embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

improved parameter inference in catchment models 1 evaluating parameter uncertainty a bayesian methodology is developed to evaluate parameter uncertainty in catchment models fitted to a hydrologic response such as runoff the goal being to improve the chance of successful regionalization the catchment model is posed as a nonlinear regression model with stochastic errors possibly being both autocorrelated and heteroscedastic the end result of this methodology which may use box cox power transformations and arma error models is the posterior distribution which summarizes what is known about the catchment model parameters this can be simplified to a multivariate normal provided a linearization in parameter space is acceptable means of checking and improving this assumption are discussed the posterior standard deviations give a direct measure of parameter uncertainty and study of the posterior correlation matrix can indicate what kinds of data are required to improve the precision of poorly determined parameters finally a case study involving a nine parameter catchment model fitted to monthly runoff and soil moisture data is presented it is shown that use of ordinary least squares when its underlying error assumptions are violated gives an erroneous description of parameter uncertainty

probabilistic optimization for conceptual rainfall runoff models a comparison of the shuffled complex evolution and simulated annealing algorithms automatic optimization algorithms are used routinely to calibrate conceptual rainfall runoff crr models the goal of calibration is to estimate a feasible and unique global set of parameter estimates that best fit the observed runoff data most if not all optimization algorithms have difficulty in locating the global optimum because of response surfaces that contain multiple local optima with regions of attraction of differing size discontinuities and long ridges and valleys extensive research has been undertaken to develop efficient and robust global optimization algorithms over the last 10 years this study compares the performance of two probabilistic global optimization methods the shuffled complex evolution algorithm sce ua and the three phase simulated annealing algorithm sa sx both algorithms are used to calibrate two parameter sets of a modified version of boughton s 1984 sfb model using data from two australian catchments that have low and high runoff yields for the reduced well identified parameter set the algorithms have a similar efficiency for the low yielding catchment but sce ua is almost twice as robust although the robustness of the algorithms is similar for the high yielding catchment sce ua is six times more efficient than sa sx when fitting the full parameter set the performance of sa sx deteriorated markedly for both catchments these results indicated that sce ua s use of multiple complexes and shuffling provided a more effective search of the parameter space than sa sx s single simplex with stochastic step acceptance criterion especially when the level of parameterization is increased examination of the response surface for the low yielding catchment revealed some reasons why sce ua outperformed sa sx and why probabilistic optimization algorithms can experience difficulty in locating the global optimum

bayesian spectral likelihood for hydrological parameter inference this paper proposes a spectral domain likelihood function for the bayesian estimation of hydrological model parameters from a time series of model residuals the spectral domain error model is based on the power density spectrum pds of the stochastic process assumed to describe residual errors the bayesian spectral domain likelihood bsl is mathematically equivalent to the corresponding bayesian time domain likelihood btl and yields the same inference when all residual error assumptions are satisfied and all residual error parameters are inferred however the bsl likelihood function does not depend on the residual error distribution in the original time domain which offers a theoretical advantage in terms of robustness for hydrological parameter inference the theoretical properties of bsl are demonstrated and compared to btl and a previously proposed spectral likelihood by montanari and toth 2007 using a set of synthetic case studies and a real case study based on the leaf river catchment in the u s the empirical analyses confirm the theoretical properties of bsl when applied to heteroscedastic and autocorrelated error models where heteroscedasticity is represented using the log transformation and autocorrelation is represented using an ar 1 process unlike mtl the use of bsl did not introduce additional parametric uncertainty compared to btl future work will explore the application of bsl to challenging modeling scenarios in arid catchments and indirect calibration with nonconcomitant input output time series

a branch and cut algorithm without binary variables for nonconvex piecewise linear optimization we give a branch and cut algorithm for solving linear programs lps with continuous separable piecewise linear cost functions plfs models for plfs use continuous variables in special ordered sets of type 2 sos2 traditionally sos2 constraints are enforced by introducing auxiliary binary variables and other linear constraints on them alternatively we can enforce sos2 constraints by branching on them thus dispensing with auxiliary binary variables we explore this approach further by studying the inequality description of the convex hull of the feasible set of lps with plfs in the space of the continuous variables and using the new cuts in a branch and cut scheme without auxiliary binary variables we give two families of valid inequalities the first family is obtained by lifting the convexity constraints the second family consists of lifted cover inequalities finally we report computational results that demonstrate the effectiveness of our cuts and that branch and cut without auxiliary binary variables is significantly more practical than the traditional mixed integer programming approach

a new evolutionary search strategy for global optimization of high dimensional problems global optimization of high dimensional problems in practical applications remains a major challenge to the research community of evolutionary computation the weakness of randomization based evolutionary algorithms in searching high dimensional spaces is demonstrated in this paper a new strategy sp uci is developed to treat complexity caused by high dimensionalities this strategy features a slope based searching kernel and a scheme of maintaining the particle population s capability of searching over the full search space examinations of this strategy on a suite of sophisticated composition benchmark functions demonstrate that sp uci surpasses two popular algorithms particle swarm optimizer pso and differential evolution de on high dimensional problems experimental results also corroborate the argument that in high dimensional optimization only problems with well formative fitness landscapes are solvable and slope based schemes are preferable to randomization based ones

uniqueness and observability of conceptual rainfall runoff model parameters the percolation process examined many researchers have expressed concerns regarding the uniqueness of parameter estimates for conceptual rainfall runoff r r models obtained through calibration recent studies sorooshian et al this issue sorooshian and gupta this issue have revealed that even though stochastic parameter estimation techniques can help the problems are not all due to inefficiencies in the calibration techniques used but are caused by the manner in which the model is structurally formulated thus even when calibrated under ideal conditions simulation studies it is often impossible to obtain unique estimates for the parameters it is possible to resolve this problem at least in part by appropriate reparameterizations of the pertinent model equations in this paper the percolation equation of the soil moisture accounting model of the national weather service river forecast system sma nwsrfs will be discussed it is shown that a logical reparameterization of this equation can result in conditions that improve the chances of obtaining unique parameter estimates it is believed that these results have implications for other conceptual r r models in which similar approaches are used in the representation of the percolation infiltration process

effective and efficient global optimization for conceptual rainfall runoff models the successful application of a conceptual rainfall runoff crr model depends on how well it is calibrated despite the popularity of crr models reports in the literature indicate that it is typically difficult if not impossible to obtain unique optimal values for their parameters using automatic calibration methods unless the best set of parameters associated with a given calibration data set can be found it is difficult to determine how sensitive the parameter estimates and hence the model forecasts are to factors such as input and output data error model error quantity and quality of data objective function used and so on results are presented that establish clearly the nature of the multiple optima problem for the research crr model sixpar these results suggest that the crr model optimization problem is more difficult than had been previously thought and that currently used local search procedures have a very low probability of successfully finding the optimal parameter sets next the performance of three existing global search procedures are evaluated on the model sixpar finally a powerful new global optimization procedure is presented entitled the shuffled complex evolution sce ua method which was able to consistently locate the global optimum of the sixpar model and appears to be capable of efficiently and effectively solving the crr model optimization problem

improving the shuffled complex evolution scheme for optimization of complex nonlinear hydrological systems application to the calibration of the sacramento soil moisture accounting model 1 an innovative algorithm shuffled complexes with principal components analysis sp uci is developed to overcome a critical deficiency of the shuffled complex evolution scheme population degeneration population degeneration means that during the evolutionary search process the population of search particles may degenerate into a subspace of the full parameter space thereby missing the capacity of fully exploring the parameter space being confined in a subspace may even lead the particle population to converge to nonstationary points which is a fatal malfunction to overcome this problem sp uci employs the principal components analysis to detect the occurrence of population degeneration and remedy the adverse effects the ensemble of calibrations of the sacramento soil moisture accounting model with the sp uci method over the leaf river basin mississippi retrieves the optimal parameter values with the lowest recorded root mean squared error of simulated daily runoff against the observation moreover the result also provides consistent narrow ranges model parameter distribution which results in a better understanding of the model s behavior given the watershed s hydrologic features

a monte carlo algorithm for free and coaxial ring extremal states of the vortex n body problem on a sphere abstract we show that the search for statistical equilibria at very low positive temperatures using a monte carlo algorithm can successfully locate dynamical equilibria of the n vortex problem on a sphere numerical results are collected to show that for a wide range of particle numbers this algorithm accurately and efficiently locates the ground state or lowest energy equilibrium the extremal states found numerically are carefully compared with well known exact configurations such as the regular polyhedra using an essential tool called the radial distribution function we state a theorem that is useful for comparing n vortex configurations which are related to one another by elements of the group o 3 it is found that by constraining the system to equally spaced latitudinal rings of vortices the computational cost may be reduced by an order of magnitude many of the results reported here apply directly to other n body problems on a sphere such as the distribution of n charges on a sphere

evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall runoff models influence of calibration data variability and length on model credibility the success of an automatic calibration procedure is highly dependent on the choice of the objective function and the nature quantity and quality of the data used the objective function should be selected on the basis of the stochastic properties of the errors present in the data and in the model also the data should be chosen so as to contain as much valuable information about the process as possible in this paper we compare the performance of two maximum likelihood estimators the amle which assumes the presence of first lag autocorrelated homogeneous variance errors and the hmle which assumes the presence of uncorrelated inhomogeneous variance errors to the commonly used simple least squares criterion sls the model calibrated was the soil moisture accounting model of the u s national weather service s river forecast system sma nwsrfs the results indicate that a properly chosen objective function can enhance the possibility of obtaining unique and conceptually realistic parameter estimates furthermore the sensitivity of the estimation results to various characteristics of the calibration data such as hydrologic variability and length are substantially reduced

a comparison of mixed integer programming models for nonconvex piecewise linear cost minimization problems we study a generic minimization problem with separable nonconvex piecewise linear costs showing that the linear programming lp relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope we also show a relationship between this result and classical lagrangian duality theory

models for the piecewise linear unsplittable multicommodity flow problems abstract in this paper we consider multicommodity flow problems with unsplittable flows and piecewise linear routing costs we first focus on the case where the piecewise linear routing costs are convex we show that this problem is np hard for the general case but polynomially solvable when there is only one commodity we then propose a strengthened mixed integer programming formulation for the problem we show that the linear relaxation of this formulation always gives the optimal solution of the problem for the single commodity case we present a wide array of computational experiments showing this formulation also produces very tight linear programming bounds for the multi commodity case finally we also adapt our formulation for the non convex case our experimental results imply that the linear programming bounds for this case are only slightly weaker than the ones of state of the art models for the splittable flow version of the problem

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

a branch cut and price algorithm for the piecewise linear transportation problem abstract in this paper we present an exact solution method for the transportation problem with piecewise linear costs this problem is fundamental within supply chain management and is a straightforward extension of the fixed charge transportation problem we consider two dantzig wolfe reformulations and investigate their relative strength with respect to the linear programming lp relaxation both theoretical and practical through tests on a number of instances based on one of the proposed formulations we derive an exact method by branching and adding generalized upper bound constraints from violated cover inequalities the proposed solution method is tested on a set of randomly generated instances and compares favorably to solving the model using a standard formulation solved by a state of the art commercial solver

comparison of newton type and direct search algorithms for calibration of conceptual rainfall runoff models an examination of the calibration aspect of conceptual rainfall runoff models was undertaken using the sacramento soil moisture accounting model and a study comparing the performance of a newton type optimization algorithm with that of a direct search algorithm results indicate that the direct search algorithm is the more robust of the two because the newton type algorithm is more susceptible to poor conditioning of the response surface graphical studies of the response surface of the model s parameter space confirmed the presence of discontinuities and a rough textured surface particularly in the derivatives

calibration of rainfall runoff models application of global optimization to the sacramento soil moisture accounting model conceptual rainfall runoff models are difficult to calibrate by means of automatic methods one major reason for this is the inability of conventional procedures to locate the globally optimal set of parameters this paper investigates the consistency with which two global optimization methods the shuffled complex evolution sce ua method developed by the authors and the multistart simplex msx method are able to find the optimal parameter set during calibration of the sacramento soil moisture accounting model sac sma of the national weather service river forecast system nwsrfs in the first phase of this study error free synthetic data are used to conduct a comparative evaluation of the algorithms under ideal conditions in 10 independent trials of each algorithm in which 13 parameters of the sac sma model were optimized simultaneously the sce ua method achieved a 100 success rate in locating the precise global optimum i e the true parameter values while the msx method failed in all trials even with more than twice the number of function evaluations in the second phase historical data from the leaf river watershed are used to conduct a comparative evaluation of the algorithms under real conditions using two different estimation criteria drms and hmle the sce ua algorithm obtained consistently lower function values and more closely grouped parameter estimates while using one third fewer function evaluations than the msx algorithm

towards compatible triangulations we state the following conjecture any two planar n point sets that agree on the number of convex hull points can be triangulated in a compatible manner i e such that the resulting two triangulations are topologically equivalent we first describe a class of point sets which can be triangulated compatibly with any other set that satisfies the obvious size and shape restrictions the conjecture is then proved true for point sets with at most three interior points finally we demonstrate that adding a small number of extraneous points the number of interior points minus three always allows for compatible triangulations the linear bound extends to point sets of arbitrary size and shape

multiobjective calibration with pareto preference ordering an application to rainfall runoff model calibration 1 automatic calibration routines for hydrologic models with multiple objective capabilities are becoming increasingly popular due to advances in computational power population based optimization techniques and the recognition that a single performance measure such as the root mean square error is no longer sufficient to characterize the complex behavior of the catchment however as more objective functions are included in the calibration the number of pareto optimal solutions as well as the number of near pareto optimal parameter sets increases the calibration problem quickly becomes a decision making problem in the practical sense users of automatic calibration routines have to face the task of selecting a set of suitable model parameters from the numerous pareto optimal sets a new method of automatic calibration is proposed which combines an effective optimization routine based on multiobjective genetic algorithms and pareto preference ordering in this case pareto optimal points that are also pareto optimal in different subspace combinations of the objective functions space are preferred the proposed method is used to calibrate the mike11 nam rainfall runoff model for a danish catchment the results indicated that the method is able to sieve through the numerous pareto optimal solutions and select a small number of preferred solutions this is extremely useful to modelers who typically are required to provide the best estimated parameter sets with good overall model performance

hydrologic consistency as a basis for assessing complexity of monthly water balance models for the continental united states methods to select parsimonious and hydrologically consistent model structures are useful for evaluating dominance of hydrologic processes and representativeness of data while information criteria appropriately constrained to obey underlying statistical assumptions can provide a basis for evaluating appropriate model complexity it is not sufficient to rely upon the principle of maximum likelihood ml alone we suggest that one must also call upon a principle of hydrologic consistency meaning that selected ml structures and parameter estimates must be constrained as well as possible to reproduce desired hydrological characteristics of the processes under investigation this argument is demonstrated in the context of evaluating the suitability of candidate model structures for lumped water balance modeling across the continental united states using data from 307 snow free catchments the models are constrained to satisfy several tests of hydrologic consistency a flow space transformation is used to ensure better consistency with underlying statistical assumptions and information criteria are used to evaluate model complexity relative to the data the results clearly demonstrate that the principle of consistency provides a sensible basis for guiding selection of model structures and indicate strong spatial persistence of certain model structures across the continental united states further work to untangle reasons for model structure predominance can help to relate conceptual model structures to physical characteristics of the catchments facilitating the task of prediction in ungaged basins

an enhanced logarithmic method for signomial programming with discrete variables abstract signomial programming problems with discrete variables spd appear widely in real life applications but they are hard to solve this paper proposes an enhanced logarithmic method to reformulate the spd problem as a mixed 0 1 linear program milp with a minimum number of binary variables and inequality constraints both of the theoretical analysis and numerical results strongly support its superior performance to other state of the art linearization methods we also extend the proposed method to linearize some more complicated problems involving product and fractional terms in discrete and continuous variables

global optimization of mixed integer quadratically constrained quadratic programs miqcqp through piecewise linear and edge concave relaxations we propose a deterministic global optimization approach whose novel contributions are rooted in the edge concave and piecewise linear underestimators to address nonconvex mixed integer quadratically constrained quadratic programs miqcqp to epsilon global optimality the facets of low dimensional n 3 edge concave aggregations dominating the termwise relaxation of miqcqp are introduced at every node of a branch and bound tree concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge concave aggregations are addressed through piecewise linear relaxations extensive computational studies are presented for point packing problems standard and generalized pooling problems and examples from globallib meeraus globallib http www gamsworld org global globallib htm

a new selection metric for multiobjective hydrologic model calibration a novel selection metric called convex hull contribution chc is introduced for solving multiobjective mo optimization problems with pareto fronts that can be accurately approximated by a convex curve the hydrologic model calibration literature shows that many biobjective calibration problems with a proper setup result in such pareto fronts the chc selection approach identifies a subset of archived nondominated solutions whose map in the objective space forms convex approximation of the pareto front the optimization algorithm can sample solely from these solutions to more accurately approximate the convex shape of the pareto front it is empirically demonstrated that chc improves the performance of pareto archived dynamically dimensioned search pa dds when solving mo problems with convex pareto fronts this conclusion is based on the results of several benchmark mathematical problems and several hydrologic model calibration problems with two or three objective functions the impact of chc on pa dds performance is most evident when the computational budget is somewhat limited it is also demonstrated that 1 000 solution evaluations limited budget in this study is sufficient for pa dds with chc based selection to achieve very high quality calibration results relative to the results achieved after 10 000 solution evaluations

a computational analysis of multidimensional piecewise linear models with applications to oil production optimization production optimization of gas lifted oil wells under facility routing and pressure constraints is a challenging problem which has attracted the interest of operations engineers aiming to drive economic gains and scientists for its inherent complexity the hardness of this problem rests on the non linear characteristics of the multidimensional well production and pressure drop functions as well as the discrete routing decisions to this end this work develops several formulations in mixed integer linear programming milp using multidimensional piecewise linear models to approximate the non linear functions with domains spliced in hypercubes and simplexes computational and simulation analyses were performed considering a synthetic but realistic oil field modeled with a multiphase flow simulator the purpose of the analyses was to assess the relative performance of the milp formulations and their impact on the simulated oil production

variable disaggregation in network flow problems with piecewise linear costs we study mixed integer programming formulations based upon variable disaggregation for generic multicommodity network flow problems with nonconvex piecewise linear costs a problem class that arises frequently in many application domains in telecommunications transportation and logistics we present several structural results for these formulations and we analyze the results of extensive experiments on a large set of instances with various characteristics in particular we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows together the theoretical and computational results allow us to suggest which formulation might be the most appropriate depending on the characteristics of the problem instances

the piecewise linear optimization polytope new inequalities and intersection with semi continuous constraints we give new facets and valid inequalities for the separable piecewise linear optimization splo knapsack polytope we also extend the inequalities to the case in which some of the variables are semi continuous finally we give computational results that demonstrate their efficiency in solving difficult instances of splo and splo with semi continuous constraints

a maximum likelihood criterion for use with data collected at unequal time intervals a new maximum likelihood nmle criterion suitable for model calibration using data which are recorded at unequal time intervals and contain autocorrelated errors is derived validation of the nmle criterion has been carried out both on a simple two parameter reservoir model using synthetically generated data and on a more complicated hillslope model using real data from the pukeiti catchment in new zealand comparison between the nmle criterion and the simple least squares criterion reveals the superiority of the former over the latter comparison between the nmle criterion and the maximum likelihood criterion for the autocorrelated case proposed by sorooshian 1978 has shown that both criteria would yield results with no practical difference if equal time interval data were used however the nmle criterion can work on variable time interval data which provide more information than equal time interval data and therefore produce better visual results in hydrologic simulations

structural investigation of piecewise linearized network flow problems in this work we study polyhedra in the context of network flow problems in which the flow value on each arc lies in one of several predefined intervals this is motivated by nonlinear problems on transportation networks where nonlinearities are handled by piecewise linear approximation or relaxation a common and established approach in many applications several methods for modeling piecewise linear functions which provide a complete description for a single network arc are known however in general this property is lost when considering multiple arcs we show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities for the case of paths of degree two nodes we give a complete description of the polyhedron projected to the integer variables our model is based on but not limited to the connection between binary variables and flow variables that is used in the multiple choice method or the convex combination method we also show how to transfe

mixed integer models for the stationary case of gas network optimization a gas network basically consists of a set of compressors and valves that are connected by pipes the problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost efficiently such that all demands of the gas network are satisfied this problem leads to a complex mixed integer nonlinear optimization problem we describe techniques for a piece wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program we study sub polyhedra linking these piece wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms suitable branching strategies complementing the separation algorithms are also presented our computational results demonstrate the success of this approach

effective and efficient algorithm for multiobjective optimization of hydrologic models 1 practical experience with the calibration of hydrologic models suggests that any single objective function no matter how carefully chosen is often inadequate to properly measure all of the characteristics of the observed data deemed to be important one strategy to circumvent this problem is to define several optimization criteria objective functions that measure different complementary aspects of the system behavior and to use multicriteria optimization to identify the set of nondominated efficient or pareto optimal solutions in this paper we present an efficient and effective markov chain monte carlo sampler entitled the multiobjective shuffled complex evolution metropolis moscem algorithm which is capable of solving the multiobjective optimization problem for hydrologic models moscem is an improvement over the shuffled complex evolution metropolis scem ua global optimization algorithm using the concept of pareto dominance rather than direct single objective function evaluation to evolve the initial population of points toward a set of solutions stemming from a stable distribution pareto set the efficacy of the moscem ua algorithm is compared with the original mocom ua algorithm for three hydrologic modeling case studies of increasing complexity

mixed integer linear optimization for optimal lift gas allocation with well separator routing abstract the lift gas allocation problem with well separator routing constraints is a mixed integer nonlinear program of considerable complexity to this end a mixed integer linear formulation compact is obtained by piecewise linearizing the nonlinear curves using binary variables to express the linearization and routing decisions a new formulation integrated combining the decisions on linearization and routing is developed by using a single binary variable the structures of both formulations are explored to generate lifted cover cuts numerical tests show that the solution of the integrated formulation using cutting plane generation is faster in spite of having more variables than the compact formulation

automatic calibration of conceptual rainfall runoff models optimization algorithms catchment conditions and model structure from 32 crr catchment cases combinations from four conceptual rainfall runoff models crr and eight catchments calibrated with either two or three optimization methods 1 the shuffle complex evolution method sce ua 2 the multiple start simplex msx and 3 the local simplex it seems that all three methods produced parameter sets of comparable local optimum quality even with comparable performance among the models some parameter values derived by the three optimization methods for the same crr catchment cases are surprisingly different from each other in addition parameter sets of sce ua or msx which often produce marginally better results than the local simplex at the calibration stage could end up with worse results at the validation stage apparently given the inherent limitations of calibration data model inadequacies and identifiability problems it is impossible to achieve global convergence in the parameter search however other than those for dry catchments such as ihimbu or bird creek the parameter sets obtained are generally feasible both sce ua and the local simplex are viable optimization tools while msx is inefficient computationally sce ua can complete the parameter search in one run while the local simplex often requires multirun operations to get good results

efficient subspace probabilistic parameter optimization for catchment models the estimation of catchment model parameters has proven to be a difficult task for several reasons which include ill posedness and the existence of multiple local optima recent work on global probabilistic search methods has developed robust techniques for locating the global optimum however these methods can be computationally intensive when the search is conducted over a large hypercube moreover specification of the hypercube may be problematic particularly if there is strong parameter interaction this study seeks to reduce the computational effort by confining the search to a subspace within which the global optimum is likely to be found the approach involves locating a local optimum using a local gradient based search it is assumed that the local optimum belongs to a set of optima which cluster about the global optimum a probabilistic search is then conducted within a hyperellipsoid defined by the second order approximation to the response surface around the local optimum a case study involving a five parameter conceptual rainfall runoff model is presented the response surface is shown to be riddled with local optima yet the second order approximation provides a not unreasonable description of parameter uncertainty the subspace search strategy provides a rational means for defining the search space and is shown to be more efficient typically twice but up to 5 times more efficient than a search over a hypercube four probabilistic search algorithms are compared shuffled complex evolution sce genetic algorithm using traditional crossover and multiple random start using either simplex or quasi newton local searches in the case study the sce algorithm was found to be robust and the most efficient the genetic algorithm although displaying initial convergence rates superior to the sce algorithm tended to flounder near the optimum and could not be relied upon to locate the global optimum

estimating epistemic and aleatory uncertainties during hydrologic modeling an information theoretic approach 1 with growing interest in understanding the magnitudes and sources of uncertainty in hydrological modeling the difficult problem of characterizing model structure adequacy is now attracting considerable attention here we examine this problem via a model structureindependent approach based in information theory in particular we a discuss how to assess and compute the information content in multivariate hydrological data b present practical methods for quantifying the uncertainty and shared information in data while accounting for heteroscedasticity c show how these tools can be used to estimate the best achievable predictive performance of a model for a system given the available data and d show how model adequacy can be characterized in terms of the magnitude and nature of its aleatory uncertainty that cannot be diminished and is resolvable only up to specification of its density and its epistemic uncertainty that can in principle be suitably resolved by improving the model an illustrative modeling example is provided using catchment scale data from three river basins the leaf and chunky river basins in the united states and the chuzhou basin in china our analysis shows that the aleatory uncertainty associated with making catchment simulations using this data set is significant 50 further estimated epistemic uncertainties of the hymod sac sma and xinanjiang model hypotheses indicate that considerable room for model structural improvements remain

fitting piecewise linear continuous functions abstract we consider the problem of fitting a continuous piecewise linear function to a finite set of data points modeled as a mathematical program with convex objective we review some fitting problems that can be modeled as convex programs and then introduce mixed binary generalizations that allow variability in the regions defining the best fit function s domain we also study the additional constraints required to impose convexity on the best fit function

modeling disjunctive constraints with a logarithmic number of binary variables and constraints many combinatorial constraints over continuous variables such as sos1 and sos2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra we give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra using these conditions we introduce mixed integer binary formulations for sos1 and sos2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables we also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions we prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations

the analysis of structural identifiability theory and application to conceptual rainfall runoff models the identification of a conceptual hydrologic model involves two stages selection of a suitable model structure and parameter estimation these two stages cannot be treated as distinct and it is important that the limitations of available parameter estimation methodologies be given serious consideration during the procedure of model structure selection this paper focuses on the concepts and tools necessary for the evaluation of whether the structure of a hydrologic model is such that its parameters can be successfully identified a definition of structural identifiability is advanced that is based purely on model properties and is independent of the stochastic nature of the output observation errors the approach is used to develop a measure of local structural identifiability which is useful in the analysis of parameter sensitivity a measure called the sensitivity ratio is introduced which is suitable for studying parameter interactions in the multidimensional parameter space also the use of allowable reparameterizations to improve identifiability is discussed the utility of the concepts discussed in this paper are demonstrated by a simulation example using a conceptual rainfall runoff model

base 2 expansions for linearizing products of functions of discrete variables this paper presents an approach for representing functions of discrete variables and their products using logarithmic numbers of binary variables given a univariate function whose domain consists of n distinct values it begins by employing a base 2 expansion to express the function in terms of the ceiling of log2n binary and n continuous variables using linear restrictions to equate the functional values with the possible binary realizations the representation of the product of such a function with a nonnegative variable is handled via an appropriate scaling of the linear restrictions products of m functions are treated in an inductive manner from i 2 to m where each step i uses such a scaling to express the product of function i and a nonnegative variable denoting a translated version of the product of functions 1 through i 1 as a newly defined variable the resulting representations both in terms of one function and many are important for reformulating general discrete variables as binary and also for linearizing mixed integer generalized geometric and discrete nonlinear programs where it is desired to economize on the number of binary variables the approach provides insight into improves upon and subsumes related linearization methods for products of functions of discrete variables

mixed integer models for nonseparable piecewise linear optimization unifying framework and extensions we study the modeling of nonconvex piecewise linear functions as mixed integer programming mip problems we review several new and existing mip formulations for continuous piecewise linear functions with special attention paid to multivariate nonseparable functions we compare these formulations with respect to their theoretical properties and their relative computational performance in addition we study the extension of these formulations to lower semicontinuous piecewise linear functions

estimating information entropy for hydrological data one dimensional case there has been a recent resurgence of interest in the application of information theory to problems of system identification in the earth and environmental sciences while the concept of entropy has found increased application little attention has yet been given to the practical problems of estimating entropy when dealing with the unique characteristics of two commonly used kinds of hydrologic data rainfall and runoff in this paper we discuss four important issues of practical relevance that can bias the computation of entropy if not properly handled the first zero effect arises when precipitation and ephemeral streamflow data must be viewed as arising from a discrete continuous hybrid distribution due to the occurrence of many zero values e g days with no rain no runoff second in the widely used bin counting method for estimation of pdf s significant error can be introduced if the bin width is not carefully selected the third measurement effect arises due to the fact that continuously varying hydrologic variables can typically only be observed discretely to some degree of precision the fourth skewness effect arises when the distribution of a variable is significantly skewed here we present an approach that can deal with all four of these issues and test them with artificially generated and real hydrological data the results indicate that the method is accurate and robust