalization is obtained for any even n and m it generalizes a transformation of wigert and can be regarded as a formula for 2 m 1 1 n applications of these transformations include a generalization of the transformation for the logarithm of dedekind eta function z zudilin and rivoal type results on transcendence of certain values and a transcendence criterion for euler s constant

probabilistically constrained linear programs and risk adjusted controller design the focal point of this paper is the probabilistically constrained linear program pclp and how it can be applied to control system design under risk constraints the pclp is the counterpart of the classical linear program where it is assumed that there is random uncertainty in the constraints and therefore the deterministic constraints are replaced by probabilistic ones it is shown that for a wide class of probability density functions called log concave symmetric densities the pclp is a convex program an equivalent formulation of the pclp is also presented which provides insight into numerical implementation this concept is applied to control system design it is shown how the results in this paper can be applied to the design of controllers for discrete time systems to obtain a closed loop system with a well defined risk of violating the so called property of superstability furthermore we address the problem of risk adjusted pole placement

simulation of stochastic systems via polynomial chaos expansions and convex optimization polynomial chaos expansions represent a powerful tool to simulate stochastic models of dynamical systems yet deriving the expansion s coefficients for complex systems might require a significant and nontrivial manipulation of the model or the computation of large numbers of simulation runs rendering the approach too time consuming and impracticable for applications with more than a handful of random variables we introduce a computationally tractable technique for computing the coefficients of polynomial chaos expansions the approach exploits a regularization technique with a particular choice of weighting matrices which allows to take into account the specific features of polynomial chaos expansions the method completely based on convex optimization can be applied to problems with a large number of random variables and uses a modest number of monte carlo simulations while avoiding model manipulations additional information on the stochastic process when available can be also incorporated in the approach by means of convex constraints we show the effectiveness of the proposed technique in three applications in diverse fields including the analysis of a nonlinear electric circuit a chaotic model of organizational behavior and finally a chemical oscillator

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

integral representations for the exceptional univariate lommel functions an elementary integral representation for the lommel function s 1 0 z is given and extended to other exceptional cases

on a secant dirichlet series and eichler integrals of eisenstein series we consider the secant dirichlet series psi s tau sum n 1 infty frac sec pi n tau n s recently introduced and studied by lalin rodrigue and rogers in particular we show as conjectured and partially proven by lalin rodrigue and rogers that the values psi 2 m sqrt r with r 0 rational are rational multiples of pi 2 m we then put the properties of the secant dirichlet series into context by showing that for even s they are eichler integrals of odd weight eisenstein series of level 4 this leads us to consider eichler integrals of general eisenstein series and to determine their period polynomials in the level 1 case these polynomials were recently shown by murty smyth and wang to have most of their roots on the unit circle we provide evidence that this phenomenon extends to the higher level case this observation complements recent results by conrey farmer and imamoglu as well as el guindy and raji on zeros of period polynomials of hecke eigenforms in the level 1 case finally we briefly revisit results of a similar type in the works of ramanujan

multiple series connected to hoffman s conjecture on multiple zeta values abstract recent results of zlobin and cresson fischler rivoal allow one to decompose any suitable p uple series of hypergeometric type into a linear combination over the rationals of multiple zeta values of depth at most p in some cases only the multiple zeta values with 2 s and 3 s are involved as in hoffman s conjecture in this text we study the depth p part of this linear combination namely the contribution of the multiple zeta values of depth exactly p we prove that it satisfies some symmetry property as soon as the p uple series does and make some conjectures on the depth p 1 part of the linear combination when p 3 our result generalizes the property that very well poised univariate hypergeometric series involve only zeta values of a given parity which is crucial in the proof by rivoal and ball rivoal that 2 n 1 is irrational for infinitely many n 1 the main feature of the proof is an algebraic approach based on representations of z 2 z p s p

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

interval predictor models identification and reliability this paper addresses the problem of constructing reliable interval predictors directly from observed data differently from standard predictor models interval predictors return a prediction interval as opposed to a single prediction value we show that in a stationary and independent observations framework the reliability of the model that is the probability that the future system output falls in the predicted interval is guaranteed a priori by an explicit and non asymptotic formula with no further assumptions on the structure of the unknown mechanism that generates the data this fact stems from a key result derived in this paper which relates at a fundamental level the reliability of the model to its complexity and to the amount of available information number of observed data

chance constrained h control for a class of time varying systems with stochastic nonlinearities the finite horizon case in this paper a new finite horizon h control problem is considered for a class of time varying systems with stochastic nonlinearities measurements degradation and chance constraints the purpose of the addressed problem is to design the time varying controller such that the closed loop system satisfies the prespecified h disturbance attenuation requirement and certain chance constraints on the controlled output vector zk i e the probability of the controlled output zk belonging to a given set is larger than a prescribed value a modified maximum volume inscribed ellipsoid mvie method is employed to convert the chance constraint into some tractable inequalities which could be conveniently handled by the recursive linear matrix inequality rlmi approach then by using stochastic control analysis method sufficient conditions are derived for the existence of the desired multi objective controller and furthermore the gains of the desired controllers are characterized by means of rlmis finally two illustrative examples and a practical system are proposed to highlight the effectiveness and applicability of the presented controller design technology

approximations to di and tri logarithms we propose hypergeometric constructions of simultaneous approximations to polylogarithms these approximations suit for computing the values of polylogarithms and satisfy 4 term apery like polynomial recursions

linking and multiplicity results for the p laplacian on unbounded cylinders abstract we consider the p laplacian problem formula on unbounded cylinders rn m rn n m 2 where pu div u p 2 u is a constant in a certain range and a ln p l is nonnegative a 0 using the principle of symmetric criticality existence and multiplicity are proved under suitable conditions on a and f

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

modular forms and period polynomials we study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group for the modular group this space is en dowed with a pairing corresponding to the petersson inner product on modular forms via a formula of haberland and with an action of hecke operators defined algebraically by zagier we generalize haberland s formula to not necessarily cuspidal modular forms for finite index subgroups and we show that it conceals two stronger formulas we extend the action of hecke operators to period polynomials of modular forms we show that the pairing on period polynomials appearing in haber land s formula is nondegenerate and we determine the adjoints of hecke operators with respect to it we give a few applications for 1 n an extension of the eichler shimura isomorphism to the entire space of modular forms the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms which are independent of the period rela tions and an explicit formula for fourier coefficients of hecke eigenforms in terms of their period polynomials generalizing the coefficients theorem of manin

a scenario approach for non convex control design randomized optimization is an established tool for control design with modulated robustness while for uncertain convex programs there exist efficient randomized approaches this is not the case for non convex problems methods based on statistical learning theory are applicable to non convex problems but they usually are conservative in achieving the desired probabilistic guarantees in this paper we derive a novel scenario approach for a wide class of random non convex programs with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program but for all feasible solutions inside a set of a priori chosen complexity we also address measure theoretic issues for uncertain convex and non convex programs among the family of non convex control design problems that can be addressed via randomization we apply our scenario approach to stochastic model predictive control for chance constrained nonlinear control affine systems

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

diophantine properties of numbers related to catalan s constant 1 as a linear form in 1 and even beta valuesonly the following lemma 2 is almost the same as the correspondingone in br and we only include a sketch of the proof we would like toemphasise the fact that it is essentially the well poisedness of the series thatenables us to produce this parity dependant behaviour when

randomized methods for design of uncertain systems in this paper we study randomized methods for feedback design of uncertain systems the first contribution is to derive the sample complexity of various constrained control problems in particular we show the key role played by the binomial distribution and related tail inequalities and compute the sample complexity this contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm these results are then applied to the analysis of worst case performance and design with robust optimization the second contribution of the paper is to introduce a general class of sequential algorithms denoted as sequential probabilistic validation spv in these sequential algorithms at each iteration a candidate solution is probabilistically validated and corrected if necessary to meet the required specifications the results we derive provide the sample complexity which guarantees that the solutions obtained with spv algorithms meet some pre specified probabilistic accuracy and confidence the performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification

stochastic model predictive control of lpv systems via scenario optimization a stochastic receding horizon control approach for constrained linear parameter varying discrete time systems is proposed in this paper it is assumed that the time varying parameters have stochastic nature and that the system s matrices are bounded but otherwise arbitrary nonlinear functions of these parameters no specific assumption on the statistics of the parameters is required by using a randomization approach a scenario based finite horizon optimal control problem is formulated where only a finite number m of sampled predicted parameter trajectories scenarios are considered this problem is convex and its solution is a priori guaranteed to be probabilistically robust up to a user defined probability level p the p level is linked to m by an analytic relationship which establishes a tradeoff between computational complexity and robustness of the solution then a receding horizon strategy is presented involving the iterated solution of a scenario based finite horizon control problem at each time step our key result is to show that the state trajectories of the controlled system reach a terminal positively invariant set in finite time either deterministically or with probability no smaller than p the features of the approach are illustrated by a numerical example

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

existence of nonzero weak solutions for a class of elliptic variational inclusions systems in rn abstract we consider the following variational inclusions system of the form u u 1 f u v in r n v v 2 f u v in r n with u v h 1 r n where f r 2 r is a locally lipschitz function and i f u v i 1 2 are the partial generalized gradients in the sense of clarke under various growth conditions on the nonlinearity f we study the existence of nonzero weak solutions of the above system in the sense of hemivariational inequalities which are critical points of an appropriate locally lipschitz function defined on h 1 r n h 1 r n the main tool used in the paper is the principle of symmetric criticality for locally lipschitz functions

zeros of combinations of the riemann ξ function and the confluent hypergeometric function on bounded vertical shifts abstract in 1914 hardy proved that infinitely many non trivial zeros of the riemann zeta function lie on the critical line using the transformation formula of the jacobi theta function recently the first author obtained an integral representation involving the riemann function and the confluent hypergeometric function linked to the general theta transformation using this result we show that a series consisting of bounded vertical shifts of a product of the riemann function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line thereby generalizing a previous result due to the first and the last authors along with roy and robles the latter itself is a generalization of hardy s theorem

a nash type solution for hemivariational inequality systems in this paper we prove an existence result for a general class of hemivariational inequality systems using the ky fan version of the kkm theorem fan 1984 10 or tarafdar fixed points tarafdar 1987 11 as application we give an infinite dimensional version for the existence result of nash generalized derivative points introduced recently by kristaly 2010 5 we also give an application to a general hemivariational inequality system

learning noisy functions via interval models this paper considers the problem of identification of an interval model for an unknown static function using a finite batch of stochastic input output data u i y i u i y i i 1 ni 1 n the criterion used for identification is that the width of the interval output of the model should be minimized while containing a given fraction of the observed outputs y i y i we show that for suitable finite nn the resulting model will be reliable that is it will explain any other unseen output up to a given and arbitrary high probability

distributionally robust control of constrained stochastic systems we investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process i e when only the first two moments of the disturbance distribution are known we consider two types of distributionally robust constraints in the first case we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments these constraints are commonly referred to as distributionally robust chance constraints in the second case we impose conditional value at risk cvar constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information such constraints are referred to as distributionally robust cvar constraints with second order moment specifications we propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems we prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust cvar and chance constrained problems the proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives

weighted divisor sums and bessel function series ii abstract on page 335 in his lost notebook ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of bessel functions in each case there are three possible interpretations for the double series in an earlier paper two of the present authors proved the first identity under one possible interpretation in the present paper the second identity is proved under a similar interpretation with one additional assumption moreover under a second interpretation entirely different proofs of both identities depending on weighted or twisted divisor sums are offered the two identities are intimately connected with the classical circle and divisor problems respectively

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost

repetitive scenario design repetitive scenario design rsd is a randomized approach to robust design based on iterating two phases a standard scenario design phase that uses n scenarios design samples followed by randomized feasibility phase that uses n o test samples on the scenario solution we give a full and exact probabilistic characterization of the number of iterations required by the rsd approach for returning a solution as a function of n n o and of the desired levels of probabilistic robustness in the solution this novel approach broadens the applicability of the scenario technology since the user is now presented with a clear tradeoff between the number n of design samples and the ensuing expected number of repetitions required by the rsd algorithm the plain one shot scenario design becomes just one of the possibilities sitting at one extreme of the tradeoff curve in which one insists in finding a solution in a single repetition this comes at the cost of possibly high n other possibilities along the tradeoff curve use lower n values but possibly require more than one repetition