solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization in this paper we consider the stochastic mathematical programs with linear complementarity constraints which include two kinds of models called here and now and lower level wait and see problems we present a combined smoothing implicit programming and penalty method for the problems with a finite sample space then we suggest a quasi monte carlo approximation method for solving a problem with continuous random variables a comprehensive convergence theory is included as well we further report numerical results with the so called picnic vender decision problem

performance limits in sensor localization in this paper we study the cramer rao lower bound crlb in single hop sensor localization using measurements derived from received signal strength rss time of arrival toa and bearing respectively from a novel perspective differently from the existing work we use a statistical sensor anchor geometry modeling method with the result that the trace of the associated crlb matrix as a scalar metric for performance limits of sensor localization becomes a random variable given a probability measure for the sensor anchor geometry the statistical properties of the metric are analyzed to demonstrate properties of sensor localization using the central limit theorems for u statistics we show that as the number of anchors increases the metric is asymptotically normal in the rss bearing case and converges to a random variable which is an affine transformation of a chi square random variable of degree 2 in the toa case we provide formulas quantitatively describing the relationship among the mean and standard deviation of the metric the number of the anchors the parameters of communication channels the noise statistics in measurements and the spatial distribution of the anchors these formulas though asymptotic in the number of the anchors in many cases turn out to be remarkably accurate in predicting performance limits even if the number is small simulations are carried out to confirm our results

euler characteristic of coherent sheaves on simplicial torics via the stanley reisner ring we combine work of cox on the total coordinate ring of a toric variety and results of eisenbud musta stillman and musta on cohomology of toric and monomial ideals to obtain a formula for computing ox d for a weil divisor d on a complete simplicial toric variety x the main point is to use alexander duality to pass from the toric irrelevant ideal which appears in the computation of ox d to the stanley reisner ideal of which is used in defining the chow ring of x

higher integrality conditions volumes and ehrhart polynomials abstract a polytope is integral if all of its vertices are lattice points the constant term of the ehrhart polynomial of an integral polytope is known to be 1 in previous work we showed that the coefficients of the ehrhart polynomial of a lattice face polytope are volumes of projections of the polytope we generalize both results by introducing a notion of k integral polytopes where 0 integral is equivalent to integral we show that the ehrhart polynomial of a k integral polytope p has the properties that the coefficients in degrees less than or equal to k are determined by a projection of p and the coefficients in higher degrees are determined by slices of p a key step of the proof is that under certain generality conditions the volume of a polytope is equal to the sum of volumes of slices of the polytope

an approximation scheme for a class of risk averse stochastic equilibrium problems we consider two models for stochastic equilibrium one based on the variational equilibrium of a generalized nash game and the other on the mixed complementarity formulation each agent in the market solves a single stage risk averse optimization problem with both here and now investment variables and production wait and see variables a shared constraint couples almost surely the wait and see decisions of all the agents an important characteristic of our approach is that the agents hedge risk in the objective functions on costs or profits of their optimization problems which has a clear economic interpretation this feature is obviously desirable but in the risk averse case it leads to variational inequalities with set valued operators a class of problems for which no established software is currently available to overcome this difficulty we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents problems such as average or conditional value at risk the smoothed variational inequalities can be tackled by the path solver for example the approximation scheme is shown to converge including the case when smoothed problems are solved approximately an interesting by product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem to assess the proposed technique numerical results are presented the first set of experiments is on randomly generated equilibrium problems for which we compare the proposed methodology with the standard smooth reformulation of average value at risk minimization using additional variables to rewrite the associated max function the second set of experiments deals with a part of the real life european gas network for which dantzig wolfe decomposition can be combined with the smoothing approach

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

the modified cramer rao bound in vector parameter estimation in this paper we extend the scalar modified cramer rao bound mcrb to the estimation of a vector of nonrandom parameters in the presence of nuisance parameters the resulting bound is denoted with the acronym mcrvb where v stands for vector as with the scalar bound the mcrvb is generally looser than the conventional crvb but the two bounds are shown to coincide in some situations of practical interest the mcrvb is applied to the joint estimation of carrier frequency phase and symbol epoch of a linearly modulated waveform corrupted by correlated impulsive noise encompassing white gaussian noise as a particular case wherein data symbols and noise power are regarded as nuisance parameters in this situation calculation of the conventional crvb is infeasible while application of the mcrvb leads to simple useful expressions with moderate analytical effort when specialized to the case of white gaussian noise the mcrvb yields results already available in the literature in fragmentary form and simplified contexts

smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications inspired by a recent work by alexander et al j bank finance 30 583 605 2006 which proposes a smoothing method to deal with nonsmoothness in a conditional value at risk problem we consider a smoothing scheme for a general class of nonsmooth stochastic problems assuming that a smoothed problem is solved by a sample average approximation method we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w p 1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem moreover under some metric regularity conditions we obtain an error bound on approximate stationary points the convergence result is applied to a conditional value at risk problem and an inventory control problem

stochastic variational inequalities residual minimization smoothing sample average approximations the stochastic variational inequality vi has been used widely in engineering and economics as an effective mathematical model for a number of equilibrium problems involving uncertain data this paper presents a new expected residual minimization erm formulation for a class of stochastic vi the objective of the erm formulation is lipschitz continuous and semismooth which helps us guarantee the existence of a solution and convergence of approximation methods we propose a globally convergent a s smoothing sample average approximation ssaa method to minimize the residual function this minimization problem is convex for the linear stochastic vi if the expected matrix is positive semidefinite we show that the erm problem and its ssaa problems have minimizers in a compact set and any cluster point of minimizers and stationary points of the ssaa problems is a minimizer and a stationary point of the erm problem a s our examples come from applications involving traffic flow problems we show that t

the grothendieck ring of varieties and piecewise isomorphisms let k0 var k be the grothendieck ring of algebraic varieties over a field k l et x y be two algebraic varieties over k which are piecewise isomorphic i e x and y admit finite partitions x1 xn y1 yn into locally closed subvarieties such that xi is iso morphic to yi for all i n then x y in k0 var k larsen and lunts ask whether the converse is true for characteristic zero and algebraically closedfield k we answer positively this question when dim x 1o rx is a smooth connected projective surface or if x contains only finitely many rational curves

on the variational equilibrium as a refinement of the generalized nash equilibrium we are concerned with a class of games in which the players strategy sets are coupled by a shared constraint a widely employed solution concept for such generalized nash games is the generalized nash equilibrium gne the variational equilibrium ve facchinei kanzow 2007 is a specific kind of gne characterized by the solution of the variational inequality formed from the common constraint and the mapping of the gradients of player objectives our contribution is a theory that provides sufficient conditions for ensuring that the existence of a gne implies the existence of a ve in such an instance the ve is said to be a refinement of the gne for certain games these conditions are shown to be necessary this theory rests on a result showing the equality of the brouwer degree of two suitably defined functions whose zeros are the gne and ve respectively this theory has a natural extension to the primal dual space of strategies and lagrange multipliers corresponding to the shared constraint our results unify some known results pertaining to such equilibria and provide mathematical substantiation for ideas that were known to be appealing to economic intuition

a class of dantzig wolfe type decomposition methods for variational inequality problems we consider a class of decomposition methods for variational inequalities which is related to the classical dantzig wolfe decomposition of linear programs our approach is rather general in that it can be used with certain types of set valued or nonmonotone operators as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single valued case also subproblems may be solved approximately convergence is established under reasonable assumptions we also report numerical experiments for computing variational equilibria of the game theoretic models of electricity markets our numerical results illustrate that the decomposition approach allows to solve large scale problem instances otherwise intractable if the widely used path solver is applied directly without decomposition

equivalence of variational inequality problems to unconstrained minimization in this paper we propose a class of merit functions for variational inequality problems vi through these merit functions the variational inequality problem is cast as unconstrained minimization problem we estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of vi

the polytope algebra let f be an ordered field and let p denote the family of all convex polytopes in the d dimensional vector space v over f the universal abelian group corresponding to the translation invariant valuations on p has generators p for p p with 0 satisfying the relations v p q p p q whenever p q p q p and t p t p for p p and t v with multiplication induced by m p q p q is almost a graded commutative algebra over f in that dr 0 r with 0 z r a vector space over f r 1 and r s r s r s 0 r 0 forr d the dilatation d p p for p p and f is such that x rx for x r and 0 negative dilatations arise from the euler map e p p f 1 dimf f the sum extending over all faces f of p since x rx for x r and 0 separating group homomorphisms for are the frame functionals which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions two isomorphisms on are described one related to cones of outer normal vectors and the other to the polytope groups obtained from by discarding polytopes of dimension less than d various applications of the polytope algebra are given including a theory of mixed polytopes which has implications for mixed valuations

epi convergent smoothing with applications to convex composite functions smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems in this note we synthesize and extend recent results due to beck and teboulle on infimal convolution smoothing for convex functions with those of x chen on gradient consistency for nonconvex functions we use epi convergence techniques to define a notion of epi smoothing that allows us to tap into the rich variational structure of the subdifferential calculus for nonsmooth nonconvex and nonfinite valued functions as an illustration of the versatility and range of epi smoothing techniques the results are applied to the general constrained optimization for which nonlinear programming is a special case

new polytope decompositions and euler maclaurin formulas for simple integral polytopes abstract we give new weighted decompositions for simple polytopes generalizing previous results of lawrence varchenko and brianchon gram we start with witten s non abelian localization principle in equivariant cohomology for the norm square of the moment map in the context of toric varieties to obtain a decomposition for delzant polytopes then by a purely combinatorial argument we show that this formula holds for any simple polytope as an application we study euler maclaurin formulas

two stage non cooperative games with risk averse players this paper formally introduces and studies a non cooperative multi agent game under uncertainty the well known nash equilibrium is employed as the solution concept of the game while there are several formulations of a stochastic nash equilibrium problem we focus mainly on a two stage setting of the game wherein each agent is risk averse and solves a rival parameterized stochastic program with quadratic recourse in such a game each agent takes deterministic actions in the first stage and recourse decisions in the second stage after the uncertainty is realized each agent s overall objective consists of a deterministic first stage component plus a second stage mean risk component defined by a coherent risk measure describing the agent s risk aversion we direct our analysis towards a broad class of quantile based risk measures and linear quadratic recourse functions for this class of non cooperative games under uncertainty the agents objective functions can be shown to be convex in their own decision variables provided that the deterministic component of these functions have the same convexity property nevertheless due to the non differentiability of the recourse functions the agents objective functions are at best directionally differentiable such non differentiability creates multiple challenges for the analysis and solution of the game two principal ones being 1 a stochastic multi valued variational inequality is needed to characterize a nash equilibrium provided that the players optimization problems are convex 2 one needs to be careful in the design of algorithms that require differentiability of the objectives moreover the resulting multi valued variational formulation cannot be expected to be of the monotone type in general the main contributions of this paper are as follows a prior to addressing the main problem of the paper we summarize several approaches that have existed in the literature to deal with uncertainty in a non cooperative game b we introduce a unified formulation of the two stage snep with risk averse players and convex quadratic recourse functions and highlight the technical challenges in dealing with this game c to handle the lack of smoothness we propose smoothing schemes and regularization that lead to differentiable approximations d to deal with non monotonicity we impose a generalized diagonal dominance condition on the players smoothed objective functions that facilitates the application and ensures the convergence of an iterative best response scheme e to handle the expectation operator we rely on known methods in stochastic programming that include sampling and approximation f we provide convergence results for various versions of the best response scheme particularly for the case of private recourse functions overall this paper lays the foundation for future research into the class of sneps that provides a constructive paradigm for modeling and solving competitive decision making problems with risk averse players facing uncertainty this paradigm is very much at an infancy stage of research and requires extensive treatment in order to meet its broad applications in many engineering and economics domains

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

diland an algorithm for distributed sensor localization with noisy distance measurements we present an algorithm for distributed sensor localization with noisy distance measurements diland that extends and makes the dlre more robust dlre is a distributed sensor localization algorithm in rm m 1 introduced in our previous work ieee trans signal process vol 57 no 5 pp 2000 2016 may 2009 diland operates when 1 the communication among the sensors is noisy 2 the communication links in the network may fail with a nonzero probability and 3 the measurements performed to compute distances among the sensors are corrupted with noise the sensors which do not know their locations lie in the convex hull of at least m 1 anchors nodes that know their own locations under minimal assumptions on the connectivity and triangulation of each sensor in the network we show that under the broad random phenomena described above diland converges almost surely a s to the exact sensor locations

an edgeworth expansion for symmetric statistics we consider asymptotically normal statistics which are symmetric functions of n i i d random variables for these statistics we prove the validity of an edgeworth expansion with remainder o n 1 under cramer s condition on the linear part of the statistic and moment assumptions for all parts of the statistic by means of a counterexample we show that it is generally not possible to obtain an edgeworth expansion with remainder o n 1 without imposing additional assumptions on the structure of the nonlinear part of the statistic

sensor network localization with imprecise distances an approach to formulate geometric relations among distances between nodes as equality constraints is introduced in this paper to study the localization problem with imprecise distance information in sensor networks these constraints can be further used to formulate optimization problems for distance estimation the optimization solutions correspond to a set of distances that are consistent with the fact that sensor nodes live in the same plane or 3d space as the anchor nodes these techniques serve as the foundation for most of the existing localization algorithms that depend on the sensors distances to anchors to compute each sensor s location 2006 elsevier b v all rights reserved

fundamental properties and performance of conventional bearings only target motion analysis this paper considers the problem of estimating the position and velocity of an object from noise corrupted bearing measurements obtained by a single moving observation platform the process is inherently nonlinear and exhibits unusual observability properties that are geometry dependent a maximum likelihood estimate mle of the target motion analysis solution is developed and its performance analyzed a comparison is drawn between the mle and two previously reported methods a nonlinear modified instrumental variable estimate miv and the pseudo linear estimate ple both the miv and ple are shown to derive from approximations to the nonlinear measurement equation and therefore share some common properties with the mle the limits on performance that can be expected from processing bearing data are detailed specifically for long range to baseline geometries approximate expressions for the cramer rao bound are derived extension of the results to the practical filters approximately predicts numerically observed behavior for less restrictive geometries bounds are presented incorporation of a target speed constraint on the mle results in a transition to a lower dimensional problem as noise level and range increases monte carlo experimental results are presented and the improvements realized by the mle techniques are evident

discrete approximation of two stage stochastic and distributionally robust linear complementarity problems in this paper we propose a discretization scheme for the two stage stochastic linear complementarity problem lcp where the underlying random data are continuously distributed under some moderate conditions we derive qualitative and quantitative convergence for the solutions obtained from solving the discretized two stage stochastic lcp slcp we explain how the discretized two stage slcp may be solved by the well known progressive hedging method phm moreover we extend the discussion by considering a two stage distributionally robust lcp drlcp with moment constraints and proposing a discretization scheme for the drlcp as an application we show how the slcp and drlcp models can be used to study equilibrium arising from two stage duopoly game where each player plans to set up its optimal capacity at present with anticipated competition for production in future

distributed sensor localization in random environments using minimal number of anchor nodes the paper introduces diloc a distributed iterative algorithm to locate m sensors with unknown locations in rm m ges 1 with respect to a minimal number of m 1 anchors with known locations the sensors and anchors nodes in the network exchange data with their neighbors only no centralized data processing or communication occurs nor is there a centralized fusion center to compute the sensors locations diloc uses the barycentric coordinates of a node with respect to its neighbors these coordinates are computed using the cayley menger determinants i e the determinants of matrices of internode distances we show convergence of diloc by associating with it an absorbing markov chain whose absorbing states are the states of the anchors we introduce a stochastic approximation version extending diloc to random environments i e when the communications among nodes is noisy the communication links among neighbors may fail at random times and the internodes distances are subject to errors we show a s convergence of the modified diloc and characterize the error between the true values of the sensors locations and their final estimates given by diloc numerical studies illustrate diloc under a variety of deterministic and random operating conditions

optimality analysis of sensor target localization geometries the problem of target localization involves estimating the position of a target from multiple noisy sensor measurements it is well known that the relative sensor target geometry can significantly affect the performance of any particular localization algorithm the localization performance can be explicitly characterized by certain measures for example by the cramer rao lower bound which is equal to the inverse fisher information matrix on the estimator variance in addition the cramer rao lower bound is commonly used to generate a so called uncertainty ellipse which characterizes the spatial variance distribution of an efficient estimate i e an estimate which achieves the lower bound the aim of this work is to identify those relative sensor target geometries which result in a measure of the uncertainty ellipse being minimized deeming such sensor target geometries to be optimal with respect to the chosen measure the optimal sensor target geometries for range only time of arrival based and bearing only localization are identified and studied in this work the optimal geometries for an arbitrary number of sensors are identified and it is shown that an optimal sensor target configuration is not in general unique the importance of understanding the influence of the sensor target geometry on the potential localization performance is highlighted via formal analytical results and a number of illustrative examples

products of cycles and the todd class of a toric variety the purpose of this paper is to show that the todd class of a simplicial toric variety has a canonical expression in terms of products of torus invariant divisors the coefficients in this expression which are generalizations of the classical dedekind sum are shown to satisfy a reciprocity relation which characterizes them uniquely we achieve these results by giving an explicit formula for the push forward of a product of cycles under a proper birational map of simplicial toric varieties since the introduction of toric varieties in the 1970s finding formulas for their todd class has been an interesting and important problem this is partly due to a well known application of the riemann roch theorem which allows a formula for the todd class of a toric variety to be translated directly into a formula for the number of lattice points in a lattice polytope cf dan an example of this application is contained in pom where a formula for the todd class of a toric variety in terms of dedekind sums is used to obtain new lattice point formulas danilov dan posed the problem of finding a formula for the todd class of a toric variety in terms of the orbit closures under the torus action specifically he asked if it is possible given a lattice to assign a rational number to each cone in the lattice such that given any fan in the lattice the todd class of the corresponding toric variety equals the sum of the orbit closures with coefficients given by these assigned rational numbers morelli mor showed that such an assignment is indeed possible in a natural way if the coefficients instead of being rational numbers are allowed to take values in the field of rational functions on a grassmannian of linear subspaces of the lattice however if it is required that the coefficients be rational numbers invariant under lattice automorphisms such an assignment is clearly impossible for example the nonsingular cone a in 22 generated by 1 0 and 0 1 when subdivided by the ray through 1 1 yields two cones a and o2 which are both lattice equivalent to a by additivity a consequence of the fact that the todd class pushes forward we deduce that the coefficient assigned to a must equal 0 which is absurd in this paper we show that there is a canonical expression for the todd class of a simplicial toric variety in terms of products of the torus invariant divisors furthermore this expression is invariant under lattice automorphisms that is the coefficient of each product depends only on the set of rays with multiplicities

sequential localization of sensor networks the sensor network localization problem with distance information is to determine the positions of all sensors in a network given the positions of some of the sensors and the distances between some pairs of sensors a definition is given for a sensor network in the plane to be sequentially localizable it is shown that the graph of a sequentially localizable network must have a bilateration ordering and a polynomial time algorithm is given for deciding whether or not a network s graph has such an ordering a provably correct algorithm is given which consists of solving a sequence of quadratic equations and it is shown that the algorithm can localize any localizable network in the plane whose graph has a bilateration ordering

benders decomposition for multi stage stochastic mixed complementarity problems applied to a global natural gas market model abstract this paper presents and implements a benders decomposition type of algorithm for large scale stochastic multi period mixed complementarity problems the algorithm is applied to various multi stage natural gas market models accounting for market power exertion by traders due to the non optimization nature of the natural gas market problem a straightforward implementation of the traditional benders decomposition is not possible the master and subproblems can be derived from the underlying optimization problems and transformed into complementarity problems however to complete the master problems optimality cuts are added using the variational inequality based method developed in gabriel and fuller 2010 in this manner an alternative derivation of benders decomposition for stochastic mcp is presented thereby making this approach more applicable to a broader audience the algorithm can successfully solve problems with up to 256 scenarios and more than 600 thousand variables and problems with over 117 thousand variables with more than two thousand first stage capacity expansion variables the algorithm is efficient for solving two stage problems the computational time reduction for other stochastic problems is considerable and would be even larger if a parallel implementation of the algorithm were used the paper concludes with a discussion of infrastructure expansion results illustrating the impact of hedging on investment timing and optimal capacity sizes

solving stochastic complementarity problems in energy market modeling using scenario reduction in this paper we analyze market equilibrium models with random aspects that lead to stochastic complementarity problems while the models presented depict energy markets the results are believed to be applicable to more general stochastic complementarity problems the contribution is the development of new heuristic scenario reduction approaches that iteratively work towards solving the full extensive form stochastic market model the methods are tested on three representative models and supporting numerical results are provided as well as derived mathematical bounds

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

sum integral interpolators and the euler maclaurin formula for polytopes a local lattice point counting formula and more generally a local euler maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space v namely the family of exponential sums s and the family of exponential integrals i parametrized by the set of rational polytopes in v the paper introduces the notion of an interpolator between these two families of meromorphic functions we prove that every rigid complement map in v gives rise to an effectively computable si interpolator and a local euler maclaurin formula an is interpolator and a reverse local euler maclaurin formula and an is 0 interpolator which interpolates between integrals and sums over interior lattice points rigid complement maps can be constructed by choosing an inner product on v or by choosing a complete flag in v the corresponding interpolators generalize and unify the work of berline vergne pommersheim thomas and morelli

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

a radar application of a modified cramer rao bound parameter estimation in non gaussian clutter in this paper we derive a lower bound on the error covariance matrix for any unbiased estimator of the parameters of a signal composed of a mixture of spherically invariant random processes sirps the proposed approach represents a special case of the global cramer rao bound for hybrid random and deterministic parameters estimation and it is particularly useful when the data conditioned on a vector of unwanted random parameters nuisance parameters with a priori known probability density function can be modeled as a gaussian vector the case of signal composed of a mixture of k distributed clutter gaussian clutter and thermal noise belongs to this set and it is regarded as a realistic radar scenario in the radar problem considered here this bound can be numerically computed in closed form whereas the computation of the true marginal cramer rao bound turns out to be infeasible the performance of some practical estimators are compared with it for two study cases

linear least squares approach for accurate received signal strength based source localization a conventional approach for passive source localization is to utilize signal strength measurements of the emitted source received at an array of spatially separated sensors the received signal strength rss information can be converted to distance estimates for constructing a set of circular equations from which the target position is determined nevertheless a major challenge in this approach lies in the shadow fading effect which corresponds to multiplicative measurement errors by utilizing the mean and variance of the squared distance estimates we devise two linear least squares lls estimators for rss based positioning in this paper the first one is a best linear unbiased estimator while the second is its improved version by exploiting the known relation between the parameter estimates the variances of the position estimates are derived and confirmed by computer simulations in particular it is proved that the performance of the improved lls estimator achieves cramer rao lower bound at sufficiently small noise conditions