semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

asymptotic analysis of perturbed mathematical programs abstract we consider a perturbed mathematical programming problem where both the objective and the constraint functions are analytical in both the underlying decision variables and in the perturbation variable parameter that is denoted by the following question arises what is the description of the solutions of such a perturbed problem when 0 we demonstrate that under weak conditions the solutions of the perturbed problems are obtained as puiseux series expansions in the results are obtained by application of the remmert stein representation theorem for complex analytic varieties

abnormal equality constrained optimization problems sensitivity theory for the equality constrained optimization problem we consider the case when the customary regularity of constraints can be violated under the assumptions substantially weaker than those previously used in the literature we develop a reasonably complete local sensitivity theory for this class of problems including upper and lower bounds for the rate of change of the optimal value function subject to parametric perturbations as well as the estimates and the description of asymptotic behavior of solutions of the perturbed problems

optimal control elliptic systems with distributed and boundary controls abstract in this paper we consider a nonlinear elliptic control system with distributed and boundary controls and with the integral cost functional some sufficient conditions under which the solutions of the elliptic systems continuously depend on boundary and distributed controls are presented the proof of the result is based on variational methods these conditions are used to obtain some existence theorem for the optimal control problem governed by nonlinear elliptic equations with distributed and boundary controls in the final part we consider the question of stability of the optimal controls and trajectories applying some results of the convergence theory we prove sufficient conditions under which small variations of parameters of given system imply small changes of optimal processes

multiaccess quantum channels shared communication channels are subject to multiple access interference transmitter and receiver design techniques that explicitly deal with this interference have been shown to improve substantially the performance of communication systems over radio frequency and other classical channels quantum multiple access communication channels on the other hand have received comparatively little attention in this paper an input output model for multiple access quantum channels relevant to optical communications is proposed the model accounts for multiaccess interference signal attenuation and random noise and can be used in the analysis and design of communication systems using a result from optimization a perturbation method is developed to find the minimum achievable error probability in small interference channels it is shown that the quantum measurement that minimizes the error probability in a no interference channel is robust in the presence of small multiaccess interference the results are illustrated with numerical examples which show that optimal quantum detectors can significantly outperform conventional detectors even for moderate levels of crosstalk

nondegeneracy and quantitative stability of parameterized optimization problems with multiple solutions this paper presents some results on quantitative stability of optimal solutions of parameterized optimization problems having nonisolated optima the analysis is based on the introduced concept of nondegeneracy which may be of independent interest examples of nonlinear and semidefinite programming are discussed

directional differentiability of the optimal value function in convex semi infinite programming in this paper directional differentiability properties of the optimal value function of a parameterized semi infinite programming problem are studied it is shown that if the unperturbed semi infinite programming problem is convex then the corresponding optimal value function is directionally differentiable under mild regularity assumptions a max min formula for the directional derivatives well known in the finite convex case is given

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

high order stability conditions for degenerate optimization problems elements of p regularity theory abstract this paper is devoted to sensitivity analysis of degenerate equality constrained optimization problems when the customary regularity of constraints may be violated under the p regularity assumptions we develop p order stability theory for this class of problems subject to small perturbations and present the estimations of asymptotic behavior of the solutions to the perturbed problems we derive conditions that guarantee the existence of a solution for perturbed optimization problems and substantiate the assertion of natural character of these conditions all results are illustrated by classical calculus of variations problems

stability of elliptic optimal control problems abstract we consider optimal problems governed by elliptical equations on parameters and give sufficient conditions for the continuous dependence of the solutions on the parameters the techniques are based on variational methods

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

perturbed optimization in banach spaces i a general theory based on a weak directional constraint qualification using a directional form of constraint qualification weaker than robinson s we derive an implicit function theorem for inclusions and use it for first and second order sensitivity analyses of the value function in perturbed constrained optimization we obtain holder and lipschitz properties and under a no gap condition first order expansions for exact and approximate solutions as an application differentiability properties of metric projections in hilbert spaces are obtained using a condition generalizing polyhedricity we also present in the appendix a short proof of a generalization of the convex duality theorem in banach spaces

on sensitivity analysis of nonlinear programs in banach spaces the approach via composite unconstrained optimization this article studies local sensitivity analysis of nonlinear programming problems in banach spaces with arbitrary sets of solutions and no a priori regularity properties this is done by first developing a corresponding theory for unconstrained optimization involving simple composite functions and then applying this theory to general nonlinear programs by way of certain reduction principles

a finite volume module for simulating global all scale atmospheric flows the paper documents the development of a global nonhydrostatic finite volume module designed to enhance an established spectral transform based numerical weather prediction nwp model the module adheres to nwp standards with formulation of the governing equations based on the classical meteorological latitude longitude spherical framework in the horizontal a bespoke unstructured mesh with finite volumes built about the reduced gaussian grid of the existing nwp model circumvents the notorious stiffness in the polar regions of the spherical framework all dependent variables are co located accommodating both spectral transform and grid point solutions at the same physical locations in the vertical a uniform finite difference discretisation facilitates the solution of intricate elliptic problems in thin spherical shells while the pliancy of the physical vertical coordinate is delegated to generalised continuous transformations between computational and physical space the newly developed module assumes the compressible euler equations as default but includes reduced soundproof pdes as an option furthermore it employs semi implicit forward in time integrators of the governing pde systems akin to but more general than those used in the nwp model the module shares the equal regions parallelisation scheme with the nwp model with multiple layers of parallelism hybridising mpi tasks and openmp threads the efficacy of the developed nonhydrostatic module is illustrated with benchmarks of idealised global weather

profile extrema for visualizing and quantifying uncertainties on excursion regions application to coastal flooding we consider the problem of describing excursion sets of a real valued function f that is the set of inputs where f is above a fixed threshold such regions are hard to visualize if the input spac

sufficient dimension reduction via direct estimation of the gradients of logarithmic conditional densities sufficient dimension reduction sdr is aimed at obtaining the low rank projection matrix in the input space such that information about output data is maximally preserved among various approaches to sdr a promising method is based on the eigendecomposition of the outer product of the gradient of the conditional density of output given input in this letter we propose a novel estimator of the gradient of the logarithmic conditional density that directly fits a linear in parameter model to the true gradient under the squared loss thanks to this simple least squares formulation its solution can be computed efficiently in a closed form then we develop a new sdr method based on the proposed gradient estimator we theoretically prove that the proposed gradient estimator as well as the sdr solution obtained from it achieves the optimal parametric convergence rate finally we experimentally demonstrate that our sdr method compares favorably with existing approaches in both accuracy and computational efficiency on a variety of artificial and benchmark data sets

the use of rational eigenvector approximations in commutative controllers commutative controllers provide a convenient means for the systematic adjustment of the gain phase characteristic of generalized nyquist diagrams due to the irrational nature of the eigenvectors of transfer function matrices practical commutative controllers are based on eigenvector approximations this paper investigates the use of rational approximations highlights stability difficulties that arise in connection with unstable fixed modes and derives the necessary and sufficient conditions that overcome these difficulties the results of the paper are

building sparse multiple kernel svm classifiers the support vector machines svms have been very successful in many machine learning problems however they can be slow during testing because of the possibly large number of support vectors obtained recently wu 2005 proposed a sparse formulation that restricts the svm to use a small number of expansion vectors in this paper we further extend this idea by integrating with techniques from multiple kernel learning mkl the kernel function in this sparse svm formulation no longer needs to be fixed but can be automatically learned as a linear combination of kernels two formulations of such sparse multiple kernel classifiers are proposed the first one is based on a convex combination of the given base kernels while the second one uses a convex combination of the so called ldquoequivalentrdquo kernels empirically the second formulation is particularly competitive experiments on a large number of toy and real world data sets show that the resultant classifier is compact and accurate and can also be easily trained by simply alternating linear program and standard svm solver

differential stability of two stage stochastic programs two stage stochastic programs with random right hand side are considered optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations respectively conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces explicit formulas for the derivatives are derived special attention is paid to the role of a lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function

optimality conditions for irregular inequality constrained problems we consider feasible sets given by conic constraints where the cone defining the constraints is convex with nonempty interior we study the case where the feasible set is not assumed to be regular in the classical sense of robinson and obtain a constructive description of the tangent cone under a certain new second order regularity condition this condition contains classical regularity as a special case while being weaker when constraints are twice differentiable assuming that the cone defining the constraints is finitely generated we also derive a special form of primal dual optimality conditions for the corresponding constrained optimization problem our results subsume optimality conditions for both the classical regular and second order regular cases while still being meaningful in the more general setting in the sense that the multiplier associated with the objective function is nonzero

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

deriving convex hulls through lifting and projection we consider convex hull descriptions for certain sets described by inequality constraints over a hypercube and propose a lifting and projection technique to construct them the general procedure obtains the convex hulls as an intersection of semi infinite families of linear inequalities each derived using lifting techniques that are interpreted using convexification tools we demonstrate that differentiability and concavity of certain perturbation functions help reduce the number of inequalities needed for this characterization each family of inequalities yields a few linear nonlinear constraints fully characterized in the space of the original problem variables when the projection problems are amenable to a closed form solution in particular we illustrate the complete procedure by constructing the convex hulls of the subsets of a compact hypercube defined by the constraints x 1 b 1 x 2 b 2 ge x 3 and x 1 x 2 b 2 le x 3 where b 1 b 2 ge 1 as a consequence we obtain a closed form description of the convex hull of the bilinear equality x 1x 2 x 3 in the presence of variable bounds as an intersection of a few linear and nonlinear inequalities this explicit characterization hitherto unknown can improve relaxation techniques for factorable functions which utilize this equality to relax products of functions with known relaxations

low complexity delay constrained beamforming for multi user mimo systems with imperfect csit we consider the delay constrained beamforming control for downlink multi user mimo mu mimo systems with imperfect channel state information at the transmitter csit the delay constrained control problem is formulated as an infinite horizon average cost partially observed markov decision process to deal with the curse of dimensionality we introduce a virtual continuous time system and derive a closed form approximate value function using perturbation analysis w r t the csit errors to deal with the challenge of the conditional packet error rate per we build a tractable closed form approximation using a bernstein type inequality based on the closed form approximations of the relative value function and the conditional per we propose a conservative formulation of the original beamforming control problem the conservative problem is non convex and we transform it into a convex problem using the semidefinite relaxation sdr technique we then propose an alternating iterative algorithm to solve the sdr problem finally we show through simulations that the proposed scheme has significant performance gain compared with various baselines

sensitivity of solutions to variational inequalities on banach spaces we analyze the sensitivity of parameterized variational inequalities for convex polyhedric sets in reflexive banach spaces we compute a generalized derivative of the solution mapping where the formula for the derivative is given in terms of the solutions to an auxiliary variational inequality these results are distinguished from other work in this area by the fact that they do not depend on the uniqueness of the solutions to the variational inequalities to obtain our results we use second order epi derivatives to analyze the second order properties of polyhedric sets we apply our results to sensitivity analyses of stationary points and kkt pairs associated with constrained infinite dimensional optimization problems

mathematical programs with complementarity constraints we study mathematical programs with complementarity constraints several stationarity concepts based on a piecewise smooth formulation are presented and compared the concepts are related to stat

conjugate duality in set valued vector optimization abstract in this paper conjugate duality results for convexlike set valued vector optimization problems are presented under closedness or boundedness hypotheses some properties of the value mapping of a set valued vector optimization problem are studied a conjugate duality result is also proved for a convex set valued vector optimization problem without the requirements of closedness and boundedness

a family of simple non parametric kernel learning algorithms previous studies of non parametric kernel learning npkl usually formulate the learning task as a semi definite programming sdp problem that is often solved by some general purpose sdp solvers however for n data examples the time complexity of npkl using a standard interior point sdp solver could be as high as o n6 5 which prohibits npkl methods applicable to real applications even for data sets of moderate size in this paper we present a family of efficient npkl algorithms termed simplenpkl which can learn non parametric kernels from a large set of pairwise constraints efficiently in particular we propose two efficient simplenpkl algorithms one is simplenpkl algorithm with linear loss which enjoys a closed form solution that can be efficiently computed by the lanczos sparse eigen decomposition technique another one is simplenpkl algorithm with other loss functions including square hinge loss hinge loss square loss that can be re formulated as a saddle point optimization problem which can be further resolved by a fast iterative algorithm in contrast to the previous npkl approaches our empirical results show that the proposed new technique maintaining the same accuracy is significantly more efficient and scalable finally we also demonstrate that the proposed new technique is also applicable to speed up many kernel learning tasks including colored maximum variance unfolding minimum volume embedding and structure preserving embedding

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

variational analysis of a composite function a formula for the lower second order epi derivative an exact formula is established for the lower second order epi derivative of a function of the form g f x where f is a smooth map from one banach space into another and g is a convex function generally not everywhere finite unconstrained minimization of such functions typically arise as an equivalent in one or another sense reduction form for many important classes of constrained optimization problems the formula is further applied to study epi differentiability of the max function x max q x q q

sufficient optimality conditions for optimal control subject to state constraints strong second order sufficient optimality conditions ssc are derived for optimal control problems of systems described by nonlinear odes subject to mixed control state and pure state constraints the obtained ssc are expressed in terms of a modified legendre clebsch condition and the associated riccati equation the role of ssc in stability analysis of solutions to parametric optimal control problems is briefly discussed

a parametric view on the mangasarian fromovitz constraint qualification in this paper we consider the behaviour of the multipliers for one parametric semi infinite programs near critical parameters where the mangasarian fromovitz constraint qualification can be violated the growth of the multipliers when the parameter approaches such a critical parameter is characterized by a parametric constraint qualification which is introduced here it is equivalent to a bound on the growth of the multipliers

online nonnegative matrix factorization with outliers we propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers our framework is particularly suited to large scale data we propose two solvers based on projected gradient descent and the alternating direction method of multipliers we prove that the sequence of objective values converges almost surely by appealing to the quasi martingale convergence theorem we also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely in addition we extend our basic problem formulation to various settings with different constraints and regularizers we also adapt the solvers and analyses to each setting we perform extensive experiments on both synthetic and real datasets these experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as parts based basis learning image denoising shadow removal and foreground background separation

second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs in this paper we study second order differential properties of an optimal value function x it is shown that under certain conditions x possesses second order directional derivatives which can be calculated by solving corresponding quadratic programs also upper and lower bounds on these derivatives are introduced under weaker assumptions in particular we show that the second order directional derivative is infinite if the corresponding quadratic program is unbounded finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

quantitative stability of variational systems ii a framework for nonlinear conditioning stability results of lipschitz and holder type are obtained for the solutions and optimal values of optimization problems when perturbations are measured in terms of the rho epi distance

sensitivity analysis for cone constrained optimization problems under the relaxed constraint qualifications we present the local sensitivity analysis for cone constrained optimization problems under the cq type conditions significantly weaker than those traditionally used in this context our basic sensitivity results are established under the first or second order sufficient optimality conditions combined with the estimate of the distance to the feasible set of the perturbed problem we demonstrate how such an estimate can be obtained under the assumptions weaker than robinson s cq and establish the corresponding sensitivity results finally we apply our results to sensitivity analysis and relaxation schemes for mathematical programs with complementarity constraints

second order necessary optimality conditions for problems without a priori normality assumptions in this article we derive second order necessary conditions of optimality for an abstract optimization problem with equality and inequality constraints and constraints in the form of an inclusion into a given closed set an important feature is that our optimality conditions dispense with any a priori normality assumptions such as robinsons constraint qualification and remain informative even for abnormal points moreover our optimality conditions take into account the second order effect of the curvature of the set in the inclusion constraints

mathematical study of very high voltage power networks i the optimal dc power flow problem the optimal power flow problem involves setting the voltage and power delivered at the nodes of an electrical network in order to minimize the loss of power over the lines this paper is the first of a series dedicated to the mathematical study of this problem we use an asymptotic analysis in which the small parameter is the inverse of the reference voltage of the network we call this scheme the very high voltage approximation here we deal with the case of direct current we obtain an analytic expansion for the optimal value and the solution

efficient methods for overlapping group lasso the group lasso is an extension of the lasso for feature selection on predefined nonoverlapping groups of features the nonoverlapping group structure limits its applicability in practice there have been several recent attempts to study a more general formulation where groups of features are given potentially with overlaps between the groups the resulting optimization is however much more challenging to solve due to the group overlaps in this paper we consider the efficient optimization of the overlapping group lasso penalized problem we reveal several key properties of the proximal operator associated with the overlapping group lasso and compute the proximal operator by solving the smooth and convex dual problem which allows the use of the gradient descent type of algorithms for the optimization our methods and theoretical results are then generalized to tackle the general overlapping group lasso formulation based on the eq norm we further extend our algorithm to solve a nonconvex overlapping group lasso formulation based on the capped norm regularization which reduces the estimation bias introduced by the convex penalty we have performed empirical evaluations using both a synthetic and the breast cancer gene expression dataset which consists of 8 141 genes organized into overlapping gene sets experimental results show that the proposed algorithm is more efficient than existing state of the art algorithms results also demonstrate the effectiveness of the nonconvex formulation for overlapping group lasso

learning sparse kernel classifiers for multi instance classification we propose a direct approach to learning sparse kernel classifiers for multi instance mi classification to improve efficiency while maintaining predictive accuracy the proposed method builds on a convex formulation for mi classification by considering the average score of individual instances for bag level prediction in contrast existing formulations used the maximum score of individual instances in each bag which leads to nonconvex optimization problems based on the convex mi framework we formulate a sparse kernel learning algorithm by imposing additional constraints on the objective function to enforce the maximum number of expansions allowed in the prediction function the formulated sparse learning problem for the mi classification is convex with respect to the classifier weights therefore we can employ an effective optimization strategy to solve the optimization problem that involves the joint learning of both the classifier and the expansion vectors in addition the proposed formulation can explicitly control the complexity of the prediction model while still maintaining competitive predictive performance experimental results on benchmark data sets demonstrate that our proposed approach is effective in building very sparse kernel classifiers while achieving comparable performance to the state of the art mi classifiers

parameterized minimax problem on lipschitz like dependence of the solution with respect to the parameter we study lipschitz continuity with respect to the parameter of the set of solutions of a parameterized minimax problem on a product banach space we present a sufficient condition ensuring that the map which to any value of the parameter assigns the set of solutions of the problem possibly multi valued and unbounded possesses aubin property

direct density derivative estimation estimating the derivatives of probability density functions is an essential step in statistical data analysis a naive approach to estimate the derivatives is to first perform density estimation and then compute its derivatives however this approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator to cope with this problem in this letter we propose a novel method that directly estimates density derivatives without going through density estimation the proposed method provides computationally efficient estimation for the derivatives of any order on multidimensional data with a hyperparameter tuning method and achieves the optimal parametric convergence rate we further discuss an extension of the proposed method by applying regularized multitask learning and a general framework for density derivative estimation based on bregman divergences applications of the proposed method to nonparametric kullback leibler divergence approximation and bandwidth matrix selection in kernel density estimation are also explored

leave one out bounds for support vector regression model selection minimizing bounds of leave one out errors is an important and efficient approach for support vector machine svm model selection past research focuses on their use for classification but not regression in this letter we derive various leave one out bounds for support vector regression svr and discuss the difference from those for classification experiments demonstrate that the proposed bounds are competitive with bayesian svr for parameter selection we also discuss the differentiability of leave one out bounds

sensitivity analysis of optimization problems under second order regular constraints we present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition this is a technical condition that is always satisfied in the case of semi definite optimization we derive lipschitz and holder expansions of approximate optimal solutions under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem we show how the theory applies to semi infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds and discuss the differentiability of metric projections as well as the moreau yosida regularization finally we show how the theory applies to semi definite optimization

on uniqueness of lagrange multipliers in optimization problems subject to cone constraints in this paper we study uniqueness of lagrange multipliers in optimization problems subject to cone constraints the main tool in our investigation of this question will be a calculus of dual polar cones we give sufficient and in some cases necessary conditions for uniqueness of lagrange multipliers in general banach spaces general results are then applied to two particular examples of the semidefinite and semi infinite programming problems respectively

on lipschitzian stability of optimal solutions of parametrized semi infinite programs we study continuity properties of optimal solutions of parametrized semi infinite programming problems the involved constraints are formulated in a form of cone constraints and then a slightly modified general result of shapiro and bonnans 1992 on lipschitzian stability of optimal solutions is applied it is shown that under certain second order sufficient conditions optimal solutions of the semi infinite programs are lipschitzian stable provided a regularity assumption related to a linearization of the considered programs is satisfied

on the marginal function in nonlinear programming for several types of finite or infinite dimensional optimization problems the marginal function or optimal value function is characterized by different local approximations such as generalized gradients generalized directional derivatives directional hadamard or dini derivatives we give estimates for these terms which are determined by multipliers satisfying necessary optimality conditions when the functions which define the optimization problem are more than once continuously differentiable then higher order necessary conditions are employed to obtain refined estimates for the marginal function as a by product we give a new equivalent formulation of clarke s multiplier rule for nonsmooth optimization problems this shows that the set of all multipliers satisfying these necessary conditions is the union of a finite number of closed convex cones

the riccati equation for optimal control problems with mixed state control constraints necessity and sufficiency the goal of this paper is to conduct a complete study of second order conditions for the optimal control problem with mixed state control constraints the conjugate point theory is presented and a necessary condition in terms of the corresponding riccati equation is obtained sufficiency criteria are developed in terms of strengthened necessary conditions including the riccati equation the results generalize the known ones for pure control constraints as well as for the mixed state control constraints

pseudopower expansion of solutions of generalized equations and constrained optimization problems we show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter i e the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter in the polyhedral case this expansion reduces to a usual taylor expansion these results are applied to the problem of regular perturbation in constrained optimization we show that if the strong regularity condition is satisfied the property of quadratic growth holds and at least locally the solutions of the optimization problem and of the associated optimality system coincide if in addition the number of inequality constraints is finite the solution and the lagrange multiplier can be expanded in taylor series if the data are analytic the solution and the multiplier are analytic functions of the perturbation parameter

optimal beamforming for video streaming in multiantenna interference networks via diffusion limit in this paper we consider queue aware beamforming control for video streaming applications in multiantenna interference network using heavy traffic approach we derive the diffusion limit for the discrete time queuing system and obtain an ergodic excess beamforming control problem for the diffusion limit system the ergodic control problem is to minimize the average power costs of the access points subject to the constraints on the playback interruption costs and buffer overflow costs of the mobile users to deal with the queue coupling challenge we utilize the weak interference coupling property in the network using the theories of calculus and perturbation techniques we derive a closed form approximate priority function of the optimality equation and the associated error bound based on this approximation we propose a low complexity queue aware beamforming control algorithm which is asymptotically optimal for sufficiently small cross channel path gain finally the proposed scheme is compared with various baselines through simulations and it is shown that significant performance gain can be achieved

second order sufficiency and quadratic growth for nonisolated minima for standard nonlinear programming problems the weak second order sufficient condition is equivalent to the quadratic growth condition as far as the set of minima consists of isolated points and some qualification hypothesis holds this kind of condition is instrumental in the study of numerical algorithms and sensitivity analysis the arm of the paper is to study the relations between various types of sufficient conditions and quadratic growth in cases when the set of minima may have nonisolated points

nonlinear unmixing of hyperspectral data based on a linear mixture nonlinear fluctuation model spectral unmixing is an important issue to analyze remotely sensed hyperspectral data although the linear mixture model has obvious practical advantages there are many situations in which it may not be appropriate and could be advantageously replaced by a nonlinear one in this paper we formulate a new kernel based paradigm that relies on the assumption that the mixing mechanism can be described by a linear mixture of endmember spectra with additive nonlinear fluctuations defined in a reproducing kernel hilbert space this family of models has clear interpretation and allows to take complex interactions of endmembers into account extensive experiment results with both synthetic and real images illustrate the generality and effectiveness of this scheme compared with state of the art methods

generalized semi infinite programming a tutorial this tutorial presents an introduction to generalized semi infinite programming gsip which in recent years became a vivid field of active research in mathematical programming a gsip problem is characterized by an infinite number of inequality constraints and the corresponding index set depends additionally on the decision variables there exist a wide range of applications which give rise to gsip models some of them are discussed in the present paper furthermore geometric and topological properties of the feasible set and in particular the difference to the standard semi infinite case are analyzed by using first order approximations of the feasible set corresponding constraint qualifications are developed then necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so called lower level problem are used finally an overview of numerical methods is given

viscosity solutions of minimization problems viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis variational convergences for sequences of functions epi convergence gamma convergence mosco convergence and for sequences of operators graph convergence provide a flexible tool for such questions it is proved in a rather large setting that the solutions of the approximate problems converge to a viscosity solution of the original problem that is a solution that is minimal among all the solutions with respect to some viscosity criteria various examples coming from mathematical programming calculus of variations semicoercive elliptic equations phase transition theory hamilton jacobi equations singular perturbations and optimal control theory are considered

asymptotic analysis of congested communication networks this paper is devoted to the mathematical study of a routing problem in telecommunication networks when the cost function is the average delay of communications we establish asymptotic expansions for the value function and solutions in the vicinity of a congested nominal problem the study is strongly related to the one of a partial inverse barrier method for linear programming

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

a note on sensitivity of value functions of mathematical programs with complementarity constraints using standard nonlinear programming nlp theory we establish formulas for first and second order directional derivatives of optimal value functions of parametric mathematical programs with complementarity constraints mpccs the main point is that under a linear independence condition on the active constraint gradients optimal value sensitivity of mpccs is essentially the same as for nonlinear programs in spite of the combinatorial nature of the mpcc feasible set unlike nlp however second order directional derivatives of the mpcc optimal value function show combinatorial structure

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal