sequential convex subdifferential calculus and sequential lagrange multipliers the aim of this paper is to establish formulas for the subdifferentials of the sum and the composition of convex functions in terms of the subdifferentials of the data functions at nearby points applications to general optimization problems lead to a new notion of sequential lagrange multipliers

strong duality in robust convex programming complete characterizations duality theory has played a key role in convex programming in the absence of data uncertainty in this paper we present a duality theory for convex programming problems in the face of data uncertainty via robust optimization we characterize strong duality between the robust counterpart of an uncertain convex program and the optimistic counterpart of its uncertain lagrangian dual we provide a new robust characteristic cone constraint qualification which is necessary and sufficient for strong duality in the sense that the constraint qualification holds if and only if strong duality holds for every convex objective function of the program we further show that this strong duality always holds for uncertain polyhedral convex programming problems by verifying our constraint qualification where the uncertainty set is a polytope we derive these results by way of first establishing a robust theorem of the alternative for parameterized convex inequality systems using conjugate analysis we also give a convex characteristic cone constraint qualification that is necessary and sufficient for strong duality between the deterministic dual pair the robust counterpart and its lagrangian dual through simple numerical examples we also provide an insightful account of the development of our duality theory

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

subsmooth semi infinite and infinite optimization problems we first consider subsmoothness for a function family and provide formulas of the subdifferential of the pointwise supremum of a family of subsmooth functions next we consider subsmooth infinite and semi infinite optimization problems in particular we provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum for such optimization problems

majorizing functions and convergence of the gauss newton method for convex composite optimization we introduce a notion of quasi regularity for points with respect to the inclusion f x in c where f is a nonlinear frechet differentiable function from mathbb r v to mathbb r m when c is the set of minimum points of a convex real valued function h on mathbb r m and f satisfies the l average lipschitz condition of wang we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the gauss newton method with quasi regular initial points for the convex composite function h circ f results are new even when the initial point is regular and f is lipschitz

regularity conditions characterizing fenchel lagrange duality and farkas type results in dc infinite programming abstract in this paper we consider a dc infinite programming problem p with inequality constraints by using the properties of the epigraph of the conjugate functions we introduce some new notions of regularity conditions for p under these new regularity conditions we completely characterize the fenchel lagrange duality and the stable fenchel lagrange duality for p similarly we also completely characterize the farkas type results and the stable farkas type results for p as applications we obtain the corresponding results for conic programming problems

the secq linear regularity and the strong chip for an infinite system of closed convex sets in normed linear spaces we consider a finite or infinite family of closed convex sets with nonempty intersection in a normed space a property relating their epigraphs with their intersection s epigraph is studied and its relations to other constraint qualifications such as the linear regularity the strong chip and jameson s g property are established with suitable continuity assumption we show how this property can be ensured from the corresponding property of some of its finite subfamilies

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

asymptotic closure condition and fenchel duality for dc optimization problems in locally convex spaces a b s t r a c t we consider the dc optimization problem p inf x f1 x f2 x g1 ax g2 ax where f1 f2 g1 and g2 are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear continuous operator from x to y adopting the standard convexification technique a fenchel dual problem of p is given by using properties of the epigraph of conjugate functions some sufficient and necessary conditions for the fenchel duality and for the stable fenchel duality of p are provided

generalizations of slater s constraint qualification for infinite convex programs in this paper we study constraint qualifications and duality results for infinite convex programs p inf f x g x s x c whereg g1 g2 ands s1 s2 si are convex cones i 1 2 c is a convex subset of a vector spacex andf andgi are respectively convex andsi convex i 1 2 in particular we consider the special case whens2 is in afinite dimensional space g2 is affine ands2 is polyhedral we show that a recently introduced simple constraint qualification and the so called quasi relative interior constraint qualification both extend to p from the special case thatg g2 is affine ands s2 is polyhedral in a finite dimensional space the so called partially finite program this provides generalized slater type conditions for p which are much weaker than the standard slater condition we exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong2 and the finite dimensionality assumption ons2 by a local compactness assumption we then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones

tangential extremal principles for finite and infinite systems of sets ii applications to semi infinite and multiobjective optimization this paper contains selected applications of the new tangential extremal principles and related results developed in mordukhovich and phan math program 2011 to calculus rules for infinite intersections of sets and optimality conditions for problems of semi infinite programming and multiobjective optimization with countable constraints

new regularity conditions for strong and total fenchel lagrange duality in infinite dimensional spaces abstract we give new regularity conditions for convex optimization problems in separated locally convex spaces we completely characterize the stable strong and strong fenchel lagrange duality then we give similar statements for the case when a solution of the primal problem is assumed as known obtaining complete characterizations for the so called total and stable totalfenchel lagrange duality respectively for particular settings the conditions that we consider turn into some constraint qualifications already used by different authors like farkas minkowski cq locally farkas minkowski cq and basic cq and we rediscover and improve some recent results from the literature

subdifferentials of a minimum time function in banach spaces in general banach space setting we study the minimum time function determined by a closed convex set k and a closed set s this function is simply the usual minkowski function of k if s is the singleton consisting of the origin in particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function

duality gap in convex programming in this paper we consider general convex programming problems and give a sufficient condition for the equality of the infimum of the original problem and the supremum of its ordinary dual this condition may be seen as a continuity assumption on the constraint sets i e on the sublevel sets of the constraint function under linear perturbation it allows us to generalize as well as treat previous results in a unified framework our main result is in fact based on a quite general constraint qualification result for quasiconvex programs involving a convex objective function proven in the setting of a real normed vector space

subdifferential calculus rules for supremum functions in convex analysis extending and improving some recent results of hantoute lopez and z linescu and others we provide characterization conditions for subdifferential formulas to hold for the supremum function of a family of convex functions on a real locally convex space

regularity conditions via quasi relative interior in convex programming we give some new regularity conditions for fenchel duality in separated locally convex vector spaces written in terms of the notion of quasi interior and quasi relative interior respectively we provide also an example of a convex optimization problem for which the classical generalized interior point conditions given so far in the literature cannot be applied while the one given by us is applicable by using a technique developed by magnanti we derive some duality results for the optimization problem with cone constraints and its lagrange dual problem and we show that a duality result recently given in the literature for this pair of problems has self contradictory assumptions

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

information loss under coarse graining a geometric approach we use information geometry in which the local distance between models measures their distinguishability from data to quantify the flow of information under the renormalization group we show that information about relevant parameters is preserved with distances along relevant directions maintained under flow by contrast irrelevant parameters become less distinguishable under the flow with distances along irrelevant directions contracting according to renormalization group exponents we develop a covariant formalism to understand the contraction of the model manifold we then apply our tools to understand the emergence of the diffusion equation and more general statistical systems described by a free energy our results give an information theoretic justification of universality in terms of the flow of the model manifold under coarse graining

a comparison of constraint qualifications in infinite dimensional convex programming in this paper the relationships between various constraint qualifications for infinite dimensional convex programs are investigated using robinson s refinement of the duality result of rockafellar it is demonstrated that the constraint qualification proposed by rockafellar provides a systematic mechanism for comparing many constraint qualifications as well as establishing new results in different topological environments

farkas type results with conjugate functions we present some new farkas type results for inequality systems involving a finite as well as an infinite number of convex constraints for this we use two kinds of conjugate dual problems namely an extended fenchel type dual problem and the recently introduced fenchel lagrange dual problem for the latter which is a combination of the classical fenchel and lagrange duals the strong duality is established

geometry of nonlinear least squares with applications to sloppy models and optimization parameter estimation by nonlinear least squares minimization is a common problem that has an elegant geometric interpretation the possible parameter values of a model induce a manifold within the space of data predictions the minimization problem is then to find the point on the manifold closest to the experimental data we show that the model manifolds of a large class of models known as sloppy models have many universal features they are characterized by a geometric series of widths extrinsic curvatures and parameter effect curvatures which we describe as a hyper ribbon a number of common difficulties in optimizing least squares problems are due to this common geometric structure first algorithms tend to run into the boundaries of the model manifold causing parameters to diverge or become unphysical before they have been optimized we introduce the model graph as an extension of the model manifold to remedy this problem we argue that appropriate priors can remove the boundaries and further improve the convergence rates we show that typical fits will have many evaporated parameters unless the data are very accurately known second bare model parameters are usually ill suited to describing model behavior cost contours in parameter space tend to form hierarchies of plateaus and long narrow canyons geometrically we understand this inconvenient parametrization as an extremely skewed coordinate basis and show that it induces a large parameter effect curvature on the manifold by constructing alternative coordinates based on geodesic motion we show that these long narrow canyons are transformed in many cases into a single quadratic isotropic basin we interpret the modified gauss newton and levenberg marquardt fitting algorithms as an euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively by adding a geodesic acceleration adjustment to these algorithms we alleviate the difficulties from parameter effect curvature improving both efficiency and success rates at finding good fits

robust conjugate duality for convex optimization under uncertainty with application to data classification in this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization extending the powerful conjugate duality technique we first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition this regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem more importantly we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem as an application we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets the support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences

existence of optimal solutions and duality results under weak conditions in this paper we consider an ordinary convex program with no qualification conditions such as slater s condition and for which the optimal set is neither required to be compact nor to be equal to the sum of a compact set and a linear space it is supposed only that the infimum is finite a very wide class of convex functions is exhibited for which the optimum is always attained and is equal to the supremum of the ordinary dual program additional results concerning the existence of optimal solutions in the non convex case are also given

new sequential lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs in this paper a new sequential lagrange multiplier condition characterizing optimality without a constraint qualification for an abstract nonsmooth convex program is presented in terms of the subdifferentials and the epsilon subdifferentials a sequential condition involving only the subdifferentials but at nearby points to the minimizer for constraints is also derived for a smooth convex program the sequential condition yields a limiting kuhn tucker condition at nearby points without a constraint qualification it is shown how the sequential conditions are related to the standard lagrange multiplier condition applications to semidefinite programs semi infinite programs and semiconvex programs are given several numerical examples are discussed to illustrate the significance of the sequential conditions

an affine analogue of the hartman nirenberg cylinder theorem let x be a smooth complete connected submanifold of dimension n n in a complex affine space a n mathbb c and r is the rank of its gauss map gamma gamma x t x x the authors prove that if 2 leq r leq n 1 n n geq 2 and in the pencil of the second fundamental forms of x there are two forms defining a regular pencil all eigenvalues of which are distinct then the submanifold x is a cylinder with n r dimensional plane generators erected over a smooth complete connected submanifold y of rank r and dimension r this result is an affine analogue of the hartman nirenberg cylinder theorem proved for x subset r n 1 and r 1 for n geq 4 and r n 1 there exist complete connected submanifolds x subset a n mathbb c that are not cylinders

constraint qualifications for convex optimization without convexity of constraints new connections and applications to best approximation abstract we study constraint qualifications and necessary and sufficient optimality conditions for a convex optimization problem with inequality constraints where the constraint functions are continuously differentiable but they are not assumed to be convex we present constraint qualifications under which the karush kuhn tucker conditions are necessary and sufficient for optimality without the convexity of the constraint functions and establish new links among various known constraint qualifications that guarantee necessary karush kuhn tucker conditions we also present a new constraint qualification which is the weakest constraint qualification for the karush kuhn tucker conditions to be necessary for optimality of the convex optimization problem consequently we present lagrange multiplier characterizations for the best approximation from a convex set in the face of nonconvex inequality constraints extending corresponding known results in the literature we finally give a table summarizing various links among the constraint qualifications

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

best approximation from the intersection of a closed convex set and a polyhedron in hilbert space weak slater conditions and the strong conical hull intersection property let x be a real hilbert space c be a closed convex subset and h i x in x mid la x h i ra le b i quad i 1 2 m be a finite collection of half spaces under the assumption that k c cap cap 1 mh i is not empty the problem of characterizing the best approximation from k to any x in x is considered the strong conical hull intersection property strong chip which was introduced by us in 1997 is shown to be both necessary and sufficient for the following perturbation property to hold for each x in x an element x 0 in k satisfies x 0 p k x if and only if x 0 p c x sum 1 m lambda ih i for some scalars lambda i ge 0 with lambda i la x 0 h i ra b i 0 for each i here p d z denotes the unique best approximation from d to z in other words determining the best approximation from the set k to any point is equivalent to the generally easier problem of determining the best approximation from the set c to a perturbation of that point moreover even when the strong chip does not hold the perturbation property still holds except now c must be replaced by a certain convex extremal subset of c we also show that the strong chip is weaker than any of the weak slater conditions that one can naturally impose on the sets in question these results generalize the main results of our 1997 paper j approx theory 90 pp 385 444 and hence those of several other papers as well

constraint qualifications for extended farkas s lemmas and lagrangian dualities in convex infinite programming for an inequality system defined by a possibly infinite family of proper functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions under the new constraint qualifications we obtain characterizations of those reverse convex inequalities which are a consequence of the constrained system and we provide necessary and or sufficient conditions for a stable farkas lemma to hold similarly we provide characterizations for constrained minimization problems to have the strong or strong stable lagrangian dualities several known results in the conic programming problem are extended and improved

the strong conical hull intersection property for convex programming the strong conical hull intersection property chip is a geometric property of a collection of finitely many closed convex intersecting sets this basic property which was introduced by deutsch et al in 1997 is one of the central ingredients in the study of constrained interpolation and best approximation in this paper we establish that the strong chip of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems we first show that a sharpened strong chip is necessary and sufficient for a complete lagrange multiplier characterization of optimality for the convex programming model problem where c is a closed convex subset of a banach space x s is a closed convex cone which does not necessarily have non empty interior y is a banach space is a continuous convex function and g x y is a continuous s convex function we also show that the strong chip completely characterizes the strong duality for partially finite convex programs where y is finite dimensional and g x ax b and s is a polyhedral convex cone global sufficient conditions which are strictly weaker than the slater type conditions are given for the strong chip and for the sharpened strong chip

fully developed isotropic turbulence nonperturbative renormalization group formalism and fixed point solution we investigate the regime of fully developed homogeneous and isotropic turbulence of the navier stokes ns equation in the presence of a stochastic forcing using the nonperturbative functional renormalization group nprg within a simple approximation based on symmetries we obtain the fixed point solution of the nprg flow equations that corresponds to fully developed turbulence both in d 2 and 3 dimensions deviations to the dimensional scalings kolmogorov in d 3 or kraichnan batchelor in d 2 are found for the two point functions to further analyze these deviations we derive exact flow equations in the large wave number limit and show that the fixed point does not entail the usual scale invariance thereby identifying the mechanism for the emergence of intermittency within the nprg framework the purpose of this work is to provide a detailed basis for nprg studies of ns turbulence the determination of the ensuing intermittency exponents is left for future work

on strong and total lagrange duality for convex optimization problems abstract we give some necessary and sufficient conditions which completely characterize the strong and total lagrange duality respectively for convex optimization problems in separated locally convex spaces we also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals in the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so called farkas minkowski and locally farkas minkowski conditions for systems of convex inequalities recently used in the literature moreover we show that our new results extend some existing ones in the literature

subdifferentials of value functions and optimality conditions for dc and bilevel infinite and semi infinite programs the paper concerns the study of new classes of parametric optimization problems of the so called infinite programming that are generally defined on infinite dimensional spaces of decision variables and contain among other constraints infinitely many inequality constraints these problems reduce to semi infinite programs in the case of finite dimensional spaces of decision variables we focus on dc infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints the main results establish efficient upper estimates of certain subdifferentials of intrinsically nonsmooth value functions in dc infinite programs based on advanced tools of variational analysis and generalized differentiation the value marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well posedness and sensitivity in this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric dc infinite programs and their remarkable specifications finally we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization the results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi infinite counterparts

lagrange multiplier characterizations of robust best approximations under constraint data uncertainty abstract in this paper we explain how to characterize the best approximation to any x in a hilbert space x from the set c x x g i x 0 i 1 2 m in the face of data uncertainty in the convex constraints g i x 0 i 1 2 m where c is a closed convex subset of x following the robust optimization approach we establish lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty this is done by characterizing the best approximation to any x from the robust counterpart of the constraints where the constraints are satisfied for all possible uncertainties within the prescribed uncertainty sets unlike the traditional lagrange multiplier characterizations without data uncertainty for constrained best approximation problems in the face uncertainty we show that the strong conical hull intersection property strong chip alone is not sufficient to guarantee the lagrange multiplier characterizations we present conditions which guarantee that the strong chip is necessary and sufficient for the multiplier characterization we also establish that the strong chip is automatically satisfied for the cases of polyhedral constraints with polytope uncertainty and linear constraints with interval uncertainty as an application we show how robust solutions of shape preserving interpolation problems under ellipsoidal and box uncertainty cases can be obtained in terms of lagrange multipliers under strict robust feasibility conditions

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

weak sharp minima for convex infinite optimization problems in normed linear spaces in the present paper we study the characterization issue of the weak sharp minima properties for convex infinite optimization problems in normed linear spaces we develop a new approach to establi

weak sharp minima on riemannian manifolds this is the first paper dealing with the study of weak sharp minima for constrained optimization problems on riemannian manifolds which are important in many applications we consider the notions of local weak sharp minima boundedly weak sharp minima and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite dimensional riemannian manifolds and hadamard manifolds a number of the results obtained in this paper are also new for the case of conventional problems in finite dimensional euclidean spaces our methods involve appropriate tools of variational analysis and generalized differentiation on riemannian and hadamard manifolds developed and efficiently implemented in this paper

convergence analysis of the gauss newton method for convex inclusion and convex composite optimization problems abstract using the convex process theory we study the convergence issues of the iterative sequences generated by the gauss newton method for the convex inclusion problem defined by a cone c and a smooth function f the derivative is denoted by f the restriction in our consideration is minimal and even in the classical case the initial point x 0 is assumed to satisfy the following two conditions f is lipschitz around x 0 and the convex process t x 0 defined by t x 0 f x 0 c is surjective our results are new in giving sufficient conditions which are weaker than the known ones and have a remarkable property being affine invariant ensuring the convergence of the iterative sequence with initial point x 0 the same study is also made for the so called convex composite optimization problem with objective function given as the composite of a convex function with a smooth map

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

stable zero duality gaps in convex programming complete dual characterisations with applications to semidefinite programs the zero duality gap that underpins the duality theory is one of the central ingredients in optimisation in convex programming it means that the optimal values of a given convex program and its associated dual program are equal it allows in particular the development of efficient numerical schemes however the zero duality gap property does not always hold even for finite dimensional problems and it frequently fails for problems with non polyhedral constraints such as the ones in semidefinite programming problems over the years various criteria have been developed ensuring zero duality gaps for convex programming problems in the present work we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions for convex semidefinite programs we establish necessary and sufficient dual conditions for stable zero duality gaps as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right hand side and convex objective function zero duality gap results for second order cone programming problems are also given our approach makes use of elegant conjugate analysis and fenchel s duality

inequality systems and global optimization abstract solvability results for infinite inequality systems involving convex and difference of convex dc functions are given generalizations of farkas lemma are obtained these results are presented in terms of epigraphs of conjugate functions applications are given for characterizing global optimality and optimality of difference of convex dc optimization problems with convex inequality constraints solvability results for cone convex inequality systems are also given

quasi slater and farkas minkowski qualifications for semi infinite programming with applications the well known farkas minkowski fm type qualification plays an important role in linear semi infinite programming and has been extensively developed by many authors in establishing optimality conditions duality and stability for semi infinite programming in this paper we introduce the concept of the quasi slater condition for a semi infinite convex inequality system and present that the slater type conditions imply the fm qualification under some appropriate continuity assumption of the set valued mapping i mapsto f i x applying these relationships we establish dual characterizations both asymptotic and nonasymptotic for set containment problems and provide some sufficient conditions for ensuring the strong lagrangian duality and farkas lemma

constraint qualifications for optimality conditions and total lagrange dualities in convex infinite programming abstract for an inequality system defined by an infinite family of proper convex functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications under the new constraint qualifications we provide necessary and or sufficient conditions for the kkt rules to hold similarly we provide characterizations for constrained minimization problems to have total lagrangian dualities several known results in the conic programming problem are extended and improved

nonlinear extensions of farkas lemma with applications to global optimization and least squares a nonlinear extension of farkas lemma for systems involving the difference of sublinear functions is presented this extension of farkas lemma is applied to give a complete characterization of global optimality for constrained global optimization problems in which the objective function is the difference of a convex and sublinear function and the constraints are systems of difference sublinear functions an application to certain fractional programming problems is also given these results are achieved with merely a stronger consistency condition which reduces to the usual feasibility requirement for problems with sublinear constraints further generalizations of farkas lemma for systems involving convex and difference sublinear functions are also presented moreover a generalized farkas lemma for certain specially structural convex inequality systems is shown to be related to the solution of appropriate constrained least squares problems

nonlinearly constrained best approximation in hilbert spaces the strong chip and the basic constraint qualification we study best approximation problems with nonlinear constraints in hilbert spaces the strong conical hull intersection property chip and the basic constraint qualification bcq condition are discussed best approximations with differentiable constraints and convex constraints are characterized the analysis generalizes some linearly constrained results of recent works f deutsch w li and j ward j approx theory 90 1997 pp 385 444 f deutsch w li and j d ward siam j optim 10 1999 pp 252 268

linear regularity and ϕ regularity of nonconvex sets in this paper we discuss some sufficient conditions for the linear regularity and bounded linear regularity and their variations of finitely many closed not necessarily convex sets in a normed vector space the accompanying necessary conditions are also given in the setting of asplund spaces

quasi relative interior type constraint qualifications ensuring strong lagrange duality for optimization problems with cone and affine constraints in this article we provide weak sufficient strong duality conditions for a convex optimization problem with cone and affine constraints stated in infinite dimensional spaces and its lagrange dual problem our results are given by using the notions of quasi relative interior and quasi interior for convex sets the main strong duality theorem is accompanied by several stronger yet easier to verify in practice versions of it as exemplification we treat a problem which is inspired from network equilibrium our results come as corrections and improvements to daniele and giuffre 2007 9

subdifferential calculus rules in convex analysis a unifying approach via pointwise supremum functions we provide a rule to calculate the subdifferential set of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space our formula is given exclusively in terms of the data functions and does not require any assumption either on the index set on which the supremum is taken or on the involved functions some other calculus rules namely chain rule formulas of standard type are obtained from our main result via new and direct proofs

stable and total fenchel duality for convex optimization problems in locally convex spaces we consider the optimization problem p a inf x in x f x g ax where f and g are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear operator from x to y by using the properties of the epigraph of the conjugated functions some sufficient and necessary conditions for the strong fenchel duality and the strong converse fenchel duality of p a are provided sufficient and necessary conditions for the stable fenchel duality and for the total fenchel duality are also derived

on best approximation by nonconvex sets and perturbation of nonconvex inequality systems in hilbert spaces by virtue of convexification techniques we study best approximations to a closed set c in a hilbert space as well as perturbation conditions relative to c and a nonlinear inequality system some results on equivalence of the best approximation and the basic constraint qualification are established

strong chip for infinite system of closed convex sets in normed linear spaces for a general possibly infinite system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property one set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior point conditions together with appropriate continuity of the associated set valued function on the topologized index set i in the special case when i is finite and x is finite dimensional one of these results reduces to a classical result of rockafellar

robust linear semi infinite programming duality under uncertainty in this paper we propose a duality theory for semi infinite linear programming problems under uncertainty in the constraint functions the objective function or both within the framework of robust optimization we present robust duality by establishing strong duality between the robust counterpart of an uncertain semi infinite linear program and the optimistic counterpart of its uncertain lagrangian dual we show that robust duality holds whenever a robust moment cone is closed and convex we then establish that the closed convex robust moment cone condition in the case of constraint wise uncertainty is in fact necessary and sufficient for robust duality in other words the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program in the case of uncertain problems with affinely parameterized data uncertainty we establish that robust duality is easily satisfied under a slater type constraint qualification consequently we derive robust forms of the farkas lemma for systems of uncertain semi infinite linear inequalities

duality gap of the conic convex constrained optimization problems in normed spaces in this paper motivated by a result due to champion math program 99 2004 we introduce a property for a conic quasi convex vector valued function in a general normed space we prove that this property characterizes the zero duality gap for a class of the conic convex constrained optimization problem in the sense that if this property is satisfied and the objective function f is continuous at some feasible point then the duality gap is zero and if this property is not satisfied then there exists a linear continuous function f such that the duality gap is positive we also present some sufficient conditions for the property

renormalization group theory for global asymptotic analysis we show with several examples that renormalization group rg theory can be used to understand singular and reductive perturbation methods in a unified fashion amplitude equations describing slow motion dynamics in nonequilibrium phenomena are rg equations the renormalized perturbation approach may be simpler to use than other approaches because it does not require the use of asymptotic matching and yields practically superior approximations

constraint qualifications for convex inequality systems with applications in constrained optimization for an inequality system defined by an infinite family of proper convex functions we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well known constraint qualifications including the basic constraint qualification studied by hiriart urrutty and lemarechal and by li nahak and singer extensions of known results to more general settings are presented and applications to particular important problems such as conic programming and approximation theory are also studied

lower semicontinuous convex relaxation in optimization we relate the argmin sets of a given function not necessarily convex or lower semicontinuous and its lower semicontinuous convex hull by means of explicit characterizations involving an appropriate concept of asymptotic functions this question is connected to the subdifferential calculus of the legendre fenchel conjugate function the final expressions which also involve a useful extension of the fenchel subdifferential introduced in r correa and a hantoute set valued var anal 18 2010 pp 405 422 are then written exclusively by means of primal objects relying on the initial function this work extends to the infinite dimensional setting of some related results given in j benoist and j b hiriart urruty siam j math anal 27 1996 pp 1661 1679

characterizations of solution sets of convex vector minimization problems complete dual characterizations of the weak and proper optimal solution sets of an infinite dimensional convex vector minimization problem are given the results are expressed in terms of subgradients lagrange multipliers and epigraphs of conjugate functions a dual condition characterizing the containment of a closed convex set defined by a cone convex inequality in a reverse convex set plays a key role in deriving the results simple lagrange multiplier characterizations of the solution sets are also derived under a regularity condition numerical examples are given to illustrate the significance of the results

an alternative formulation for a new closed cone constraint qualification abstract we give an alternative formulation for the so called closed cone constraint qualification cccq related to a convex optimization problem in banach spaces recently introduced in the literature this new formulation allows to prove in a simple way that cccq is weaker than some generalized interior point constraint qualifications given in the past by means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature

necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities abstract in this paper we present constraint qualifications which completely characterize the farkas minkowski and the locally farkas minkowski convex possibly infinite inequality systems posed in topological vector spaces the number of constraints and the dimension of the linear space are arbitrary possibly infinite the constraint qualifications considered in this paper are expressed in terms of the solvability of certain parametric convex linear systems and the uniform strong duality or the uniform min max duality relative to the lagrange haar dual problems of suitable convex linear parametric optimization problems

from the farkas lemma to the hahn banach theorem this paper provides new versions of the farkas lemma characterizing those inequalities of the form f x geq 0 which are consequences of a composite convex inequality s circ g x leq 0 on a closed convex subset of a given locally convex topological vector space x where f is a proper lower semicontinuous convex function defined on x s is an extended sublinear function and g is a vector valued s convex function in parallel associated versions of a stable farkas lemma considering arbitrary linear perturbations of f are also given these new versions of the farkas lemma and their corresponding stable forms are established under the weakest constraint qualification conditions the so called closedness conditions and they are actually equivalent to each other as well as equivalent to an extended version of the so called hahn banach lagrange theorem and its stable version correspondingly it is shown that any of them implies analytic and algebraic versions of the hahn banach t

constraint qualification the strong chip and best approximation with convex constraints in banach spaces several fundamental concepts such as the basic constraint qualification bcq the strong conical hull intersection property chip and the perturbations for convex systems of inequalities in banach spaces over mathbb r or mathbb c are extended and studied here the systems are not necessarily finite their relationships with each other in connection with the best approximations are investigated as applications we establish results on the unconstrained reformulation of best approximations with infinitely many constraints in hilbert spaces also we give several characterizations of best restricted range approximations in c q under quite general constraints

on constraint qualification for an infinite system of convex inequalities in a banach space for a general infinite system of convex inequalities in a banach space we study the basic constraint qualification and its relationship with other fundamental concepts including various versions of conditions of slater type the mangasarian fromovitz constraint qualification as well as the pshenichnyi levin valadier property introduced by li nahak and singer applications are given in the restricted range approximation problem constrained optimization problems as well as in the approximation problem with constraints by conditionally positive semidefinite functions

geometrical aspects of statistical mechanics investigation of the geometry of thermodynamic state space based upon the differential geometric approach to parametric statistics developed by chentsov statistical decision rules and optimal inference nauka moscow 1972 efron ann stat 3 1189 1975 amari ann stat 10 357 1982 and others provides a deeper understanding of the mathematical structure of statistical thermodynamics in the present paper the riemannian geometrical approach to statistical mechanical systems due to janyszek j phys a 23 477 1990 is applied to various models including the van der waals gas and magnetic models the scalar curvature for these models is shown to diverge not only at the critical points but also along the entire spinodal curve the critical behavior of the curvature derived from the fisher information metric turns out to coincide with that derived from the entropy differential metric by ruppeiner phys rev a 20 1608 1979

complete characterizations of stable farkas lemma and cone convex programming duality we establish necessary and sufficient conditions for a stable farkas lemma we then derive necessary and sufficient conditions for a stable duality of a cone convex optimization problem where strong duality holds for each linear perturbation of a given convex objective function as an application we obtain stable duality results for convex semi definite programs and convex second order cone programs

weak sharp minima revisited part ii application to linear regularity and error bounds the notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems it has been studied extensively by several authors this paper is the second of a series on this subject where the basic results on weak sharp minima in part i are applied to a number of important problems in convex programming in part ii we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming we obtain both new results and reproduce several existing results from a fresh perspective

on the stability of the feasible set in optimization problems this paper provides stability theorems for the feasible set of optimization problems posed in locally convex topological vector spaces the problems considered in this paper have an arbitrary number of inequality constraints and one constraint set different models are discussed depending on the properties of the constraint functions linear or not convex or not but at least lower semicontinuous and one closed constraint set but not necessarily convex the parameter space is formed by systems of the same type as the nominal one with the same space of variables and the same number of constraints where the constraint set can be perturbed or not equipped with the metric of the uniform convergence on the positive multiples of a fixed barrelled neighborhood of zero in finite dimensions this topology describes the uniform convergence on compact sets and in the particular case that the constraints are linear the uniform convergence of the vector coefficients the paper examines in a unified way the lower and upper semicontinuity and the closedness of the feasible set mapping the stable consistency of the constraint system with respect to arbitrary and right hand side perturbations tuy and robinson regularities and other desirable stability properties of the feasible set

some convex programs without a duality gap an important issue in convex programming concerns duality gap various conditions have been developed over the years that guarantee no duality gap including one developed by rockafellar network flows and monotropic programming wiley interscience new york 1984 involving separable objective function and affine constraints we show that this sufficient condition can be further relaxed to allow the constraint functions to be separable we also refine a sufficient condition involving weakly analytic functions by allowing them to be extended real valued

statistical inference occam s razor and statistical mechanics on the space of probability distributions the task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions using the techniques of low temperature expansions i arrive at a systematic series for the bayesian posterior probability of a model family that significantly extends known results in the literature in particular i arrive at a precise understanding of how occam s razor the principle that simpler models should be preferred until the data justify more complex models is automatically embodied by probability theory these results require a measure on the space of model parameters and i derive and discuss an interpretation of jeffreys prior distribution as a uniform prior over the distributions indexed by a family finally i derive a theoretical index of the complexity of a parametric family relative to some true distribution that i call the razor of the model the form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics

a geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints we first establish a relaxed version of dines theorem associated to quadratic minimization problems with finitely many linear equality and a single nonconvex quadratic inequality constraints the case of unbounded optimal valued is also discussed then we characterize geometrically the strong duality and some relationships with the conditions employed in finsler theorem are established furthermore necessary and sufficient optimality conditions with or without the slater assumption are derived our results can be used to situations where none of the results appearing elsewhere are applicable in addition a revisited theorem due to frank and wolfe along with that due to eaves is established for asymptotically linear sets

infinite dimensional duality and applications the usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear in this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi relative interior which in all the concrete problems considered is nonempty we apply this theory to solve the until now unsolved problem of finding in the infinite dimensional case the lagrange multipliers associated to optimization problems or to variational inequalities as an example we find the lagrange multiplier associated to a general elastic plastic torsion problem