a superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming we present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the mangasarian fromovitz constraint qualification and the quadratic growth condition this convergence result is obtained despite the potential lack of a locally convex augmented lagrangian the algorithm solves a succession of subproblems that have quadratic objectives and quadratic constraints both possibly nonconvex by the use of a trust region constraint we guarantee that any stationary point of the subproblem induces superlinear convergence which avoids the problem of computing a global minimum we compare this algorithm with sequential quadratic programming algorithms on several degenerate nonlinear programs

sensitivity analysis of solutions to optimization problems in hilbert spaces with applications to optimal control and estimation abstract this research provides sufficient conditions for lipschitz continuity and directional differentiability of the solutions and the lagrange multipliers of constraint nonlinear programming problems in hilbert spaces the abstract results are applied to problems in optimal control and parameter estimation

a quasi newton method for estimating the parameter in a nonlinear hyperbolic system a quasi newton method qnm in infinite dimensional spaces for estimating parameters involved in distributed parameter systems is presented in this paper next the global convergence of a sequence generated by the algorithm qnm is also proved we apply the algorithm qnm to an identification problem of a distributed parameter system governed by a nonlinear hyperbolic partial differential equation

perturbed optimization in banach spaces iii semi infinite optimization this paper is devoted to the study of perturbed semi infinite optimization problems i e minimization over er n with an infinite number of inequality constraints we obtain the second order expansion of the optimal value function and the first order expansion of approximate optimal solutions in two cases i when the number of binding constraints is finite and ii when the inequality constraints are parametrized by a real scalar these results are partly obtained by specializing the sensitivity theory for perturbed optimization developed in part i cf siam j control optim 34 1996 pp 1151 1171 and deriving specific sharp lower estimates for the optimal value function which take into account the curvature of the positive cone in the space c omega of continuous real valued functions

the extreme points of the set of decreasing failure rate distributions since the class of extended decreasing failure rate edfr life distributions i e distributions with support in 0 is compact and convex it follows from choquet s theorem that every edfr life distribution can be represented as a mixture of extreme points of the edfr class we identify the extreme points of this class and of the standard class of decresing failure rate dfr life distributions further we show that even though the convex class of dfr life distributions is not compact every dfr life distribution can be represented as a mixture of extreme points of the dfr class

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

maximum likelihood estimation for two parameter decreasing hazard rate distributions using censored data abstract problems of maximum likelihood estimation are discussed for shape and scale parameters from certain decreasing hazard rate distributions typically either mixed exponential or work hardened sufficient conditions on the mixing distribution are given that guarantee regular behavior of the hazard rate and that ensure even with highly censored data that the mle s exist from such dhr distributions whenever the sample satisfies a certain condition otherwise a constant hazard rate is estimated as a limiting case some computational methods are given and applications made

an algorithm for degenerate nonlinear programming with rapid local convergence this paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution the framework is constructed from three main algorithmic ingredients the first is any conventional method for nonlinear programming that produces estimates of the lagrange multipliers at each iteration the second is a technique for estimating the set of active constraint indices the third is a stabilized lagrange newton algorithm with rapid local convergence properties results concerning rapid local convergence and global convergence of the proposed framework are proved the approach improves on existing approaches in that less restrictive assumptions are needed for convergence and or the computational workload at each iteration is lower

sensitivity analysis of parametrized programs via generalized equations this paper investigates local behavior of optimal solutions of parametrized optimization problems with cone constraints in banach spaces the corresponding first order optimality conditions are formulated in a form of generalized equations variational inequalities and solutions of these generalized equations are studied it is shown that under certain second order sufficient optimality conditions and a regularity assumption related to the associated lagrange multipliers the considered optimal solutions are lipschitzian stable this is compared with a similar result in shapiro and bonnans siam j control optim 30 1992 pp 1409 1422 under the additional assumption of uniqueness of the lagrange multipliers first order expansions of the optimal solutions are given in terms of solutions of auxiliary optimization problems finally as an example semi infinite programming problems are discussed

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

properties of the log barrier function on degenerate nonlinear programs we examine the sequence of local minimizers of the log barrier function for a nonlinear program near a solution at which second order sufficient conditions and the mangasarian fromovitz constraint qualification are satisfied but the active constraint gradients are not necessarily linearly independent when a strict complementarity condition is satisfied we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution and we obtain a semiexplicit characterization of this point when strict complementarity does not hold we obtain several other interesting characterizations in particular an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution

free random levy matrices using the theory of free random variables and the coulomb gas analogy we construct stable random matrix ensembles that are random matrix generalizations of the classical one dimensional stable levy distributions we show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit we solve these equations in a number of cases and show that the eigenvalue distributions exhibit levy tails for the analytically known levy measures we explicitly construct the density of states using the method of orthogonal polynomials we show that the levy tail distributions are characterized by a different novel form of microscopic universality

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

convergence analysis of a regularized interior point algorithm for the barrier problems with singular solutions abstract wachter and biegler 2006 25 presented and implemented an interior point filter line search method called ipopt for large scale nonlinear programming to make ipopt efficient and robust they add some safeguards in their implementation one crucial safeguard is to apply the inertia correction technique to modify the kkt matrix we focus on the degenerate problems in which the kkt matrix might be singular to complement theoretical foundation of this technique in the degenerate case we propose and analyze a regularized interior point method for the barrier problems with singular solutions which largely relies on ipopt based on the inertia correction technique the matrix arising in the kkt equation is corrected to be a nonsingular one so that the newton step can be calculated at each iteration even in the degenerate case we emphasize that the special inertia correction for the kkt matrix does not impose any negative impact on the fast convergence of our algorithm under some local error bound condition we prove that the rate of quadratic convergence is achieved when the second order condition or some constraint qualification condition fails to hold or both

rigorous mean field model for coherent potential approximation anderson model with free random variables a model of a randomly disordered system with site diagonal random energy fluctuations is introduced it is an extension of wegner sn orbital model to arbitrary eigenvalue distribution in the electronic level space the new feature is that the random energy values are not assumed to be independent at different sites but free freeness of random variables is an analog of the concept of independence for noncommuting random operators a possible realization is the ensemble of randomly rotated matrices at different lattice sites the one and two particle green functions of the proposed hamiltonian are calculated exactly the eigenstates are extended and the conductivity is novanishing everywhere inside the band the long range behavior and the zero frequency limit of the two particle green function are universal with respect to the eigenvalue distribution in the electronic level space the solutions solve the cpa equation for the one and two particle green function of the corresponding anderson model thus our multisite model is a rigorous mean field model for the single site cpa we show how the lloyd model is included in our model and treat various kinds of noises

general random matrix approach to account for the effect of static disorder on the spectral properties of light harvesting systems we develop a random matrix model approach to study static disorder in pigment protein complexes in photosynthetic organisms as a case study we examine the ring of b850 bacteriochlorophylls in the peripheral light harvesting complex of rhodospirillum molischianum formulated in terms of an effective hamiltonian describing the collective electronic excitations of the system we numerically examine and compare various models of disorder and observe that both the density of states and the absorption spectrum of the model show remarkable spectral universality for the case of unitary disorder we develop a method to analytically evaluate the density of states of the ensemble using the supersymmetric formulation of random matrix theory succinct formulas that can be readily applied in future studies are provided in an appendix

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

inexact subgradient methods for quasi convex optimization problems in this paper we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi convex constrained optimization problem the inexactness stems from computation errors and noise which come from practical considerations and applications assuming that the computational errors and noise are deterministic and bounded we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima in both cases using the constant and diminishing stepsize rules we describe convergence results in both objective values and iterates and finite convergence to approximate optimality we also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the cobb douglas production efficiency problem the numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm especially for the large scale problems

perturbation analysis of second order cone programming problems we discuss first and second order optimality conditions for nonlinear second order cone programming problems and their relation with semidefinite programming problems for doing this we extend in an abstract setting the notion of optimal partition then we state a characterization of strong regularity in terms of second order optimality conditions this is the first time such a characterization is given for a nonpolyhedral conic problem

on using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints we investigate the possibility of solving mathematical programs with complementarity constraints mpccs using algorithms and procedures of smooth nonlinear programming although mpccs do not satisfy a constraint qualification we establish sufficient conditions for their lagrange multiplier set to be nonempty mpccs that have nonempty lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter in this context the elastic mode transforms mpcc into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem which in turn makes it approachable by sequential quadratic programming sqp algorithms one such algorithm is shown to achieve local linear convergence once the problem is relaxed under stronger conditions we also prove superlinear convergence to the solution of an mpcc using an adaptive elastic mode approach for an sqp algorithm recently analyzed in an mpcc context in r fletcher s leyffer s sholtes and d ralph local convergence of sqp methods for mathematical programs with equilibrium constraints tech report na 210 university of dundee dundee uk 2002 our assumptions are more general since we do not use a critical assumption from that reference in addition we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

an interior affine scaling cubic regularization algorithm for derivative free optimization subject to bound constraints abstract in this paper we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function we first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method at each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique the global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions preliminary numerical results are reported to show the effectiveness of the proposed algorithm

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

sensitivity analysis of control constrained optimal control problems for distributed parameter systems the differential stability of solutions to control constrained quadratic optimal control problems for distributed parameter systems is investigated in this paper the form of the right derivatives of optimal controls for such problems with respect to the real parameter is derived the differential sensitivity of optimal controls with respect to the perturbations of the coefficients of the state equation as well as to the deformations of the domain of integration is considered the right derivative of an optimal control with respect to the parameter is obtained in the form of an optimal control to the auxiliary control constrained optimal control problem

free random variables and molecular spectra abstract we briefly review the method of free random variables and its relation to random matrices to demonstrate use of the approach we apply the formalism to study the effect of static disorder on the density of states in a model hamiltonian system

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 convergence rates of linearized proximal algorithms for convex composite optimization with applications in the present paper we investigate a linearized proximal algorithm lpa for solving a convex composite optimization problem each iteration of the lpa is a proximal minimization of the convex composite function with the inner function being linearized at the current iterate the lpa has the attractive computational advantage that the solution of each subproblem is a singleton which avoids the difficulty as in the gauss newton method gnm of finding a solution with minimum norm among the set of minima of its subproblem while still maintaining the same local convergence rate as that of the gnm under the assumptions of local weak sharp minima of order p p in 1 2 and a quasi regularity condition we establish a local superlinear convergence rate for the lpa we also propose a globalization strategy for the lpa based on a backtracking line search and an inexact version of the lpa we further apply the lpa to solve a possibly nonconvex feasibility problem as well as a sensor network localiza

degenerate nonlinear programming with a quadratic growth condition we show that the quadratic growth condition and the mangasarian fromovitz constraint qualification mfcq imply that local minima of nonlinear programs are isolated stationary points as a result when started sufficiently close to such points an l infty exact penalty sequential quadratic programming algorithm will induce at least r linear convergence of the iterates to such a local minimum we construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the mfcq but for which no positive semidefinite augmented lagrangian exists we present numerical results obtained using several nonlinear programming packages on this example and discuss its implications for some algorithms

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

karush kuhn tucker systems regularity conditions error bounds and a class of newton type methods we consider optimality systems of karush kuhn tucker kkt type which arise for example as primal dual conditions characterizing solutions of optimization problems or variational inequalities in particular we discuss error bounds and newton type methods for such systems an exhaustive comparison of various regularity conditions which arise in this context is given we obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for kkt systems such as quasi regularity or semistability equivalently the r0 property error bounds are useful among oth er things for identifying active constraints and developing efficient local algorithms we propose a family of local newton type algorithms this family contains some known active set newton methods as well as some new methods regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth newton methods and sequential quadratic programming methods

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

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

on the behavior of certain maximum likelihood estimators from large randomly censored samples abstract in the case the survival distribution of a life length x is of the form ln 1 fx x q x for x 0 where the hazard function q is known but or or both are unknown the asymptotic distribution of the maximum likelihood estimators of the shape and scale parameters is found for large randomly censored samples the covariance matrix of is expressed in terms of the distribution of the random variable that can be observed namely y min x t here t is the random censoring time independent of x sufficient conditions are found such as t having only finite support to ensure that the estimates are strongly consistent and asymptotically normal application is made to decreasing hazard rate distributions

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

directional input adaptation in parametric optimal control problems this paper deals with input adaptation in dynamic processes in order to guarantee feasible and optimal operation despite the presence of uncertainty for those optimal control problems having terminal and mixed control state path constraints two orthogonal sets of adaptation directions can be distinguished in the input space the sensitivity seeking directions along which a small variation from an optimal nominal solution will not affect the respective active constraints and the complementary constraint seeking directions along which a variation will affect the respective constraints it follows that selective input adaptation strategies can be defined namely adaptation in the sensitivity and constraint seeking directions this paper proves the important result that for small parametric perturbations the cost variation resulting from adaptation in the sensitivity seeking directions over no input adaptation is typically smaller than the cost variation due to adaptation in the constraint seeking

characterizations of weak sharp minima for lower c1 functions abstract we derive characterizations of weak sharp minimizers of order one for the class of lower c 1 functions the characterizations of such minimizers are obtained for a nonlinear programming problem with an abstract set constraint the first characterization is formulated in terms of the proximal normals to a given set relative to the abstract set constraint and the directional derivative of the objective function two examples are given to illustrate this characterization the other characterizations are extensions of the main characterizations of weak sharp minima for convex functions recently proved by burke and deng 2002 13 theorems 2 2 and 2 3 to the class of lower c 1 functions

on the aubin property of critical points to perturbed second order cone programs two gaps were found in the proof of the main theorems theorems 21 and 26 of the paper on the aubin property of critical points to perturbed second order cone programs siam j optim 21 2011 3 pp 798 823 by j v outrata and h ramirez c in this note both these gaps will be filled as to the second one a new technical result will be employed which may possibly be used also in other situations

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

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools