a new exchange method for convex semi infinite programming in this paper we propose a new exchange method for solving convex semi infinite programming csip problems we introduce a new dropping rule in the proposed exchange algorithm which only keeps those active constraints with positive lagrange multipliers moreover we exploit the idea of looking for eta infeasible indices of the lower level problem as the adding rule in our algorithm hence the algorithm does not require to solve a maximization problem over the index set at each iteration it only needs to find some points such that a certain computationally easy criterion is satisfied under some reasonable conditions the new adding dropping rule guarantees that our algorithm provides an approximate optimal solution for the csip problem in a finite number of iterations in the numerical experiments we apply the proposed algorithm to solve some test problems from the literature including some medium sized problems from complex approximation theory and fir filter design we compare our algorithm with an existing central cutting plane algorithm and with the semi infinite solver fseminf in matlab toolbox and we find that our algorithm solves the csip problem much faster for the fir filter design problem we show that our algorithm solves the problem better than some algorithms that were technically established for the problem

multiple objectives satisficing under uncertainty we propose a class of functions called multiple objective satisficing mos criteria for evaluating the level of compliance of a set of objectives in meeting their targets collectively under uncertainty the mos criteria include the joint targets achievement probability joint success probability criterion as a special case and also extend to situations when the probability distributions are not fully characterized we focus on a class of mos criteria that favors diversification which has the potential to mitigate severe shortfalls in scenarios when any objective fails to achieve its target naturally this class excludes joint success probability we further propose the shortfall aware mos criterion s mos which is inspired by the probability measure and is diversification favoring we also show how to build tractable approximations of the s mos criterion because the s mos criterion maximization is not a convex optimization problem we propose improvement algorithms via solving sequences of convex optimization problems we report encouraging computational results on a blending problem in meeting specification targets even in the absence of full probability distribution description

a general stochastic outer approximations method optimization problems involving an infinite number of constraints are considered this paper presents a general stochastic outer approximations method which incorporates mechanisms for active search of relevant constraints and for dropping of irrelevant constraints the method extracts the characteristic features of several stochastic outer approximations algorithms suggested by wardi j optim theory appl 56 1988 pp 285 311 j optim theory appl 64 1990 pp 615 640 and furthermore develops the approach to get advantages of the eaves zangwill scheme similarly to gonzaga and polak siam j control optim 17 1979 pp 477 493 the method is based on the use of quasi optimality functions satisfying some general unrestricted assumptions these functions are usually employed in the stopping criteria of numerical techniques for solving simpler problems it is shown that the method s trajectories almost surely converge to the quasi optimal set following the proposed approach a stochastic algorithm for solving the approximation problem is constructed and studied the proposed general method can be considered as a developed eaves zangwill method applying the multistart technique at each iteration for the search of relevant constraints parameters

a norm relaxed method of feasible directions for finely discretized problems from semi infinite programming abstract in this paper a class of finely discretized semi infinite programming sip problems is discussed combining the idea of the norm relaxed method of feasible directions mfd and the technique of updating discretization index set we present a new algorithm for solving the discretized semi infinite dsi problems from sip at each iteration the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem i e a quadratic program and some appropriate constraints are chosen to reduce the computational cost a high order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints under weak conditions such as mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses weak global convergence moreover the superlinear convergence is obtained under linearly independent constraint qualification licq and other assumptions in the end some elementary numerical experiments are reported

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

on approximate solutions of infinite systems of linear inequalities abstract this paper extends a j hoffman s results on approximate solutions of finite systems of linear inequalities to infinite systems of linear inequalities it is shown that for a given infinite system of linear inequalities satisfying certain conditions the euclidean distance from a vector x to the solution set of the system is equivalent to the biggest violation by x of the system thus if a vector x almost satisfies the system then x is close to a solution of the system

distributionally robust optimization and its tractable approximations in this paper we focus on a linear optimization problem with uncertainties having expectations in the objective and in the set of constraints we present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible next we develop new piecewise linear decision rules that allow a more flexible reformulation of the original problem the reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables finally we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations these bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds

a nonsmooth levenberg marquardt method for solving semi infinite programming problems in this paper we first transform the semi infinite programming problem into the kkt system by the techniques in d h li l qi j tam s y wu a smoothing newton method for semi infinite programming j global optim 30 2004 169 194 l qi s y wu g l zhou semismooth newton methods for solving semi infinite programming problems j global optim 27 2003 215 232 then a nonsmooth and inexact levenberg marquardt method is proposed for solving this kkt system based on h dan n yamashita m fukushima convergence properties of the inexact levenberg marquardt method under local error bound conditions optimim methods softw 11 2002 605 626 this method is globally and superlinearly even quadratically convergent finally some numerical results are given

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

a duality theorem for semi infinite convex programs and their finite subprograms in this paper we first establish a general recession condition under which a semi infinite convex program and its formal lagrangian dual have the same value we go on to show that under this condition the following hold first every finite subprogram with enough of the given constraints has the same value as its lagrangian dual second the weak value of the primal program is equal to the optimal value of the primal

an exact penalty function for semi infinite programming this paper introduces a global approach to the semi infinite programming problem that is based upon a generalisation of the l1 exact penalty function the advantages are that the ensuing penalty function is exact and the penalties include all violations the merit function requires integrals for the penalties which provides a consistent model for the algorithm the discretization is a result of the approximate quadrature rather than an a priori aspect of the model

robust solutions of linear programming problems contaminated with uncertain data optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed we demonstrate this phenomenon by studying 90 lps from the well known netlib collection we then apply the robust optimization methodology ben tal and nemirovski 1 3 el ghaoui et al 5 6 to produce robust solutions of the above lps which are in a sense immuned against uncertainty surprisingly for the netlib problems these robust solutions nearly lose nothing in optimality

an unconstrained convex programming approach to linear semi infinite programming in this paper an unconstrained convex programming dual approach for solving a class of linear semi infinite programming problems is proposed both primal and dual convergence results are established under some basic assumptions numerical examples are also included to illustrate this approach

some outer approximation methods for semi infinite optimization problems abstract the paper starts with a simple model and convergence theorem for outer approximation methods this general framework is used to unifyingly derive and modify certain exchange methods cutting methods and discretization methods for semi infinite programming problems by that in particular a cutting plane method for convex semi infinite programs is developed for a practically reasonable specification the method is more generally stated the subproblems in the given algorithm are moderately sized quadratic problems and each step of the algorithm can be performed by means of finitely many operations

convexity and decomposition of mean risk stochastic programs traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion a common approach to addressing risk in decision making problems is to consider a weighted mean risk objective where some dispersion statistic is used as a measure of risk we investigate the computational suitability of various mean risk objective functions in addressing risk in stochastic programming models we prove that the classical mean variance criterion leads to computational intractability even in the simplest stochastic programs on the other hand a number of alternative mean risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms we propose decomposition based parametric cutting plane algorithms to generate mean risk efficient frontiers for two particular classes of mean risk objectives

approximation in normal linear spaces abstract a historical account is given of the development of methods for solving approximation problems set in normed linear spaces approximation of both real functions and real data is considered with particular reference to lp or lp and chebyshev norms as well as coverage of methods for the usual linear problems an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters and special attention is paid to some particular nonlinear approximating families

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

optimization of convex risk functions we consider optimization problems involving convex risk functions by employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models and optimality and duality theory for problems with convex risk functions

a new proof of the strong duality theorem for semidefinite programming semidefinite programs are convex optimization problems arising in a wide variety of applications and are the extension of linear programming most methods for linear programming have been generalized to semidefinite programs just as in linear programming duality theorem plays a basic and an important role in theory as well as in algorithmics based on the discretization method and convergence property this paper proposes a new proof of the strong duality theorem for semidefinite programming which is different from other common proofs and is more simple

modifications of the first remez algorithm this paper proves the convergence of several modifications of the first remez algorithm for the solution of linear and nonlinear chebyshev approximation problems on compact b subset mathbb r s while the first remez algorithm in its original form requires the determination of the global maximum of the error function on all of b in each iteration the algorithms given here are based on its being sufficient to compute the maximum of the kth error function on a grid b k 1 where b k k geqq 0 is a prescribed sequence of finite point sets in b with density tending to zero interpreted differently some results on the discretization of chebyshev approximation problems which do not use full grids b k in b but only small subsets of them are provided the paper concludes with some numerical examples for the solution of linear multivariate problems

a linear decision based approximation approach to stochastic programming stochastic optimization especially multistage models is well known to be computationally excruciating moreover such models require exact specifications of the probability distributions of the underlying uncertainties which are often unavailable in this paper we propose tractable methods of addressing a general class of multistage stochastic optimization problems which assume only limited information of the distributions of the underlying uncertainties such as known mean support and covariance one basic idea of our methods is to approximate the recourse decisions via decision rules we first examine linear decision rules in detail and show that even for problems with complete recourse linear decision rules can be inadequate and even lead to infeasible instances hence we propose several new decision rules that improve upon linear decision rules while keeping the approximate models computationally tractable specifically our approximate models are in the forms of the so called second order cone soc programs which could be solved efficiently both in theory and in practice we also present computational evidence indicating that our approach is a viable alternative and possibly advantageous to existing stochastic optimization solution techniques in solving a two stage stochastic optimization problem with complete recourse

a globally convergent sqp method for semi infinite nonlinear optimization abstract a new approach for semi infinite programming problems is presented which belongs to the class of successive quadratic programming sqp methods with trust region technique the proposed algorithm employs the exact l penalty function as a criterion function and incorporates an appropriate scheme for estimating active constraints it is proved that the algorithm is globally convergent under some assumptions numerical experiments show that the algorithm is very promising in practice

on the numerical treatment of linearly constrained semi infinite optimization problems abstract we consider the application of two primal algorithms to solve linear semi infinite programming problems depending on a real parameter combining a simplex type strategy with a feasible direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently the second algorithm runs a feasible direction method first and then switches to the purification procedure the linear programming subproblems that yield the search direction involve only a small subset of the constraints these subsets are updated at each iteration using a multi local optimization algorithm numerical test examples taken from the literature in order to compare the numerical effort with other methods show the efficiency of the proposed algorithms

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

discretization method for semi definite programming semi definite programs are convex optimization problems arising in a wide variety of applications and the extension of linear programming most methods for linear programming have been generalized to semi definite programs this paper discusses the discretization method in semi definite programming the convergence and the convergent rate of error between the optimal value of the semi definite programming problems and the optimal value of the discretized problems are obtained an approximately optimal division is given under certain assumptions with the significance of the convergence property the duality result in semi definite programs is proved in a simple way which is different from the other common proofs

optimization under probabilistic envelope constraints chance constraints are an important modeling tool in stochastic optimization providing probabilistic guarantees that a solution succeeds in satisfying a given constraint although they control the probability of success they provide no control whatsoever in the event of a failure that is they do not distinguish between a slight overshoot or undershoot of the bounds and more catastrophic violation in short they do not capture the magnitude of violation of the bounds this paper addresses precisely this topic focusing on linear constraints and ellipsoidal gaussian like uncertainties we show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi infinite optimization problem we provide conditions that guarantee polynomial time solvability of the resulting semi infinite formulation we show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature as a byproduct we provide tight probabilistic bounds for comprehensive robust optimization thus analogously to the connection between chance constraints and robust optimization we provide a broader connection between probabilistic envelope constraints and comprehensive robust optimization

conformal modulus and its reduction via mean curvatures abstract this note shows that the conformal modulus of a condenser and the reduced conformal modulus of a compact subset in r n 2 are naturally but optimally tied with the corresponding graphic mass and the semi diameter through mean curvatures

from cvar to uncertainty set implications in joint chance constrained optimization we review and develop different tractable approximations to individual chance constrained problems in robust optimization on a variety of uncertainty sets and show their interesting connections with bounds on the conditional value at risk cvar measure we extend the idea to joint chance constrained problems and provide a new formulation that improves upon the standard approach our approach builds on a classical worst case bound for order statistics problems and is applicable even if the constraints are correlated we provide an application of the model on a network resource allocation problem with uncertain demand

existence of classical solutions to a free boundary problem for the p laplace operator i the exterior convex case in this paper we prove under convexity assumptions for the data the existence of classical solutions for a bernoulli type free boundary problem with the p laplacian as the governing operator the method employed here originates from a pioneering work of a beurling where he proves the existence for the harmonic case in the plane though with no geometrical restrictions

a note on an implementation of a method for quadratic semi infinite programming for convex quadratic semi infinite programming problems afortran package is described a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements powell s 1983 numerically stable convex quadratic programming implementation is used to solve the qp subproblems

a note on the computation of an orthonormal basis for the null space of a matrix a highly regarded method to obtain an orthonormal basis z for the null space of a matrix a t is the qr decomposition of a where q is the product of householder matrices in several optimization contexts a x varies continuously with x and it is desirable the z x vary continuously also in this note we demonstrate that the standard implementation of the qr decomposition does not yield an orthonormal basis z x whose elements vary continuously with x we suggest three possible remedies

a globally convergent algorithm with adaptively refined discretization for semi infinite optimization problems arising in engineering design although most of the algorithms that have been proposed for the solution of semi infinite optimization problems make use at each iteration of a set of local maximizers over the range of the independent parameter the question of suitably approximating such maximizers is generally left aside it has been suggested that this issue can be addressed by means of an adaptively refined discretization of the interval of variation of the independent parameter the algorithm proposed in the paper makes use of such a technique and by means of a certain memory mechanism avoids the potential lack of convergence suffered by an existing algorithm while requiring a relatively small number of gradient evaluations

geometrical logarithmic capacitance abstract this paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance a very key notion in mathematical physics quasiconformal geometry and variational calculus through 1 identifying with the reduced conformal module 2 evaluating the minimal log potential energy 3 relating to both the volume radius and the surface radius 4 linking with the n harmonic radius and the log capacity of the kevin image of a compact surface 5 finding the minkowski inequality and the general variational formula for the log capacity 6 pinching the log isocapacitary inequality from left and solving the left prescribed problem for the normalized log capacitary curvature measure 7 pinching the log isocapacitary inequality from right and handling the right prescribed problem for the normalized log capacitary curvature measure 8 handling an overdetermination of the n equilibrium potential of a given convex body via the log capacitary concavity

a projected lagrangian algorithm for semi infinite programming a globally convergent algorithm is presented for the solution of a wide class of semi infinite programming problems the method is based on the solution of a sequence of equality constrained quadratic programming problems and usually has a second order convergence rate numerical results illustrating the method are given

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose

on the capacity of surfaces in manifolds with nonnegative scalar curvature given a surface in an asymptotically flat 3 manifold with nonnegative scalar curvature we derive an upper bound for the capacity of the surface in terms of the area of the surface and the willmore functional of the surface the capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at even in the special case of 3 this is a new estimate more generally equality holds precisely for a spherically symmetric sphere in a spatial schwarzschild 3 manifold as applications we obtain inequalities relating the capacity of the surface to the hawking mass of the surface and the total mass of the asymptotically flat manifold

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

an approximation technique for the numerical solution of a stefan problem a nonlinear approximation technique for the numerical solution of certain free boundary problems is proposed the method is shown for a degenerate one dimensional stefan problem for this problem an error estimate which is independent of the used algorithm is derived numerical examples are discussed

solving air traffic conflict problems via local continuous optimization this paper first introduces an original trajectory model using b splines and a new semi infinite programming formulation of the separation constraint involved in air traffic conflict problems a new continuous optimization formulation of the tactical conflict resolution problem is then proposed it involves very few optimization variables in that one needs only one optimization variable to determine each aircraft trajectory encouraging numerical experiments show that this approach is viable on realistic test problems not only does one not need to rely on the traditional discretized combinatorial optimization approaches to this problem but moreover local continuous optimization methods which require relatively fewer iterations and thereby fewer costly function evaluations are shown to improve the performance of the overall global optimization of this non convex problem

an accelerated central cutting plane algorithm for linear semi infinite programming an algorithm for linear semi infinite programming is presented which accelerates the convergence of the central cutting plane algorithm first proposed in 4 compared with other algorithms the algorithm in 4 has the advantage of being applicable under mild conditions and of providing feasible solutions however its convergence has been shown to be rather slow in practical instances the algorithm proposed in this paper introduces a simple acceleration scheme which gives faster convergence as confirmed by several examples as well as an interval of prefixed length containing the optimum value it is also shown that the algorithm provides a solution of the dual problem and that it can be used for convex semi infinite programming too

n capacity n harmonic radius and n harmonic transplantation abstract for the n laplacian n on a regular domain the n green s function exists this allows us to define the n robin s function and the n harmonic radius we show their basic properties and extend the method of the harmonic transplantation to that of the n harmonic transplantation the method is used to estimate the first eigenvalue of the n laplacian n we also give another proof of the isoperimetric inequality for the n capacity and give the isoperimetric inequality for the n harmonic radius

the equality case of the penrose inequality for asymptotically flat graphs we prove the equality case of the penrose inequality in all dimensions for asymptotically flat hypersurfaces it was re cently proven by g lam 19 that the penrose inequality holds for asymptotically flat graphical hypersurfaces in euclidean space with non negative scalar curvature and with a minimal boundary our main theorem states that if the equality holds then the hy persurface is a schwarzschild solution as part of our proof we show that asymptotically flat graphical hypersurfaces with a min imal boundary and non negative scalar curvature must be mean convex using the argument that we developed in 15 this en ables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation

the maximum entropy method on the mean applications to linear programming and superresolution in this paper we give two different results we propose new methods to solve classical optimization problems in linear programming we also obtain precise quantitative results for the superresolution phenomenon as observed earlier by practical searchers on specific algorithms the common background of our work is the generalized moment problem which is known to be connected with linear programming and superresolution we describe the maximum entropy method on the mean that provides solution to the problem and leads to computational criteria to decide the existence of solutions or not

a one phase algorithm for semi infinite linear programming we present an algorithm for solving a large class of semi infinite linear programming problems this algorithm has several advantages it handles feasibility and optimality together it has very weak restrictions on the constraints it allows cuts that are not near the most violated cut and it solves the primal and the dual problems simultaneously we prove the convergence of this algorithm in two steps first we show that the algorithm can find ane optimal solution after finitely many iterations then we use this result to show that it can find an optimal solution in the limit we also estimate how good ane optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding ane optimal solution under some assumptions this algorithm is generalized to solve a class of nonlinear semi infinite programming problems applications to convex programming are discussed

a comparative study of several semi infinite nonlinear programming algorithms abstract a computational comparison of several recent semi infinite nonlinear programming algorithms is presented because of the importance of this area there have recently been proposed a number of algorithms having a global convergence property the algorithms used in this study include a feasible direction method and some variants of successive quadratic programming methods robustness and relative efficiency of the algorithms are examined on the basis of the results for some standard test examples chosen from earlier literature

goal scoring coherent loss and applications to machine learning motivated by the binary classification problem in machine learning we study in this paper a class of decision problems where the decision maker has a list of goals from which he aims to attain the maximal possible number of goals in binary classification this essentially means seeking a prediction rule to achieve the lowest probability of misclassification and computationally it involves minimizing a difficult non convex 0 1 loss function to address the intractability previous methods consider minimizing the cumulative loss the sum of convex surrogates of the 0 1 loss of each goal we revisit this paradigm and develop instead an axiomatic framework by proposing a set of salient properties on functions for goal scoring and then propose the coherent loss approach which is a tractable upper bound of the loss over the entire set of goals we show that the proposed approach yields a strictly tighter approximation to the total loss i e the number of missed goals than any convex cumulative loss approach while preserving the convexity of the underlying optimization problem moreover this approach applied to for binary classification also has a robustness interpretation which builds a connection to robust svms

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

a robust optimization perspective on stochastic programming in this paper we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations these deviation measures capture distributional asymmetry and lead to better approximations of chance constraints using a linear decision rule we also propose a tractable approximation approach for solving a class of multistage chance constrained stochastic linear optimization problems an attractive feature of the framework is that we convert the original model into a second order cone program which is computationally tractable both in theory and in practice we demonstrate the framework through an application of a project management problem with uncertain activity completion time

an implementation of a discretization method for semi infinite programming for linear semi infinite programming problems a discretization method is presented a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements a numerically stable simplex algorithm is used to solve the lp subproblems numerical examples from bivariate chebyshev approximation are presented

goal driven optimization we develop a goal driven stochastic optimization model that considers a random objective function in achieving an aspiration level target or goal our model maximizes the shortfall aware aspiration level criterion which encompasses the probability of success in achieving the aspiration level and an expected level of underperformance or shortfall the key advantage of the proposed model is its tractability we can obtain its solution by solving a small collection of stochastic linear optimization problems with objectives evaluated under the popular conditional value at risk cvar measure using techniques in robust optimization we propose a decision rule based deterministic approximation of the goal driven optimization problem by solving subproblems whose number is a polynomial with respect to the accuracy with each subproblem being a second order cone optimization problem socp we compare the numerical performance of the deterministic approximation with sampling based approximation and report the computational insights on a multiproduct newsvendor problem

a caching game with infinitely divisible hidden material we consider a caching game in which a unit amount of infinitely divisible material is distributed among n geq 2 locations a searcher chooses how to distribute his search effort r about the locations so as to maximize the probability she will find a given minimum amount bar m 1 m leq r of the material if the search effort y i invested by the searcher in a given location i is at least as great as the amount of material x i located there she finds all of it otherwise the amount she finds is only y i in other words she finds min left x i y i right in location i we seek the randomized distribution of search effort that maximizes the probability of success for the searcher in the worst case hence we model the problem as a zero sum win lose game between the searcher and a malevolent hider who wishes to keep more than m of the material we show that in the case r bar m the game has a geometric interpretation that for n 2 corresponds to a problem posed by w h ruckle in his monograph geometric games and their applications pitman boston 1983 we give solutions for the geometric game when n 3 for certain values of m and bounds on the value for other values of m in the more general case r geq bar m we show that for n 2 the game reduces to ruckle s game

chance constrained uncertain classification via robust optimization this paper studies the problem of constructing robust classifiers when the training is plagued with uncertainty the problem is posed as a chance constrained program ccp which ensures that the uncertain data points are classified correctly with high probability unfortunately such a ccp turns out to be intractable the key novelty is in employing bernstein bounding schemes to relax the ccp as a convex second order cone program whose solution is guaranteed to satisfy the probabilistic constraint prior to this work only the chebyshev based relaxations were exploited in learning algorithms bernstein bounds employ richer partial information and hence can be far less conservative than chebyshev bounds due to this efficient modeling of uncertainty the resulting classifiers achieve higher classification margins and hence better generalization methodologies for classifying uncertain test data points and error measures for evaluating classifiers robust to uncertain data are discussed experimental results on synthetic and real world datasets show that the proposed classifiers are better equipped to handle data uncertainty and outperform state of the art in many cases

solving quadratic semi infinite programming problems by using relaxed cutting plane scheme this paper studies the cutting plane approach for solving quadratic semi infinite programming problems several relaxation techniques and their combinations are proposed and discussed a flexible convergence proof is provided to cover different settings of a relaxation scheme the implementation issues are addressed with some numerical experiments to illustrate the computational behavior of each different combination

robust convex quadratically constrained programs in this paper we study robust convex quadratically constrained programs a subset of the class of robust convex programs introduced by ben tal and nemirovski 4 in contrast to 4 where it is shown that such robust problems can be formulated as semidefinite programs our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second order cone programs socp we propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit socp and present examples where these classes of uncertainty sets are natural

relaxation based bounds for semi infinite programs finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi infinite programs sips involving smooth potentially nonconvex objective function and constraints the lower bounding problem is obtained by a formulation that combines the first and second order kkt necessary conditions of the lower level problem with a discretization of the parameter set the resulting mathematical program with equilibrium constraints mpec is a relaxation of the original sip and furnishes valid lower bounds if the parameter set is subdivided the optimal solution value of the lower bounding problem converges to the optimal solution value of the sip the upper bounding problem is based on convex and linear relaxations of the lower level problem resulting in a restriction of the sip if the parameter set is subdivided the constructed relaxations converge to the original lower level program the existence of sip slater points ensures convergence of the upper bounding problems to the optimal solution value of the sip several alternatives for the upper bounding problem are presented and discussed numerical results are presented for a number of test problems from the literature

a nonmonotonic trust region algorithm for a class of semi infinite minimax programming based on the discretization methods for solving semi infinite programming problems this paper presents a new nonmonotonic trust region algorithm for a class of semi infinite minimax programming problem under some mild assumptions the global convergence of the proposed algorithm is given numerical tests are reported that show the efficiency of the proposed method