dual dynamic programing with cut selection convergence proof and numerical experiments abstract we consider convex optimization problems formulated using dynamic programing equations such problems can be solved using the dual dynamic programing algorithm combined with the level 1 cut selection strategy or the territory algorithm to select the most relevant benders cuts we propose a limited memory variant of level 1 and show the convergence of ddp combined with the territory algorithm level 1 or its variant for nonlinear optimization problems in the special case of linear programs we show convergence in a finite number of iterations numerical simulations illustrate the interest of our variant and show that it can be much quicker than a simplex algorithm on some large instances of portfolio selection and inventory problems

regularized decomposition of high dimensional multistage stochastic programs with markov uncertainty we develop a quadratic regularization approach for the solution of high dimensional multistage stochastic optimization problems characterized by a potentially large number of time periods stages e g hundreds a high dimensional resource state variable and a markov information process the resulting algorithms are shown to converge to an optimal policy after a finite number of iterations under mild technical assumptions computational experiments are conducted using the setting of optimizing energy storage over a large transmission grid which motivates both the spatial and temporal dimensions of our problem our numerical results indicate that the proposed methods exhibit significantly faster convergence than their classical counterparts with greater gains observed for higher dimensional problems

complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

weighted average importance sampling and defensive mixture distributions importance sampling uses observations from one distribution to estimate for another distribution by weighting the observations including the target distribution as one component of a mixture distribution bounds the weights and makes importance sampling more reliable the usual importance sampling estimate is a weighted average with weights that do not sum to 1 we discuss simple normalization and other more efficient normalization methods these innovations make importance sampling useful in a wider variety of problems we demonstrate with a case study of oil inventory reliability at a large utility

continuous procrustes distance between two surfaces the procrustes distance is used to quantify the similarity or dissimilarity of three dimensional shapes and extensively used in biological morphometrics typically each normalized shape is represented by n landmark points chosen to be homologous as far as possible and the procrustes distance is then computed as where the minimization is over all euclidean transformations and the correspondences are picked in an optimal way the discrete procrustes distance has the drawback that each shape is represented by only a finite number of points which may not capture all the geometric aspects of interest a need has been expressed for alternatives that are still easy to compute we propose in this paper the concept of continuous procrustes distance and prove that it provides a true metric for two dimensional surfaces embedded in three dimensions we also propose an efficient algorithm to calculate approximations to this new distance 2012 wiley periodicals inc

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

distributionally robust stochastic dual dynamic programming we consider a multistage stochastic linear program that lends itself to solution by stochastic dual dynamic programming sddp in this context we consider a distributionally robust variant of the

sampling based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures we define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it the approach is based on dynamic programming equations written for a risk averse formulation of the problem this formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures dual representations of such risk functionals are given and used to derive conditions of coherence in the one period case conditions for convexity and consistency with second order stochastic dominance are also provided the risk averse dynamic programming equations are specialized considering convex combinations of one period extended polyhedral risk measures such as spectral risk measures to implement the proposed policy the approximation of the risk averse recourse functions for stochastic linear programs is discussed in this context we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk averse p

stochastic lipschitz dynamic programming we propose a new algorithm for solving multistage stochastic mixed integer linear programming milp problems with complete continuous recourse in a similar way to cutting plane methods we construct nonlinear lipschitz cuts to build lower approximations for the non convex cost to go functions an example of such a class of cuts are those derived using augmented lagrangian duality for milps the family of lipschitz cuts we use is milp representable so that the introduction of these cuts does not change the class of the original stochastic optimization problem we illustrate the application of this algorithm on two case studies comparing our approach with the convex relaxation of the problems for which we can apply sddp and for a discretized approximation applying sddip

minimum cardinality non anticipativity constraint sets for multistage stochastic programming we consider multistage stochastic programs in which decisions can adapt over time i e at each stage in response to observation of one or more random variables uncertain parameters the case that the time at which each observation occurs is decision dependent known as stochastic programming with endogeneous observation of uncertainty presents particular challenges in handling non anticipativity although such stochastic programs can be tackled by using binary variables to model the time at which each endogenous uncertain parameter is observed the consequent conditional non anticipativity constraints form a very large class with cardinality in the order of the square of the number of scenarios however depending on the properties of the set of scenarios considered only very few of these constraints may be required for validity of the model here we characterize minimal sufficient sets of non anticipativity constraints and prove that their matroid structure enables sets of minimum cardinality to be found efficiently under general conditions on the structure of the scenario set

a continuous linear optimal transport approach for pattern analysis in image datasets we present a new approach to facilitate the application of the optimal transport metric to pattern recognition on image databases the method is based on a linearized version of the optimal transport metric which provides a linear embedding for the images hence it enables shape and appearance modeling using linear geometric analysis techniques in the embedded space in contrast to previous work we use monge s formulation of the optimal transport problem which allows for reasonably fast computation of the linearized optimal transport embedding for large images we demonstrate the application of the method to recover and visualize meaningful variations in a supervised learning setting on several image datasets including chromatin distribution in the nuclei of cells galaxy morphologies facial expressions and bird species identification we show that the new approach allows for high resolution construction of modes of variations and discrimination and can enhance classification accuracy in a variety of image discrimination problems highlightsa continuous version of the lot framework is described that bypasses many of the difficulties associated with the discrete formulation using continuous transport maps a forward and inverse transform operation is defined for images an improved reference average image estimation algorithm is proposed the range within which points in lot space are invertible according to the continuous formulation is calculated we show that our method significantly speeds up the computation of the lot embedding

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

approaches to multistage one shot decision making abstract in this research multistage one shot decision making under uncertainty is studied in such a decision problem a decision maker has one and only one chance to make a decision at each stage with possibilistic information based on the one shot decision theory approaches to multistage one shot decision making are proposed in the proposed approach a decision maker chooses one state amongst all the states according to his her attitude about satisfaction and possibility at each stage the payoff at each stage is associated with the focus points at the succeeding stages based on the selected states focus points the sequence of optimal decisions is determined by dynamic programming the proposed method is a fundamental alternative for multistage decision making under uncertainty because it is scenario based instead of lottery based as in the other existing methods the one shot optimal stopping problem is analyzed where a decision maker has only one chance to determine stopping or continuing at each stage the theoretical results have been obtained

algorithms to automatically quantify the geometric similarity of anatomical surfaces we describe approaches for distances between pairs of two dimensional surfaces embedded in three dimensional space that use local structures and global information contained in interstructure geometric relationships we present algorithms to automatically determine these distances as well as geometric correspondences this approach is motivated by the aspiration of students of natural science to understand the continuity of form that unites the diversity of life at present scientists using physical traits to study evolutionary relationships among living and extinct animals analyze data extracted from carefully defined anatomical correspondence points landmarks identifying and recording these landmarks is time consuming and can be done accurately only by trained morphologists this necessity renders these studies inaccessible to nonmorphologists and causes phenomics to lag behind genomics in elucidating evolutionary patterns unlike other algorithms presented for morphological correspondences our approach does not require any preliminary marking of special features or landmarks by the user it also differs from other seminal work in computational geometry in that our algorithms are polynomial in nature and thus faster making pairwise comparisons feasible for significantly larger numbers of digitized surfaces we illustrate our approach using three datasets representing teeth and different bones of primates and humans and show that it leads to highly accurate results

the earth mover s distance as a metric for image retrieval we investigate the properties of a metric between two distributions the earth mover s distance emd for content based image retrieval the emd is based on the minimal cost that must be paid to transform one distribution into the other in a precise sense and was first proposed for certain vision problems by peleg werman and rom for image retrieval we combine this idea with a representation scheme for distributions that is based on vector quantization this combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods the emd is based on a solution to the transportation problem from linear optimization for which efficient algorithms are available and also allows naturally for partial matching it is more robust than histogram matching techniques in that it can operate on variable length representations of the distributions that avoid quantization and other binning problems typical of histograms when used to compare distributions with the same overall mass the emd is a true metric in this paper we focus on applications to color and texture and we compare the retrieval performance of the emd with that of other distances

a sequential sampling procedure for stochastic programming we develop a sequential sampling procedure for a class of stochastic programs we assume that a sequence of feasible solutions with an optimal limit point is given as input to our procedure such a sequence can be generated by solving a series of sampling problems with increasing sample size or it can be found by any other viable method our procedure estimates the optimality gap of a candidate solution from this sequence if the point estimate of the optimality gap is sufficiently small according to our termination criterion then we stop otherwise we repeat with the next candidate solution from the sequence under an increased sample size we provide conditions under which this procedure i terminates with probability one and ii terminates with a solution that has a small optimality gap with a prespecified probability

optimal mass transport for registration and warping image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times in this paper we present a method for computing elastic registration and warping maps based on the monge kantorovich theory of optimal mass transport this mass transport method has a number of important characteristics first it is parameter free moreover it utilizes all of the grayscale data in both images places the two images on equal footing and is symmetrical the optimal mapping from image a to image b being the inverse of the optimal mapping from b to a the method does not require that landmarks be specified and the minimizer of the distance functional involved is uniques there are no other local minimizers finally optimal transport naturally takes into account changes in density that result from changes in area or volume although the optimal transport method is certainly not appropriate for all registration and warping problems this mass preservation property makes the monge kantorovich approach quite useful for an interesting class of warping problems as we show in this paper our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the l2 kantorovich wasserstein or earth mover s distance under a mass preservation constraint we show how this approach leads to practical algorithms and demonstrate our method with a number of examples including those from the medical field we also extend this method to take into account changes in intensity and show that it is well suited for applications such as image morphing

multistage stochastic decomposition a bridge between stochastic programming and approximate dynamic programming multistate stochastic programs pose some of the more challenging optimization problems because such models can become rather intractable in general it is important to design algorithms that can provide approximations which in the long run yield solutions that are arbitrarily close to an optimum in this paper we propose such a sequential sampling method which is applicable to multistage stochastic linear programs and we refer to it as the multistage stochastic decomposition msd algorithm this algorithm represents a dynamic extension of a regularized version of stochastic decomposition sd while the method allows general correlation structures specialized streamlined versions are also possible for special cases of stagewise independent and autoregressive processes commonly incorporated in stochastic programming as with its two stage counterpart the msd algorithm is shown to provide an asymptotically optimal solution with probability one as a by product of this study we also show that sd al

bounds on risk averse mixed integer multi stage stochastic programming problems with mean cvar abstract risk averse mixed integer multi stage stochastic programming forms a class of extremely challenging problems since the problem size grows exponentially with the number of stages the problem is non convex due to integrality restrictions and the objective function is nonlinear in general we propose a scenario tree decomposition approach namely group subproblem approach to obtain bounds for such problems with an objective of dynamic mean conditional value at risk mean cvar our approach does not require any special problem structure such as convexity and linearity therefore it can be applied to a wide range of problems we obtain lower bounds by using different convolution of mean cvar risk measures and different scenario partition strategies the upper bounds are obtained through the use of optimal solutions of group subproblems using these lower and upper bounds we propose a solution algorithm for risk averse mixed integer multi stage stochastic problems with mean cvar risk measures we test the performance of the proposed algorithm on a multi stage stochastic lot sizing problem and compare different choices of lower bounds and partition strategies comparison of the proposed algorithm to a commercial solver revealed that on the average the proposed algorithm yields 1 13 stronger bounds the commercial solver requires additional running time more than a factor of five on the average to reach the same optimality gap obtained by the proposed algorithm

risk neutral and risk averse stochastic dual dynamic programming method abstract in this paper we discuss risk neutral and risk averse approaches to multistage linear stochastic programming problems based on the stochastic dual dynamic programming sddp method we give a general description of the algorithm and present computational studies related to planning of the brazilian interconnected power system

analysis of stochastic dual dynamic programming method abstract in this paper we discuss statistical properties and convergence of the stochastic dual dynamic programming sddp method applied to multistage linear stochastic programming problems we assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original true distribution is generated and consequently the sddp algorithm is applied to the constructed sample average approximation saa problem then we proceed to analysis of the sddp solutions of the saa problem and their relations to solutions of the true problem finally we discuss an extension of the sddp method to a risk averse formulation of multistage stochastic programs we argue that the computational complexity of the corresponding sddp algorithm is almost the same as in the risk neutral case

dynamic stochastic approximation for multi stage stochastic optimization in this paper we consider multi stage stochastic optimization problems with convex objectives and conic constraints at each stage we present a new stochastic first order method namely the dynamic stochastic approximation dsa algorithm for solving these types of stochastic optimization problems we show that dsa can achieve an optimal mathcal o 1 epsilon 4 rate of convergence in terms of the total number of required scenarios when applied to a three stage stochastic optimization problem we further show that this rate of convergence can be improved to mathcal o 1 epsilon 2 when the objective function is strongly convex we also discuss variants of dsa for solving more general multi stage stochastic optimization problems with the number of stages t 3 the developed dsa algorithms only need to go through the scenario tree once in order to compute an epsilon solution of the multi stage stochastic optimization problem as a result the memory required by dsa only grows linearly with respect to the number of stages to the best of our knowledge this is the first time that stochastic approximation type methods are generalized for multi stage stochastic optimization with t ge 3

multi stage stochastic optimization applied to energy planning this paper presents a methodology for the solution of multistage stochastic optimization problems based on the approximation of the expected cost to go functions of stochastic dynamic programming by piecewise linear functions no state discretization is necessary and the combinatorial explosion with the number of states the well known curse of dimensionality of dynamic programming is avoided the piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to benders cuts in a stochastic multistage decomposition framework a case study of optimal stochastic scheduling for a 39 reservoir system is presented and discussed

discontinuous piecewise quadratic lyapunov functions for planar piecewise affine systems abstract for planar piecewise affine systems this paper proposes sufficient stability conditions based on discontinuous lyapunov functions the monotonicity condition for discontinuous functions at switching instants is presented based on the behavior of state trajectories on the switching surfaces first the stability conditions are derived for a typical multiple lyapunov function and then these conditions are formulated as a set of linear matrix inequalities for piecewise quadratic lyapunov functions the implementation of the proposed method is illustrated by an example

graph connection laplacian methods can be made robust to noise recently several data analytic techniques based on graph connection laplacian gcl ideas have appeared in the literature at this point the properties of these methods are starting to be understood in the setting where the data is observed without noise we study the impact of additive noise on these methods and show that they are remarkably robust as a by product of our analysis we propose modifications of the standard algorithms that increase their robustness to noise we illustrate our results in numerical simulations

conditional risk mappings we introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties in particular we prove a representation theorem for conditional risk mappings in terms of conditional expectations we also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings

revised cpa method to compute lyapunov functions for nonlinear systems the cpa method uses linear programming to compute continuous and piecewise affine lyapunov function for nonlinear systems with asymptotically stable equilibria in it was shown that the method always succeeds in computing a cpa lyapunov function for such a system the size of the domain of the computed cpa lyapunov function is only limited by the equilibrium s basin of attraction however for some systems an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori this is necessary if the equilibrium is not exponentially stable because the existence of a cpa lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in however if the equilibrium is exponentially stable then this was an artifact of the method in this paper we overcome this artifact by developing a revised cpa method we show that this revised method is always able to compute a cpa lyapunov function for a system with an exponentially stable equilibrium the only conditions on the system are that it is c and autonomous the domain of the cpa lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction whereas in a previous paper we have shown these results for planar systems in this paper we cover general n dimensional systems

polyhedral risk measures in stochastic programming we consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures thereby we place emphasis on dynamic models i e multistage stochastic programs with multiperiod risk measures in this context we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right hand side of the dynamic constraints dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence as examples of polyhedral risk measures we propose multiperiod extensions of the conditional value at risk

on solving multistage stochastic programs with coherent risk measures we consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized a general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy by computing an inner approximation to future cost functions we can evaluate an upper bound on the cost of an optimal policy and an outer approximation delivers a lower bound the approach we describe is particularly useful in sampling based algorithms and a numerical example is provided to show the efficacy of the methodology when used in conjunction with stochastic dual dynamic programming

efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form for hybrid systems described by switched linear difference equations linear threshold conditions automata and propositional logic conditions described in mixed logical dynamical form this note describes two algorithms for transforming such systems into an equivalent piecewise affine form where equivalent means that for the same initial conditions and input sequences the trajectories of the system are identical the proposed techniques exploit ideas from mixed integer programming and multiparametric programming

evaluating policies in risk averse multi stage stochastic programming we consider a risk averse multi stage stochastic program using conditional value at risk as the risk measure the underlying random process is assumed to be stage wise independent and a stochastic dual dynamic programming sddp algorithm is applied we discuss the poor performance of the standard upper bound estimator in the risk averse setting and propose a new approach based on importance sampling which yields improved upper bound estimators modest additional computational effort is required to use our new estimators our procedures allow for significant improvement in terms of controlling solution quality in sddp style algorithms in the risk averse setting we give computational results for multi stage asset allocation using a log normal distribution for the asset returns

conformal wasserstein distances comparing surfaces in polynomial time abstract we present a constructive approach to surface comparison realizable by a polynomial time algorithm we determine the similarity of two given surfaces by solving a mass transportation problem between their conformal densities this mass transportation problem differs from the standard case in that we require the solution to be invariant under global mobius transformations we present in detail the case where the surfaces to compare are disk like we also sketch how the approach can be generalized to other types of surfaces

asymptotic stability of piecewise affine systems with filippov solutions via discontinuous piecewise lyapunov functions asymptotic stability of continuous time piecewise affine systems defined over a polyhedral partition of the state space with possible discontinuous vector field on the boundaries is considered in the first part of the paper the feasible filippov solution concept is introduced by characterizing single mode caratheodory sliding mode and forward zeno behaviors then a global asymptotic stability result through a possibly discontinuous piecewise lyapunov function is presented the sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing unreachable and caratheodory boundaries it is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic lyapunov functions and using the cone copositivity approach several examples illustrate the theoretical arguments and the effectiveness of the stability result

optimal mass transport for shape matching and comparison surface based 3d shape analysis plays a fundamental role in computer vision and medical imaging this work proposes to use optimal mass transport map for shape matching and comparison focusing on two important applications including surface registration and shape space the computation of the optimal mass transport map is based on monge brenier theory in comparison to the conventional method based on monge kantorovich theory this method significantly improves the efficiency by reducing computational complexity from o n 2 to o n for surface registration problem one commonly used approach is to use conformal map to convert the shapes into some canonical space although conformal mappings have small angle distortions they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis this work proposes to compose the conformal map with the optimal mass transport map to get the unique area preserving map which is intrinsic to the riemannian metric unique and diffeomorphic for shape space study this work introduces a novel riemannian framework conformal wasserstein shape space by combing conformal geometry and optimal mass transport theory in our work all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping which pushes the area element on the surface to a probability measure on the disk the optimal mass transport provides a map from the shape space of all topological disks with metrics to the wasserstein space of the disk and the pullback wasserstein metric equips the shape space with a riemannian metric we validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach

convergence analysis of sampling based decomposition methods for risk averse multistage stochastic convex programs we consider a class of sampling based decomposition methods to solve risk averse multistage stochastic convex programs we prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions this formula can be used to obtain an efficient implementation of stochastic dual dynamic programming applied to convex nonlinear problems we prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds we also prove the almost sure convergence of these algorithms when applied to risk averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption the analysis is first done assuming the underlying stochastic process is interstage independent and discrete with a finite set of possible realizations at each stage we then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent

a parametrization of piecewise linear lyapunov functions via linear programming in this paper we present a parametrization of piecewise linear pwl lyapunov functions to this end we consider the class of all continuous pwl functions defined over a simplicial partition we take advantage of a recently developed high level canonical pwl hl cpwl representation which expresses the pwl function in a compact and closed form once the parametrization problem is properly stated we focus on its application to the stabiilty analysis of dynamic systems we consider uncertain non linear systems and extend the sector condition obtained by ohta et al in addition we propose a method of selecting an optimal candidate one of the main advantages of this approach is that the parametrization and choice of the lyapunov candidate as well as the stability analysis result in linear programming problems

complexity of stochastic dual dynamic programming stochastic dual dynamic programming is a cutting plane type algorithm for multi stage stochastic optimization originated about 30 years ago in spite of its popularity in practice there does not exist any analysis on the convergence rates of this method in this paper we first establish the number of iterations i e iteration complexity required by a basic dual dynamic programming method for solving single scenario multi stage optimization problems by introducing novel mathematical tools including the saturation of search points we then refine these basic tools and establish the iteration complexity for an explorative dual dynamic programing method proposed herein and the classic stochastic dual dynamic programming method for solving more general multi stage stochastic optimization problems under the standard stage wise independence assumption our results indicate that the complexity of these methods mildly increases with the number of stages t in fact linearly dependent on t for discounted problems therefore they are efficient for strategic decision making which involves a large number of stages but with a relatively small number of decision variables in each stage without explicitly discretizing the state and action spaces these methods might also be pertinent to the related reinforcement learning and stochastic control areas

passivity analysis and passification of discrete time hybrid systems for discrete time hybrid systems in piecewise affine or piece wise polynomial pwp form this note proposes sufficient passivity analysis and synthesis criteria based on the computation of piecewise quadratic or pwp storage functions by exploiting linear matrix inequality techniques and sum of squares decomposition methods passivity analysis and synthesis of passifying controllers can be carried out through standard semidefinite programming packages providing a tool particularly important for stability of interconnected heterogeneous dynamical systems

problem driven scenario generation an analytical approach for stochastic programs with tail risk measure scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program most approaches to scenario generation are distribution driven that is they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty on the other hand a problem driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty in this paper we propose an analytic approach to problem driven scenario generation this approach applies to stochastic programs where a tail risk measure such as conditional value at risk is applied to a loss function since tail risk measures only depend on the upper tail of a distribution standard methods of scenario generation which typically spread their scenarios evenly across the support of the random vector struggle to adequately represent tail risk our scenario generation approach works by targeting the construction of scenarios in areas of the distribution corresponding to the tails of the loss distributions we provide conditions under which our approach is consistent with sampling and as proof of concept demonstrate how our approach could be applied to two classes of problem namely network design and portfolio selection numerical tests on the portfolio selection problem demonstrate that our approach yields better and more stable solutions compared to standard monte carlo sampling

worst case expectation approach to optimization under uncertainty in this paper we discuss multistage programming with the data process subject to uncertainty we consider a situation where the data process can be naturally separated into two components one can be modeled as a random process with a specified probability distribution and the other one can be treated from a robust worst case point of view we formulate this in a time consistent way and derive the corresponding dynamic programming equations to solve the obtained multistage problem we develop a variant of the stochastic dual dynamic programming method we give a general description of the algorithm and present computational studies related to planning of the brazilian interconnected power system

improving the performance of stochastic dual dynamic programming this paper is concerned with tuning the stochastic dual dynamic programming algorithm to make it more computationally efficient we report the results of some computational experiments on a large scale hydrothermal scheduling model developed for brazil we find that the best improvements in computation time are obtained from an implementation that increases the number of scenarios in the forward pass with each iteration and selects cuts to be included in the stage problems in each iteration this gives an order of magnitude decrease in computation time with little change in solution quality presents two tree traversing strategies for sddp presents three cut selection algorithms to improve sddp performance considers the whole brazilian power system in our computational results significant reduction in computational time without compromising the policy

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

combining sampling based and scenario based nested benders decomposition methods application to stochastic dual dynamic programming nested benders decomposition is a widely used and accepted solution methodology for multi stage stochastic linear programming problems motivated by large scale applications in the context of hydro thermal scheduling in 1991 pereira and pinto introduced a sampling based variant of the benders decomposition method known as stochastic dual dynamic programming sddp in this paper we embed the sddp algorithm into the scenario tree framework essentially combining the nested benders decomposition method on trees with the sampling procedure of sddp this allows for the incorporation of different types of uncertainties in multi stage stochastic optimization while still maintaining an efficient solution algorithm we provide an illustration of the applicability of our method towards a least cost hydro thermal scheduling problem by examining an illustrative example combining both fuel cost with inflow uncertainty and by studying the panama power system incorporating both electricity demand and inflow uncertainties

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility

hyperbolic harmonic mapping for surface registration automatic computation of surface correspondence via harmonic map is an active research field in computer vision computer graphics and computational geometry it may help document and understand physical and biological phenomena and also has broad applications in biometrics medical imaging and motion capture industries although numerous studies have been devoted to harmonic map research limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints this work conquers this problem by changing the riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints the computational algorithms are based on ricci flow and nonlinear heat diffusion methods the approach is general and robust we employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications experimental results demonstrate that by changing the riemannian metric the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards

surface fluid registration of conformal representation application to detect disease burden and genetic influence on hippocampus in this paper we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1 forms inverse consistent surface fluid registration and multivariate tensor based morphometry mtbm first we conformally map a surface onto a planar rectangle space with holomorphic 1 forms second we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface third we align the feature image with a chosen template image via the fluid image registration algorithm which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization the inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images this alignment induces a corresponding deformation on the surface we tested the system on alzheimer s disease neuroimaging initiative adni baseline dataset to study ad symptoms on hippocampus in our system by modeling a hippocampus as a 3d parametric surface we nonlinearly registered each surface with a selected template surface then we used mtbm to analyze the morphometry difference between diagnostic groups experimental results show that the new system has better performance than two publicly available subcortical surface registration tools first and spharm we also analyzed the genetic influence of the apolipoprotein e 4 allele apoe4 which is considered as the most prevalent risk factor for ad our work successfully detected statistically significant difference between apoe4 carriers and non carriers in both patients of mild cognitive impairment mci and healthy control subjects the results show evidence that the apoe genotype may be associated with accelerated brain atrophy so that our work provides a new mri analysis tool that may help presymptomatic ad research

automatic alignment of genus zero surfaces a new algorithm is presented that provides a constructive way to conformally warp a triangular mesh of genus zero to a destination surface with minimal metric deformation as well as a means to compute automatically a measure of the geometric difference between two surfaces of genus zero the algorithm takes as input a pair of surfaces that are topological 2 spheres each surface given by a distinct triangulation the algorithm then constructs a map f between the two surfaces first each of the two triangular meshes is mapped to the unit sphere using a discrete conformal mapping algorithm the two mappings are then composed with a mobius transformation to generate the function f the mobius transformation is chosen by minimizing an energy that measures the distance of f from an isometry we illustrate our approach using several real life data sets we show first that the algorithm allows for accurate automatic and landmark free nonrigid registration of brain surfaces we then validate our approach by comparing shapes of proteins we provide numerical experiments to demonstrate that the distances computed with our algorithm between low resolution surface based representations of proteins are highly correlated with the corresponding distances computed between high resolution atomistic models for the same proteins

morphometry of anatomical shape complexes with dense deformations and sparse parameters we propose a generic method for the statistical analysis of collections of anatomical shape complexes namely sets of surfaces that were previously segmented and labeled in a group of subjects the method estimates an anatomical model the template complex that is representative of the population under study its shape reflects anatomical invariants within the dataset in addition the method automatically places control points near the most variable parts of the template complex vectors attached to these points are parameters of deformations of the ambient 3d space these deformations warp the template to each subject s complex in a way that preserves the organization of the anatomical structures multivariate statistical analysis is applied to these deformation parameters to test for group differences results of the statistical analysis are then expressed in terms of deformation patterns of the template complex and can be visualized and interpreted the user needs only to specify the topology of the template complex and the number of control points the method then automatically estimates the shape of the template complex the optimal position of control points and deformation parameters the proposed approach is completely generic with respect to any type of application and well adapted to efficient use in clinical studies in that it does not require point correspondence across surfaces and is robust to mesh imperfections such as holes spikes inconsistent orientation or irregular meshing the approach is illustrated with a neuroimaging study of down syndrome ds the results demonstrate that the complex of deep brain structures shows a statistically significant shape difference between control and ds subjects the deformation based modelingis able to classify subjects with very high specificity and sensitivity thus showing important generalization capability even given a low sample size we show that the results remain significant even if the number of control points and hence the dimension of variables in the statistical model are drastically reduced the analysis may even suggest that parsimonious models have an increased statistical performance the method has been implemented in the software deformetrica which is publicly available at www deformetrica org

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

two stage linear decision rules for multi stage stochastic programming multi stage stochastic linear programs mslps are notoriously hard to solve in general linear decision rules ldrs yield an approximation of an mslp by restricting the decisions at each stage to be an affine function of the observed uncertain parameters finding an optimal ldr is a static optimization problem that provides an upper bound on the optimal value of the mslp and under certain assumptions can be formulated as an explicit linear program similarly as proposed by kuhn et al math program 130 1 177 209 2011 a lower bound for an mslp can be obtained by restricting decisions in the dual of the mslp to follow an ldr we propose a new approximation approach for mslps two stage ldrs the idea is to require only the state variables in an mslp to follow an ldr which is sufficient to obtain an approximation of an mslp that is a two stage stochastic linear program 2slp we similarly propose to apply ldr only to a subset of the variables in the dual of the mslp which yields a 2slp approximation of the dual that provides a lower bound on the optimal value of the mslp although solving the corresponding 2slp approximations exactly is intractable in general we investigate how approximate solution approaches that have been developed for solving 2slp can be applied to solve these approximation problems and derive statistical upper and lower bounds on the optimal value of the mslp in addition to potentially yielding better policies and bounds this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static ldr approach a computational study on two example problems demonstrates that using a two stage ldr can yield significantly better primal policies and modestly better dual policies than using policies based on a static ldr

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

the tps direct transport a new method for transporting deformations in the size and shape space modern shape analysis allows the fine comparison of shape changes occurring between different objects very often the classic machineries of generalized procrustes analysis and principal component analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks however if size and shape data are structured in different groups thus constituting different morphological trajectories a data centering is needed if one wants to compare solely the deformation representing the trajectories to do that inter individual variation must be filtered out this maneuver is rarely applied in studies using simulated or real data a geometrical procedure named parallel transport that can be based on various connection types is necessary to perform such kind of data centering usually the levi civita connection is used for interpolation of curves in a riemannian space it can also be used to transport a deformation we demonstrate that this procedure does not preserve some important characters of the deformation even in the affine case we propose a novel procedure called tps direct transport which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools we recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study

lyapunov stability for piecewise affine systems via cone copositivity cone copositive piecewise quadratic lyapunov functions pwq lfs for the stability analysis of continuous time piecewise affine pwa systems are proposed the state space is assumed to be partitioned into a finite number of convex possibly unbounded polyhedra preliminary conditions on pwq functions for their sign in the polyhedra and continuity over the common boundaries are provided the sign of each quadratic function is studied by means of cone constrained matrix inequalities which are translated into linear matrix inequalities lmis via cone copositivity the continuity is guaranteed by adding equality constraints over the polyhedra intersections an asymptotic stability result for pwa systems is then obtained by finding a continuous pwq lf through the solution of a set of constrained lmis the effectiveness of the proposed approach is shown by analyzing an opinion dynamics model and two saturating control systems

a lyapunov method for stability analysis of piecewise affine systems over non invariant domains this paper analyses stability of discrete time piecewise affine systems defined on possibly non invariant domains taking into account the possible presence of multiple dynamics in each of the polytopic regions of the system an algorithm based on linear programming is proposed in order to prove exponential stability of the origin and to find a positively invariant estimate of its region of attraction the results are based on the definition of a piecewise affine lyapunov function which is in general discontinuous on the boundaries of the regions the proposed method is proven to lead to feasible solutions in a broader range of cases as compared to a previously proposed approach two numerical examples are shown among which a case where the proposed method is applied to a closed loop system to which model predictive control was applied without a priori guarantee of stability