a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

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

near equality near convexity sums of maximally monotone operators and averages of firmly nonexpansive mappings we study nearly equal and nearly convex sets ranges of maximally monotone operators and ranges and fixed points of convex combinations of firmly nonexpansive mappings the main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings a striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular throughout examples are provided to illustrate the theory we also obtain detailed information on the domain and range of the resolvent average

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

a new matrix free algorithm for the large scale trust region subproblem we present a new method for the large scale trust region subproblem the method is matrix free in the sense that only matrix vector products are required we recast the trust region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter we then find the solution of the trust region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter the new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so called hard case we show that the new iteration is well defined and convergent at a superlinear rate we present computational results to illustrate convergence properties and robustness of the method

a weaker regularity condition for subdifferential calculus and fenchel duality in infinite dimensional spaces abstract in this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping this condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior point regularity conditions given so far in the literature moreover it provides a weak sufficient condition for fenchel duality regarding convex optimization problems in infinite dimensional spaces as an application we discuss the strong conical hull intersection property chip for a finite family of closed convex sets

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

a unifying framework in vector valued reproducing kernel hilbert spaces for manifold regularization and co regularized multi view learning this paper presents a general vector valued reproducing kernel hilbert spaces rkhs framework for the problem of learning an unknown functional dependency between a structured input space and a structured output space our formulation encompasses both vector valued manifold regularization and co regularized multi view learning providing in particular a unifying framework linking these two important learning approaches in the case of the least square loss function we provide a closed form solution which is obtained by solving a system of linear equations in the case of support vector machine svm classification our formulation generalizes in particular both the binary laplacian svm to the multi class multi view settings and the multi class simplex cone svm to the semi supervised multi view settings the solution is obtained by solving a single quadratic optimization problem as in standard svm via the sequential minimal optimization smo approach empirical results obtained on the task of object recognition using several challenging data sets demonstrate the competitiveness of our algorithms compared with other state of the art methods

distributed self localization for relative position sensing networks in 2d space this paper studies the 2d localization problem of a sensor network given anchor node positions in a common global coordinate frame and relative position measurements in local coordinate frames between node pairs it is assumed that the local coordinate frames of different sensors have different orientations and the orientation difference with respect to the global coordinate frame are not known in terms of graph connectivity a necessary and sufficient condition is obtained for self localizability that leads to a fully distributed localization algorithm moreover a distributed verification algorithm is developed to check the graph connectivity condition which can terminate successfully when the sensor network is self localizable finally a fully distributed linear and iterative algorithm based on the complex valued laplacian associated with the sensor network is proposed which converges globally and gives the correct localization result

robust target localization in the absence of signal propagation models this paper considers a problem of target or source localization based on the measurements from a mobile sensor or a network of sensors the assumption is that the received signal strength is strictly monotonic with the distance between the target and the sensor no explicit signal propagation model neither the structure nor the mathematical description is assumed or used in localization the contribution of the paper is to show that this mere knowledge of monotonicity suffices in locating the unknown target in the absence of noise further in the presence of noise robust localization algorithms are developed and analyzed by exploiting various forms of angular temporal and spatial averages the asymptotic convergence results have been established in the presence of noise and further the finite step performance results are developed

translation semigroups and their linearizations on spaces of integrable functions of concern is the unbounded operator abf f with nonlinear domain d av f e w li f o 4 f which is considered on the banach space e of bochner integrable functions on an interval with values in a banach space f under the assumption that 4 is a lipschitz continuous operator from e to f it is shown that as generates a strongly continuous translation semigroup tb t 1 0 for linear operators 4 several properties such as essential compactness positivity and irreducibility of the semigroup tob t i o depending on the operator 4 are studied it is shown that if f is a banach lattice with order continuous norm then tlbl t 0 is the modulus semigroup of td t o finally spectral properties of as are studied and the spectral bound s a0p is determined this leads to a result on the global asymptotic behavior in the case where 4 is linear and to a local stability result in the case where 4 is frechet differentiable

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

an iterative algorithm for large size least squares constrained regularization problems abstract in this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems the regularization problem is formulated as a constrained optimization problem the dual lagrangian problem is iteratively solved to compute an approximate solution before starting the iterations the algorithm computes the necessary smoothing parameters and the error tolerances from the data the numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency

a trust region approach to nonlinear systems of equalities and inequalities in this paper two new trust region algorithms for the numerical solution of systems of nonlinear equalities and inequalities are introduced the formulation is free of arbitrary parameters and possesses sufficient smoothness to exploit the robustness of the trust region approach the proposed algorithms are one sided least squares trust region algorithms the first algorithm is a single model algorithm and the second one is a multimodel algorithm where the cauchy point computation is a model selection procedure global convergence analysis for the two algorithms is presented our analysis generalizes to nonlinear systems of equalities and inequalities the well developed theory for nonlinear least squares problems numerical experiments on the two algorithms are also presented the performance of the two algorithms is reported the numerical results validate the effectiveness of our approach

dynamics of structured populations with inherited properties abstract a mathematical model of an age structured proliferating cell population is analyzed the model allows the inheritance of cycle length from mother cells to daughter cells it is proved that the solutions have the property of asynchronous exponential growth provided the transition kernel of inheritance satisfies smoothness and positivity requirements the method of proof uses the theory of semigroups of operators and the theory of positive operators

asymptotic analysis of a cell cycle model based on unequal division we provide an analysis of the asymptotic behavior of a novel cell cycle model offering a uniform description of the processes of rna production and division and explaining the cell generation time variability at least in certain cell lines we prove that the distribution m t x of the rna level x in the dividing cells at a given time t tends to exp lambda t mu x as t tends to infinity i e that there exists the so called exponential steady state for our model and that perturbed cell populations reach this state in the limit the result is obtained by applying the semigroup theory to a functional integral equation describing the evolution of m t x in time for a variety of reasons the analysis requires explicit characterization of the semigroup spectrum our asymptotic result may be viewed as a generalization of similar results for the generalized branching processes with continuous time under less restrictive assumptions also we discuss related models and asymp

sensor network localization on the group of three dimensional displacements we consider the problem of estimating relative configurations of nodes in a sensor network based on noisy measurements by exploiting the cyclic constraints induced by the sensing topology to the network we derive a constrained optimization on se 3 n for the case of a network with a single cyclic constraint we present a closed form solution we show that in certain cases namely restriction to pure rotation and pure translation this solution is independent of the particular representation of the constraint function and is the unique constrained minimizer of an appropriate cost for sensing topologies represented by a general weakly connected digraph we present a solution method which is based on the limits of the solution curves of a continuous time ordinary differential equation we show that solutions obtained by our method satisfy all semicycle generalized cycle constraints induced by the sensing topology of the network further we show through numerical simulation that for gaussian like

a new condition for maximal monotonicity via representative functions abstract in this paper we give a weaker sufficient condition for the maximal monotonicity of the operator s a t a where s x x t y y are two maximal monotone operators a x y is a linear continuous mapping and x y are reflexive banach spaces we prove that our condition is weaker than the generalized interior point conditions given so far in the literature this condition is formulated using the representative functions of the operators involved in particular we rediscover some sufficient conditions given in the past using the so called fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator respectively

a singular perturbation in an age structured population model the aim of this work is to study a model of age structured population with two time scales the first one is slow and corresponds to the demographic process and the second one is comparatively fast and describes the migration process between different spatial patches from a mathematical point of view the model is a linear system of partial differential equations where the state variables are the population densities in each spatial patch together with a boundary condition of integral type the birth equation due to the two different time scales the system depends on a small parameter varepsilon and can be thought of as a singular perturbation problem the main results of the work are that for varepsilon 0 small enough the solutions of the system can be approximated by means of the solutions of a scalar problem where the fast process has been avoided by supposing it has attained an equilibrium the state variable of the scalar system represents the global density of the population the birth

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

distributed localization with mixed measurements under switching topologies abstract this paper investigates the distributed localization problem of sensor networks with mixed measurements each node holds a local coordinate system without a common orientation and is capable of measuring only one type of information either distance bearing or relative position to near by nodes thus three types of measurements are mixed in the sensor networks moreover the communication topologies in the sensor networks may be time varying due to unreliable communications this paper develops a fully distributed algorithm called bcdl barycentric coordinate based distributed localization where each node starts from a random initial guess about its true coordinate and converges to the true coordinate via only local node interactions the key idea in bcdl is to establish a unified linear equation constraints for the sensor coordinates by using the barycentric coordinates of each node with respect to its neighbors though the sensor nodes may have different types of measurements then a distributed iterative algorithm is proposed to solve the linear equations under time varying communication networks a necessary and sufficient graphical condition is obtained to ensure global convergence of the distributed algorithm

an iterative lagrange method for the regularization of discrete ill posed inverse problems in this paper an iterative method is presented for the computation of regularized solutions of discrete ill posed problems in the proposed method the regularization problem is formulated as an equality constrained minimization problem and an iterative lagrange method is used for its solution the lagrange iteration is terminated according to the discrepancy principle the relationship between the proposed approach and classical tikhonov regularization is discussed results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method

the procrustes problem for orthogonal stiefel matrices in this paper we consider the procrustes problem on the manifold of orthogonal stiefel matrices given matrices cal a in bbb r m times k cal b in bbb r m times p m ge p ge k we seek the minimum of cal a cal b q 2 for all matrices q in bbb r p times k q tq i k times k we introduce a class of relaxation methods for generating sequences of approximations to a minimizer and offer a geometric interpretation of these methods results of numerical experiments illustrating the convergence of the methods are given

a customized douglas rachford splitting algorithm for separable convex minimization with linear constraints we consider applying the douglas rachford splitting method drsm to the convex minimization problem with linear constraints and a separable objective function the dual application of drsm has been well studied in the literature resulting in the well known alternating direction method of multipliers admm in this paper we show that the primal application of drsm in combination with an appropriate decomposition can yield an efficient structure exploiting algorithm for the model under consideration whose subproblems could be easier than those of admm both the exact and inexact versions of this customized drsm are studied and their numerical efficiency is demonstrated by some preliminary numerical results

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

guaranteeing practical convergence in algorithms for sensor and source localization this paper considers localization of a source or a sensor from distance measurements we argue that linear algorithms proposed for this purpose are susceptible to poor noise performance instead given a set of sensors anchors of known positions and measured distances of the source sensor to be localized from them we propose a potentially nonconvex weighted cost function whose global minimum estimates the location of the source sensor one seeks the contribution of this paper is to provide nontrivial ellipsoidal and polytopic regions surrounding these sensors anchors of known positions such that if the object to be localized is in this region localization occurs by globally exponentially convergent gradient descent in the noise free case exponential convergence in the noise free case represents practical convergence as it ensures graceful performance degradation in the presence of noise these results guide the deployment of sensors anchors so that small subsets can be made responsible for practical localization in geographical areas determined by our approach

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

enlargements of positive sets in this paper we introduce the notion of enlargement of a positive set in ssd spaces to a maximally positive set a we associate a family of enlargements e a and characterize the smallest and biggest element in this family with respect to the inclusion relation we also emphasize the existence of a bijection between the subfamily of closed enlargements of e a and the family of so called representative functions of a we show that the extremal elements of the latter family are two functions recently introduced and studied by stephen simons in this way we extend to ssd spaces some former results given for monotone and maximally monotone sets in banach spaces

distributed localization for 2 d sensor networks with bearing only measurements under switching topologies this paper investigates the problem of bearing measurement based distributed localization for sensor networks that are undirected and switching each node holds a local coordinate system with no knowledge about the global coordinate system and measures the bearing angle information about its neighbors in its local coordinate system a novel scheme for localization is developed using a complex laplacian to overcome the challenges due to the absence of a global coordinate system and the presence of topology switching in communication first by using bearing only measurements an algorithm is proposed to establish linear equation constraints for the coordinates of sensor nodes in the global coordinate frame the main idea is that each node uses its own bearing and its neighbors bearing information to construct a similar configuration though it is not able to recover the true configuration by using only bearing measurements second a distributed iterative algorithm is proposed such that all the sensor nodes can cooperatively find the true coordinates of themselves it is shown that the algorithm exponentially converges provided that the communication network jointly satisfies certain connectivity properties the simulation results validate our proposed algorithm

tikhonov regularization with a solution constraint many numerical methods for the solution of linear ill posed problems apply tikhonov regularization this paper presents a modification of a numerical method proposed by golub and von matt for quadratically constrained least squares problems and applies it to tikhonov regularization of large scale linear discrete ill posed problems the method is based on partial lanczos bidiagonalization and gauss quadrature computed examples illustrating its performance are presented

variational analysis of an extended eigenvalue problem abstract for a symmetric matrix b rn n and a vector a rn the maximal extended eigenvalue a max x r n s t b i x a x t x 1 is known to arise in optimality conditions for the mathematical programming problemp a given by max x t b x 2 a t x x t x 1 as well as in extended rayleigh ritz type results pertaining to the one parameter family of nonsymmetric border perturbations ofb given by a t a b a t a t nonsmooth analysis is employed in order to describe the function with special emphasis on its sensitivity near the origin further connections with p a are drawn

asynchronous exponential growth of semigroups of nonlinear operators the property of asynchronous exponential growth is analyzed for the abstract nonlinear differential equation z t az t f z t t 0 z 0 x x where a is the infinitesimal generator of a semigroup of linear operators in the banach space x and f is a nonlinear operator in x asynchronous exponential growth means that the nonlinear semigroup s t t 0 associated with this problem has the property that there exists 0 and a nonlinear operator q in x such that the range of q lies in a one dimensional subspace of x and limt e ts t x qx for all x x it is proved that if the linear semigroup generated by a has asynchronous exponential growth and f satisfies f x f x x where f r r and f r r dr then the nonlinear semigroup s t t 0 has asynchronous exponential growth the method of proof is a linearization about infinity examples from structured population dynamics are given to illustrate the results

a continuous time linear adaptive source localization algorithm robust to persistent drift the problem of source localization has assumed increased importance in recent years in this paper we formulate a continuous time adaptive localization algorithm that permits a mobile agent to estimate the location of a stationary source using only the measured distance from the source the algorithm is shown to be exponentially asymptotically stable under a persistent excitation condition that has an appealing interpretation we quantify the fact that exponential asymptotic stability endows the algorithm with the ability to track slow bounded but potentially persistent and nontrivial drift in the source

α and β curves sister sister and mother daughter correlations in cell population dynamics abstract the and curves and sister sister and mother daughter cycle length correlations are analyzed for a general model of age and size structured cell population dynamics some basic facts concerning these concepts are established a criterion is given for log a and log a to have asymptotically equal slopes the significance of size dependence upon sister sister and mother daughter correlations is demonstrated some numerical examples are given to illustrate the concepts

a trust region approach to the regularization of large scale discrete forms of ill posed problems we consider large scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill posed problem and the right hand side contains noise special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution we pose the regularization problem as a quadratically constrained least squares problem this formulation is equivalent to tikhonov regularization and we note that it is also a special case of the trust region subproblem from optimization we analyze the trust region subproblem in the regularization case and we consider the nontrivial extensions of a recently developed method for general large scale subproblems that will allow us to handle this case the method relies on matrix vector products only has low and fixed storage requirements and can handle the singularities arising in ill posed problems we present numerical results on test problems on an inverse interpolation problem with field data and on a model seismic inversion problem with field data

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

some spectral properties for the asymptotic behavior of semigroups connected to population dynamics this paper gives a short review of the mathematics involved in the study of asymptotic behavior of semigroups arising from structured models of population dynamics first are considerations on the

robust inversion dimensionality reduction and randomized sampling we consider a class of inverse problems in which the forward model is the solution operator to linear odes or pdes this class admits several dimensionality reduction techniques based on data averaging or sampling which are especially useful for large scale problems we survey these approaches and their connection to stochastic optimization the data averaging approach is only viable however for a least squares misfit which is sensitive to outliers in the data and artifacts unexplained by the forward model this motivates us to propose a robust formulation based on the student s t distribution of the error we demonstrate how the corresponding penalty function together with the sampling approach can obtain good results for a large scale seismic inverse problem with 50 corrupted data

hidden convexity in some nonconvex quadratically constrained quadratic programming we consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints under a suitable simultaneous diagonalization assumption which trivially holds for trust region type problems we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints we then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints which is also shown to be equivalent to a minimax convex problem in both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts special cases and applications are also discussed we outline interior point polynomial time algorithms for the solution of the equivalent convex programs

on the generalized parallel sum of two maximal monotone operators of gossez type d abstract the generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping in this article by assuming that the operators are maximal monotone of gossez type d we provide sufficient conditions of both interiority and closedness type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of gossez type d too this result will follow as a particular instance of a more general one concerning the maximal monotonicity of gossez type d of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals

duality for constrained robust sum optimization problems given an infinite family of extended real valued functions f i i in i and a family mathcal h of nonempty finite subsets of i the mathcal h partial robust sum of f i i in i is the supremum for j in mathcal h of the finite sums sum j in j f j these infinite sums arise in a natural way in location problems as well as in functional approximation problems and include as particular cases the well known sup function and the so called robust sum function corresponding to the set mathcal h of all nonempty finite subsets of i whose unconstrained minimization was analyzed in previous papers of three of the authors https doi org 10 1007 s11228 019 00515 2 and https doi org 10 1007 s00245 019 09596 9 in this paper we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given mathcal h partial robust sum under constraints as well as closedness and convex criteria for the formulas on the subdifferential of the sup function