tauberian theorems and the central limit theorem 1 results let xn be independent and identically distributed random variables with law p mean 0 and variance a2 00 s 1 xk write p qn 1 for the laws of sn sn nyi a n and the standard normal n o 1 thus qn 1d weak convergence by the global central limit theorem we shall need the local central limit theorem see e g ibragimov and linnik 1971 theorems 4 4 1 4 2 1 4 2 2 we shall assume either llt i some pk has an absolutely continuous component in which case qn 11 3 0 as n x oo variation norm or llt ii p is lattice supported by some arithmetic progression a hk with k integer and h maximal we shall suppose for simplicity that a 0 and h 1 when writing pnk for pn k

a discrete schrodinger equation via optimal transport on graphs abstract in 1966 edward nelson presented an interesting derivation of the schrodinger equation using brownian motion recently this derivation is linked to the theory of optimal transport which shows that the schrodinger equation is a hamiltonian system on the probability density manifold equipped with the wasserstein metric in this paper we consider similar matters on a finite graph by using discrete optimal transport and its corresponding nelson s approach we derive a discrete schrodinger equation on a finite graph the proposed system is quite different from the commonly referred discretized schrodinger equations it is a system of nonlinear ordinary differential equations odes with many desirable properties several numerical examples are presented to illustrate the properties

a framework for robustness analysis of constrained finite receding horizon control a framework for robustness analysis of input constrained finite receding horizon control is presented under the assumption of quadratic upper bounds on the finite horizon costs we derive sufficient conditions for robust stability of the standard discrete time linear quadratic receding horizon control formulation this is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms lending itself to s procedure tools which are used to convert robustness questions to tractable convex conditions robustness with respect to plant model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities simple examples demonstrate the approach

nonlinear control of constrained linear systems via predictive reference management a method based on conceptual tools of predictive control is described for solving set point tracking problems wherein pointwise in time input and or state inequality constraints are present it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time a virtual sequence among a family of linearly parameterized command sequences by solving a convex constrained quadratic optimization problem and feeds the primal system according to a receding horizon control philosophy the overall system is proved to fulfill the constraints be asymptotically stable and exhibit an offset free tracking behavior provided that an admissibility condition on the initial state is satisfied though the cg can be tailored for the application at hand by appropriately choosing the available design knobs the required online computational load for the usual case of affine constraints is well tempered by the related relatively simple convex quadratic programming problem

general regular variation popa groups and quantifier weakening abstract we introduce general regular variation a theory of regular variation containing the existing karamata bojanic karamata de haan and beurling theories as special cases the unifying theme is the popa groups of our title viewed as locally compact abelian ordered topological groups together with their haar measure and fourier theory the power of this unified approach is shown by the simplification it brings to the whole area of quantifier weakening so important in this field

a direct method for sparse least squares problems with lower and upper bounds a direct method is developed for solving linear least squares problems mathop min left ax b right 2 limits x wherea is large and sparse and the solution is subject to lower and upper boundsl x u the problem is initially transformed to upper triangular form by a sparseqr factorization an active set algorithm is then used the key step is the stable updating of ther factor associated with the columns ofa corresponding to the free variables when theq factor is not available for this a new method is developed which uses the semi normal equations and iterative refinement

estimation of positive definite m matrices and structure learning for attractive gaussian markov random fields abstract consider a random vector with finite second moments if its precision matrix is an m matrix then all partial correlations are non negative if that random vector is additionally gaussian the corresponding markov random field gmrf is called attractive we study estimation of m matrices taking the role of inverse second moment or precision matrices using sign constrained log determinant divergence minimization we also treat the high dimensional case with the number of variables exceeding the sample size the additional sign constraints turn out to greatly simplify the estimation problem we provide evidence that explicit regularization is no longer required to solve the resulting convex optimization problem we propose an algorithm based on block coordinate descent in which each sub problem can be recast as non negative least squares problem illustrations on both simulated and real world data are provided

fast nonnegative matrix factorization an active set like method and comparisons nonnegative matrix factorization nmf is a dimension reduction method that has been widely used for numerous applications including text mining computer vision pattern discovery and bioinformatics a mathematical formulation for nmf appears as a nonconvex optimization problem and various types of algorithms have been devised to solve the problem the alternating nonnegative least squares anls framework is a block coordinate descent approach for solving nmf which was recently shown to be theoretically sound and empirically efficient in this paper we present a novel algorithm for nmf based on the anls framework our new algorithm builds upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes a limitation of the active set method we introduce ideas that efficiently extend the block principal pivoting method within the context of nmf computation our algorithm inherits the convergence property of the anls framework and can easily be extended to other constrained nmf formulations extensive computational comparisons using data sets that are from real life applications as well as those artificially generated show that the proposed algorithm provides state of the art performance in terms of computational speed

maximum correntropy criterion for robust face recognition in this paper we present a sparse correntropy framework for computing robust sparse representations of face images for recognition compared with the state of the art l1norm based sparse representation classifier src which assumes that noise also has a sparse representation our sparse algorithm is developed based on the maximum correntropy criterion which is much more insensitive to outliers in order to develop a more tractable and practical approach we in particular impose nonnegativity constraint on the variables in the maximum correntropy criterion and develop a half quadratic optimization technique to approximately maximize the objective function in an alternating way so that the complex optimization problem is reduced to learning a sparse representation through a weighted linear least squares problem with nonnegativity constraint at each iteration our extensive experiments demonstrate that the proposed method is more robust and efficient in dealing with the occlusion and corruption problems in face recognition as compared to the related state of the art methods in particular it shows that the proposed method can improve both recognition accuracy and receiver operator characteristic roc curves while the computational cost is much lower than the src algorithms

a new approach to stability analysis for constrained finite receding horizon control without end constraints we present a new approach to the stability analysis of finite receding horizon control applied to constrained linear systems by relating the final predicted state to the current state through a bound on the terminal cost it is shown that knowledge of upper and lower bounds for the finite horizon costs is sufficient to determine the stability of a receding horizon controller this analysis is valid for receding horizon schemes with arbitrary positive definite terminal weights and does not rely on the use of stabilizing constraints the result is a computable test for stability and two simple examples are used to illustrate its application

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

regular variation without limits karamata theory n h bingham et al 1987 8 ch 1 explores functions f for which the limit function g f x f x exists as x and for which g subject to mild regularity assumptions on f further karamata theory n h bingham et al 1987 8 ch 2 explores functions f for which the upper limit as x remains bounded here the usual regularity assumptions invoke boundedness of f on a baire non meagre measurable non null set with f baire measurable and the conclusions assert uniformity over compact sets implying upper bounds of the form f x f x k for all large x we give unifying combinatorial conditions which include the two classical cases deriving them from a combinatorial semigroup theorem we examine character degradation in the passage from f to f using some standard descriptive set theory and thus identify natural classes in which the theory may be established

optimization algorithms for the solution of the frictionless normal contact between rough surfaces this paper revisits the fundamental equations for the solution of the frictionless unilateral normal contact problem between a rough rigid surface and a linear elastic half plane using the boundary element method bem after recasting the resulting linear complementarity problem lcp as a convex quadratic program qp with nonnegative constraints different optimization algorithms are compared for its solution i a greedy method based on different solvers for the unconstrained linear system conjugate gradient cg gauss seidel cholesky factorization ii a constrained cg algorithm iii the alternating direction method of multipliers admm and iv the non negative least squares nnls algorithm possibly warm started by accelerated gradient projection steps or taking advantage of a loading history the latter method is two orders of magnitude faster than the greedy cg method and one order of magnitude faster than the constrained cg algorithm finally we propose another type of warm start based on a refined criterion for the identification of the initial trial contact domain that can be used in conjunction with all the previous optimization algorithms this method called cascade multi resolution cmr takes advantage of physical considerations regarding the scaling of the contact predictions by changing the surface resolution the method is very efficient and accurate when applied to real or numerically generated rough surfaces provided that their power spectral density function is of power law type as in case of self affine fractal surfaces

computational complexity certification for real time mpc with input constraints based on the fast gradient method this paper proposes to use nesterov s fast gradient method for the solution of linear quadratic model predictive control mpc problems with input constraints the main focus is on the method s a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre specified suboptimality is obtained for any possible state of the system we investigate cold and warm starting strategies and provide an easily computable lower iteration bound for cold starting and an asymptotic characterization of the bounds for warm starting moreover we characterize the set of mpc problems for which small iteration bounds and thus short solution times are expected the theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal dual interior point method

continuous on rays solutions of an equation of the golab schinzel type abstract let x be a real linear space we characterize continuous on rays solutions f g x r of the equation f x g x y f x f y our result refers to papers of j chudziak 2006 14 and j brzd k 2003 11

on the quotient stability of a family of functional equations abstract we show some stability results for a family of functional equations that contains the exponential and the golab schinzel functional equations

the density manifold and configuration space quantization on developpe la structure geometrique differentielle d une variete de frechet de densites ceci fournit un cadre geometrique a la quantification reliee a la mecanique stochastique de nelson on etudie les structures riemanniennes et symplectiques de la variete densite et on deduit l equation de schrodinger a partir d un principe variationnel

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

on the generalized hyers ulam stability of multi quadratic mappings a mapping f v email protected w where v is a commutative group w is a linear space and n 2 is an integer is called multi quadratic if it is quadratic in each variable in this paper we prove the generalized hyers ulam stability of multi quadratic mappings in banach spaces and complete non archimedean spaces

exact complexity certification of active set methods for quadratic programming active set methods are recognized to often outperform other methods in terms of speed and solution accuracy when solving small size quadratic programming qp problems making them very attractive in embedded linear model predictive control mpc applications a drawback of active set methods is the lack of tight bounds on the worst case number of iterations a fundamental requirement for their implementation in a real time system extensive simulation campaigns provide an indication of the expected worst case computation load but not a complete guarantee this paper solves such a certification problem by proposing an algorithm to compute the exact bound on the maximum number of iterations and floating point operations required by a state of the art dual active set qp solver the algorithm is applicable to a given qp problem whose linear term of the cost function and right hand side of the constraints depend linearly on a vector of parameters as in the case of linear mpc in addition a new solver is presented that combines explicit and implicit mpc ideas guaranteeing improvements of the worst case computation time the ability of the approach to exactly quantify memory and worst case computation requirements is tested on a few mpc examples also highlighting when online optimization should be preferred to explicit mpc

computational complexity certification for dual gradient method application to embedded mpc abstract in this paper we analyze the computational complexity of the dual gradient method for solving linearly constrained convex problems when it is difficult to project on the primal feasible set described by linear constraints we use the lagrangian relaxation to handle the complicated constraints and then we apply the dual gradient algorithm for solving the corresponding dual we give a unified convergence rate analysis for the dual gradient algorithm we provide sublinear or linear estimates on the primal suboptimality and feasibility violation of the generated approximate primal solutions our analysis relies on the lipschitz property of the dual function or an error bound property furthermore the iteration complexity analysis is based on two types of approximate primal solutions an average primal sequence or the last primal iterate sequence we also discuss complexity certifications and implementation aspects of the dual gradient algorithm on constrained mpc problems for embedded linear systems

optimal complexity reduction of polyhedral piecewise affine systems this paper focuses on the np hard problem of reducing the complexity of piecewise polyhedral systems e g polyhedral piecewise affine pwa systems the results are fourfold firstly the paper presents two computationally attractive algorithms for optimal complexity reduction that under the assumption that the system is defined over the cells of a hyperplane arrangement derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra the algorithms are based on the cells and the markings of the hyperplane arrangement in particular the first algorithm yields a set of disjoint non overlapping merged polyhedra by executing a branch and bound search on the markings of the cells the second approach leads to non disjoint overlapping polyhedra by formulating and solving an equivalent and well studied logic minimization problem secondly the results are extended to systems defined on general polyhedral partitions and not on cells of hyperplane arrangements thirdly the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable fourthly the paper shows that based on the notion of the hyperplane arrangement pwa state feedback control laws can be implemented efficiently three examples including a challenging industrial problem illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude

technical communique improvements in the efficiency of linear mpc a model predictive control mpc strategy based on augmented autonomous predictions enables a highly efficient online optimization by imposing a terminal constraint at the current time near optimal performance is obtained by delaying the imposition of the terminal constraint by one sampling period however under certain conditions the degree of optimality can be affected an extension is proposed to remove this difficulty yielding significant improvements in the degree of optimality and achieving this at modest computational cost

a gauss seidel type solver for the fast computation of input constrained control systems abstract an important class of nonlinear control systems can be represented as the feedback interconnection of two parts a linear time invariant system and a block of decentralized nonlinearities when the linear time invariant part has a nontrivial feedthrough term or the nonlinearity has a feedback gain control computation involves the online implementation of a multivariable algebraic loop which must be resolved at each sampling instant the requirements for such online computation may result in several implementation issues especially in real time and embedded control applications this paper considers the implementation of such algebraic loops arising from several input constrained systems the proposed solution algorithm is globally convergent for a very large class of feedthrough or feedback gains and shows promise for real time and embedded control applications where a fast but approximate solution is of the essence

stability problem for the golab schinzel type functional equations let x be a vector space over a field k of real or complex numbers n n and k 0 we study the stability problem for the golab schinzel type functional equations f x f x ny f x f y in the class of functions f x k such that the set x x f x 0 has an algebraically interior point

stabilizing linear model predictive control under inexact numerical optimization this note describes a model predictive control mpc formulation for discrete time linear systems with hard constraints on control and state variables under the assumption that the solution of the associated quadratic program is neither optimal nor satisfies the inequality constraints this is common in embedded control applications for which real time constraints and limited computing resources dictate restrictions on the possible number of on line iterations that can be performed within a sampling period the proposed approach is rather general in that it does not refer to a particular optimization algorithm and is based on the definition of an alternative mpc problem that we assume can only be solved within bounded levels of suboptimality and violation of the inequality constraints by showing that the inexact solution is a feasible suboptimal one for the original problem asymptotic or exponential stability is guaranteed for the closed loop system based on the above general results we focus on a specific dual accelerated gradient projection method to obtain a stabilizing mpc law that only requires a predetermined maximum number of on line iterations

on solutions of a common generalization of the golab schinzel equation and of the addition formulae abstract under some additional assumptions we determine solutions of the equation f x m f x y f x f y where f r r is lebesgue measurable or baire measurable m r r and r 2 r all three functions are unknown

two stage nonnegative sparse representation for large scale face recognition this paper proposes a novel nonnegative sparse representation approach called two stage sparse representation tsr for robust face recognition on a large scale database based on the divide and conquer strategy tsr decomposes the procedure of robust face recognition into outlier detection stage and recognition stage in the first stage we propose a general multisubspace framework to learn a robust metric in which noise and outliers in image pixels are detected potential loss functions including l1 l2 1 and correntropy are studied in the second stage based on the learned metric and collaborative representation we propose an efficient nonnegative sparse representation algorithm to find an approximation solution of sparse representation according to the l1 ball theory in sparse representation the approximated solution is unique and can be optimized efficiently then a filtering strategy is developed to avoid the computation of the sparse representation on the whole large scale dataset moreover theoretical analysis also gives the necessary condition for nonnegative least squares technique to find a sparse solution extensive experiments on several public databases have demonstrated that the proposed tsr approach in general achieves better classification accuracy than the state of the art sparse representation methods more importantly a significant reduction of computational costs is reached in comparison with sparse representation classifier this enables the tsr to be more suitable for robust face recognition on a large scale dataset

approximate explicit receding horizon control of constrained nonlinear systems an algorithm for the construction of an explicit piecewise linear state feedback approximation to nonlinear constrained receding horizon control is given it allows such controllers to be implemented via an efficient binary tree search avoiding real time optimization this is of significant benefit in applications that requires low real time computational complexity or low software complexity the method has a priori guarantee of asymptotic stability with region of attraction being a close inner approximation to the stabilizable set this is achieved by ensuring that the approximation error does not exceed the stability margin

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

a comparison of block pivoting and interior point algorithms for linear least squares problems with nonnegative variables in this paper we discuss the use of block principal pivoting and predictor corrector methods for the solution of large scale linear least squares problems with nonnegative variables nvlsq we also describe two implementations of these algorithms that are based on the normal equations and corrected seminormal equations csne approaches we show that the method of normal equations should be employed in the implementation of the predictor corrector algorithm this type of approach should also be used in the implementation of the block principal pivoting method but a switch to the csne method may be useful in the last iterations of the algorithm computational experience is also included in this paper and shows that both the predictor corrector and the block principal pivoting algorithms are quite efficient to deal with large scale nvlsq problems

dynamic programming for constrained optimal control of discrete time linear hybrid systems in this paper we study the solution to optimal control problems for constrained discrete time linear hybrid systems based on quadratic or linear performance criteria the aim of the paper is twofold first we give basic theoretical results on the structure of the optimal state feedback solution and of the value function second we describe how the state feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming

degeneracy in the presence of roundoff errors abstract a technique is described for resolving degeneracy in the simplex method for linear programming it is shown that this technique enables a guarantee of termination to be given not only for exact arithmetic but also for inexact arithmetic when roundoff errors are present the method is recursive and each level of recursion is obtained by localizing and dualizing the problem at a lower level the existence of a cost function at any level is exploited to provide the guarantee a number of interesting theoretical properties of the method are given data structures are described which enable the method to be implemented with very little overhead beyond what is normally required for the simplex method there is no difficulty in using any of the currently popular techniques for representing and updating matrix factors the technique is extended to handle degeneracy in l1 linear programming problems including linear l1 approximation and it is shown that all the above properties can be preserved some comparative numerical experiments are described

on the stability of a pexiderized gol ab schinzel equation in this paper we establish the stability of the pexiderized email protected b schinzel functional equation f x yf x g x h y under the condition that lim t 0h tx exists and is non zero

bounds on the density of states for schrodinger operators we establish bounds on the density of states measure for schrodinger operators these are deterministic results that do not require the existence of the density of states measure or equivalently of the integrated density of states the results are stated in terms of a density of states outer measure that always exists and provides an upper bound for the density of states measure when it exists we prove log holder continuity for this density of states outer measure in one two and three dimensions for schrodinger operators and in any dimension for discrete schrodinger operators

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

fokker planck equations for a free energy functional or markov process on a graph the classical fokker planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a euclidean space corresponding to a stochastic process there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy in recent years it has been shown that the fokker planck equation is the gradient flow of the free energy functional defined on the riemannian manifold of probability distributions whose inner product is generated by a 2 wasserstein distance in this paper we consider analogous matters for a free energy functional or markov process defined on a graph with a finite number of vertices and edges if n 2 is the number of vertices of the graph we show that the corresponding fokker planck equation is a system of nnonlinear ordinary differential equations defined on a riemannian manifold of probability distributions however in contrast to stochastic processes defined on euclidean spaces the situation is more subtle for discrete spaces we have different choices for inner products on the space of probability distributions resulting in different fokker planck equations for the same process it is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical fokker planck equation on euclidean spaces furthermore both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the riemannian manifolds of probability distributions with different metrics some examples are also discussed

existence and convergence of solutions for nonlinear biharmonic equations on graphs abstract in this paper we first prove some propositions of sobolev spaces defined on a locally finite graph g v e which are fundamental when dealing with equations on graphs under the variational framework then we consider a nonlinear biharmonic equation 2 u u a 1 u u p 2 u on g v e under some suitable assumptions we prove that for any 1 and p 2 the equation admits a ground state solution u moreover we prove that as the solutions u converge to a solution of the equation 2 u u u u p 2 u in u 0 on where x v a x 0 is the potential well and denotes the boundary of

technical communique evaluation of piecewise affine control via binary search tree we present an algorithm for generating a binary search tree that allows efficient computation of piecewise affine pwa functions defined on a polyhedral partition this is useful for pwa control approaches such as explicit model predictive control as it allows the controller to be implemented online with small computational effort the computation time is logarithmic in the number of regions in the pwa partition

christensen measurable solutions of some functional equation abstract let x be a separable f space over the field k of reals or complex numbers we characterize solutions of the equation f x m f x y f x f y in the class of functions m k k and christensen measurable functions f x k

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

a multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares model predictive control mpc is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables to circumvent the main drawback of mpc i e the need to solve a quadratic program qp on line to compute the control action explicit mpc was proposed in the past to precompute the control law off line using multiparametric qp mpqp the resulting form of the mpc law is piecewise affine which is extremely easy to code can be computed online by simple arithmetic operations and requires a maximum number of iterations that can be exactly determined a priori on the other hand the offline computations to solve the mpqp problem require detecting emptiness full dimensionality and minimal hyperplane representations of polyhedra and other computational geometric operations while most of the existing methods solve such operations via linear programming the approach proposed in this paper relies on a nonnegative least squares nnls solver that is very simple to code and fast to execute in addition the new approach exploits qp duality to identify and construct critical regions and to handle degeneracy issues

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

group action and shift compactness shift compactness has recently been found to be the foundation stone of classical as well as topological regular variation most recently it has come again to prominence in new proofs of the effros open mapping principle of group action another ingredient of topological regular variation using the real line under the euclidean and density topologies as a paradigm we develop group action versions of shift compactness theorems for baire groups acting on baire spaces under metrizable topologies and under certain refinements of these one aim is to pursue constructive approaches rather than rely on plain baire category methods so keeping more to the banach mazur strategic approach along the way we uncover three new coarse topologies for groups of homeomorphisms a second purpose is to establish limitations of the shift compactness methodology

brief paper fast large scale model predictive control by partial enumeration partial enumeration pe is presented as a method for treating large linear model predictive control applications that are out of reach with available mpc methods pe uses both a table storage method and online optimization to achieve this goal versions of pe are shown to be closed loop stable pe is applied to an industrial example with more than 250 states 32 inputs and a 25 sample control horizon the performance is less than 0 01 suboptimal with average speedup factors in the range of 80 220 and worst case speedups in the range of 4 9 39 2 compared to an existing mpc method small tables with only 25 200 entries were used to obtain this performance while full enumeration is intractable for this example

a weighted gram schmidt method for convex quadratic programming range space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases in this paper we describe a range space method based upon updating a weighted gram schmidt factorization of the constraints in the active set the updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active set strategy

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

energy efficient secure communication over decode and forward relay channels in this paper we raise the concern on energy efficient secure communication over a decode and forward relay network with potential eavesdroppers our objective is maximizing the secure energy efficiency ee which is defined as the ratio of secrecy rate to total power subject to the given individual power relay decoding rate and target secrecy rate constraints the problem is formulated as a unified mixed integer nonlinear optimization adapting to different assumptions of channel state information csi to deal with this combinatorial optimization a suboptimal solution scheme is developed upon some mathematical methods as decoupling of mixed integer programming fractional programming dual decomposition and difference of convex functions programming the core of the scheme is to transform the primal problem into simple subproblems step by step and then convex programming can be exploited finally the proposed scheme for the case with statistical eavesdropper s csi may be conditionally extended to the scenario with full csi in which a local optimal solution may be obtained in certain cases finally the performance and achievable secure ee of the proposed algorithm are demonstrated by simulations where the tradeoff between secure ee and secrecy rate is also revealed

a quadratic programming algorithm based on nonnegative least squares with applications to embedded model predictive control this technical note proposes an active set method based on nonnegative least squares nnls to solve strictly convex quadratic programming qp problems such as those that arise in model predictive control mpc the main idea is to rephrase the qp problem as a least distance problem ldp that is solved via a nnls reformulation while the method is rather general for solving strictly convex qp s subject to linear inequality constraints it is particularly useful for embedded mpc because i is very fast compared to other existing state of the art qp algorithms ii is very simple to code requiring only basic arithmetic operations for computing ldl t decompositions recursively to solve linear systems of equations iii contrary to iterative methods provides the solution or recognizes infeasibility in a finite number of steps

on approximation of approximately linear mappings by linear mappings abstract assume a is a normed linear space b is a banach space and f a b is a mapping approximately linear we solve the following ulam problem give conditions in order for a linear mapping near an approximately linear mapping to exist

approximate ternary jordan derivations on banach ternary algebras let a be a banach ternary algebra over a scalar field r or c and x be a ternary banach a module a linear mapping d a a x x is called a ternary jordan derivation if d xxx a d x xx x xd x x x xxd x x for all x a in this paper we investigate ternary jordan derivations on banach ternary algebras associated with the following functional equation f x y z 4 f 3x y 4z 4 f 4x 3z 4 2f x moreover we prove the generalized ulam hyers stability of ternary jordan derivations on banach ternary algebras

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

on the stability of the golab schinzel functional equation abstract motivated by the problem of r ger we consider the stability of the golab schinzel functional equation we show that in the class of functions satisfying some weak regularity assumption the equation is superstable

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

the non parametric sub pixel local point spread function estimation is a well posed problem most medium to high quality digital cameras dslrs acquire images at a spatial rate which is several times below the ideal nyquist rate for this reason only aliased versions of the cameral point spread function psf can be directly observed yet it can be recovered at a sub pixel resolution by a numerical method since the acquisition system is only locally stationary this psf estimation must be local this paper presents a theoretical study proving that the sub pixel psf estimation problem is well posed even with a single well chosen observation indeed theoretical bounds show that a near optimal accuracy can be achieved with a calibration pattern mimicking a bernoulli 0 5 random noise the physical realization of this psf estimation method is demonstrated in many comparative experiments we use an algorithm to accurately estimate the pattern position and its illumination conditions once this accurate registration is obtained the local psf can be directly computed by inverting a well conditioned linear system the psf estimates reach stringent accuracy levels with a relative error of the order of 2 to 5 to the best of our knowledge such a regularization free and model free sub pixel psf estimation scheme is the first of its kind

a numerically stable solver for positive semidefinite quadratic programs based on nonnegative least squares this paper proposes a new algorithm for solving convex quadratic programs qp subject to linear inequality and equality constraints the method extends an approach recently proposed by the author for solving strictly convex qp s using nonnegative least squares by making it numerically more robust and able to handle also the nonstrictly convex case equality constraints and warm starting from an initial guess robustness is achieved by introducing an outer proximal point iteration scheme that regularizes the problem without altering the solution and by adaptively weighting the least squares problems encountered while solving the problem the performance of the resulting qp solver in terms of speed and robustness in the single precision arithmetic is assessed against other optimization algorithms tailored to embedded model predictive control applications

the pexiderized golab schinzel functional equation let x be a linear space over a commutative field k we characterize a general solution f g h k x k of the pexiderized golab schinzel equation f x g x y h x k y as well as real continuous solutions of the equation

forces nlp an efficient implementation of interior point methods for multistage nonlinear nonconvex programs abstractreal time implementation of optimisation based control and trajectory planning can be very challenging for nonlinear systems as a result if an implementation based on a fixed linearisatio

wasserstein hamiltonian flows abstract we establish kinetic hamiltonian flows in density space embedded with the l 2 wasserstein metric tensor we derive the euler lagrange equation in density space which introduces the associated hamiltonian flows we demonstrate that many classical equations such as vlasov equation schrodinger equation and schrodinger bridge problem can be rewritten as the formalism of hamiltonian flows in density space