surrogate programming and multipliers in quasi convex programming a result due to luenberger on the existence of multipliers in a quasi convex programming problem is extended to the case of constraints given by an arbitrary convex cone under a constraint qualification condition more general than slater s condition the existence of solutions is not assumed we point out links with even convexity in the sense of fenchel and quasi subdifferentiability in the sense of greenberg pierskalla and we observe that the couples of primal dual optimal solutions reduce to saddle points of a suitable lagrangian function

dual and bidual problems for a lipschitz optimization problem based on quasi conjugation abstract in this paper we consider a lipschitz optimization problem lop constrained by linear functions in r n in general since it is hard to solve lop directly lop is transformed into a certain problem mp constrained by a ball in r n 1 despite there is no guarantee that the objective function of mp is quasi convex by using the idea of the quasi conjugate function defined by thach 1991 1 we can construct its dual problem dp as a quasi convex maximization problem we show that the optimal value of dp coincides with the multiplication of the optimal value of mp by 1 and that each optimal solution of the primal and dual problems can be easily obtained by the other moreover we formulate a bidual problem bdp for mp that is a dual problem for dp we show that the objective function of bdp is a quasi convex function majorized by the objective function of mp and that both optimal solution sets of mp and bdp coincide furthermore we propose an outer approximation method for solving dp

isometries and isometric equivalence of hermitian operators on a1 p x let x be a separable complex banach space with no nontrivial l1 projections and not isometrically isomorphic to lp 0 1 x where 1 p p 2 the space a1 p x is defined to be the set of all absolutely continuous functions f 0 1 x such that dfdx exist a e on 0 1 and belongs to lp 0 1 x if f a1 p x the norm of f on this space is defined to be f f 0 x f lp 0 1 x we prove that if t is a surjective isometry t of a1 p x then t is given by tf x t0f 0 0xu f t dt where t0 is a surjective isometry of x and u is a surjective isometry of lp 0 1 x we also give the form of a hermitian operators on a1 p x in addition if we assume that x is not the lp direct sum of two nonzero banach spaces for the same p we obtain the conditions of isometric equivalence of two hermitian operators on a1 p x

a method for globally minimizing concave functions over convex sets a method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear the algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region the algorithm does not involve cuts of the feasible region requires only simplex pivot operations and univariate search computations to be performed allows the objective function to be lower semicontinuous and nonseparable and is guaranteed to converge to the global solution computational aspects of the algorithm are discussed

complete monotone quasiconcave duality we introduce a notion of complete monotone quasiconcave duality motivated by some economic applications we show that this duality holds for important classes of quasiconcave functions

conditionally evenly convex sets and evenly quasi convex maps abstract evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces we introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem this notion is then applied to obtain the dual representation of conditionally evenly quasi convex maps which turns out to be a fundamental ingredient in the study of quasi convex dynamic risk measures

the lorentzian splitting theorem without the completeness assumption on demontre le theoreme de decomposition lorentzien pour des espace temps globalement hyperboliques obeissant a la condition d energie forte sans hypothese de completude geodesique du genre temps

spectral projections for hermitian operators abstract the behavior of operators which are both scalar type spectral and hermitian and this includes all hermitian operators on finite dimensional spaces is studied for example it is shown that the resolution of the identity of such an operator being hermitian is equivalent to all the powers of the operator being hermitian or to the operator having a contractive borel functional calculus a sufficient condition for these properties to hold on a finite dimensional space is that the eigenvalues of the operator are rationally independent

capital allocation a la aumann shapley for non differentiable risk measures abstract we study capital allocation rules satisfying suitable properties for convex and quasi convex risk measures by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the aumann shapley allocation principle these rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non gateaux differentiable risk measures we also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi convex context

characterizations of evenly convex sets and evenly quasiconvex functions the aim of this paper is to present a geometric characterization of even convexity in separable banach spaces which is not expressed in terms of dual functionals or separation theorems as an application an analytic equivalent definition for the class of evenly quasiconvex functions is derived

generalized bicircular projections on jb triples abstract let a be a jb triple and let p a a be a linear projection it is proved that p id p is an isometry for some modulus one complex number 1 if and only if either 1 or p is hermitian it is also proved that every rank one bicontractive projection on a is hermitian the particular case when a is a c algebra is discussed through several examples

portfolio optimization with quasiconvex risk measures in this paper we focus on the portfolio optimization problem associated with a quasiconvex risk measure satisfying some additional assumptions for coherent convex risk measures the portfolio optimization problem has been already studied in the literature following the approach of ruszczynski and shapiro ruszczynski a shapiro a 2006 optimization of convex risk functions math oper res 31 3 433 452 but by means of quasiconvex analysis and notions of subdifferentiability we characterize optimal solutions of the portfolio problem associated with quasiconvex risk measures the shape of the efficient frontier in the mean risk space and some particular cases are also investigated

on linear systems containing strict inequalities this paper deals with systems of an arbitrary possibly infinite number of both weak and strict linear inequalities we analyze the existence of solutions for such kind of systems and show that the large class of convex sets which admit this kind of linear representations i e the so called evenly convex sets enjoys most of the well known properties of the subclass of the closed convex sets we also show that it is possible to obtain geometrical information on these sets from a given linear representation finally we discuss the theory and methods for those linear optimization problems which contain strict inequalities as constraints

conjugate duality for vector maximization problems in this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n dimensional orthant using this duality we obtain a zero gap duality for a vector maximization problem

geometric analysis of lorentzian distance function on spacelike hypersurfaces some analysis on the lorentzian distance in a spacetime with controlled sectional or ricci curvatures is done in particular we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the omori yau maximum principle as a consequence and under appropriate hypotheses on the sectional or ricci curvatures of the ambient spacetime we obtain sharp estimates for the mean curvature of those hypersurfaces moreover we also give a sufficient condition for its hyperbolicity

hermitian projections on some banach spaces and related topics abstract in the setting of various complex banach spaces we consider the questions of when a square of a hermitian operator is also hermitian and when a hermitian operator has a hermitian square root we also describe the structure of hermitian projections and hermitian square roots of the identity operator

risk preferences and their robust representation to address the plurality of interpretations of the subjective notion of risk we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets this representation covers most instruments related to risk and allows for a differentiated interpretation depending on the underlying context that is illustrated in different settings for random variables risk perception can be interpreted as model risk and we compute among others the robust representation of the economic index of riskiness for lotteries risk perception can be viewed as distributional risk and we study the value at risk for consumption patterns which excerpt an intertemporality dimension in risk perception we provide an interpretation in terms of discounting risk and discuss some examples

a generalized dantzig wolfe decomposition principle for a class of nonconvex programming problems since dantzig wolfe s pioneering contribution the decomposition approach using a pricing mechanism has been developed for a wide class of mathematical programs for convex programs a linear space of lagrangean multipliers is enough to define price functions for general mathematical programs the price functions could be defined by using a subclass of nondecreasing functions however the space of nondecreasing functions is no longer finite dimensional in this paper we consider a specific nonconvex optimization problem min f x hj x g x j 1 m x x wheref hj andg are finite convex functions andx is a closed convex set we generalize optimal price functions for this problem in such a way that the parameters of generalized price functions are defined in a finite dimensional space combining convex duality and a nonconvex duality we can develop a decomposition method to find a globally optimal solution

global minimization of a linearly constrained concave function by partition of feasible domain the problem of minimizing a concave function subject to linear inequality constraints may have many local solutions therefore finding the global constrained minimum is a computationally difficult problem a computational method is described which finds the global minimum of a smooth concave function over a polyhedron in rn the feasible domain is partitioned into a rectangular domain which can be excluded from further consideration and r 2n subdomains at least one of which contains the global minimum a known algorithm can be applied sequentially or in parallel to each of these r subdomains to compute the global minimum a method is also presented appendix b for the construction of nontrivial test problems for which the global minimum point is known given an arbitrary polyhedron and a selected vertex it is shown how to determine a concave quadratic function generally with many local minima with its global minimum at the selected vertex

a finite cutting plane method for solving linear programs with an additional reverse convex constraint abstract the paper deals with linear programs with an additional reverse convex constraint if the feasible region of this problem is nonempty and the objective function to be minimized is bounded below on it then the problem has a finite minimum obtained on an at most one dimensional face of the polyhedron as well this paper presents a finite cutting plane method using convexity and disjunctive cuts for solving the considered problem the method is based upon a procedure which for a given nonnegative integer q either finds such an at most q dimensional face of the original polyhedron which has a point feasible to the cuts generated previously or proves that there exists no such a face computational experience is also provided

quasiconjugates of functions duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint and applications abstract in this paper we introduce a concept of quasiconjugate for functions defined on rn whose values are in r the conjugacy correspondence between functions and their quasiconjugates is one to one and symmetric in a class of quasiconvex functions whose minimizer on rn is located at the origin by using the concept of quasiconjugate we obtain a duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint this duality relationship allows us to establish a primal dual pair in a class of nonconvex optimization problems without the duality gap several applications are given

splitting theorems for spatially closed space times a lorentzian splitting theorem is obtained for spatially closed spacetimes the proof employs and extends some recent results of bartnik and gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space times a splitting theorem for spatially closedtime periodic space times which generalizes a result first considered by avez is derived as a corollary

some relations between dualities polarities coupling functionals and conjugations on etablit certaines relations entre dualites au sens de evers et van maaren 1981 polarites comme celles utilisees dans griffin araoz et edmonds 1982 fonctionnelles de couplage au sens de moreau 1966 1967 1970 et conjugaisons comme celles introduites par l auteur 1983

surrogate duality for robust optimization abstract robust optimization problems which have uncertain data are considered we prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min max duality theorems for robust convex optimization problems we give necessary and sufficient constraint qualifications for surrogate duality and surrogate min max duality and show some examples at which such duality results are used effectively moreover we obtain a surrogate duality theorem and a surrogate min max duality theorem for semi definite optimization problems in the face of data uncertainty

subdifferential calculus for a quasiconvex function with generator abstract recently we discussed optimality conditions for quasiconvex programming by introducing q subdifferential which is a notion of differential of quasiconvex functions in this paper we investigate basic and fundamental properties of the q subdifferential especially we show results of a chain rule for composition with non decreasing functions monotonicity of the q subdifferential mean value theorem a sufficient condition for a global minimizer for quasiconvex programming and the calculus of the q subdifferential of the supremum of quasiconvex functions

optimality conditions and the basic constraint qualification for quasiconvex programming abstract in this paper we consider optimality conditions and a constraint qualification for quasiconvex programming for this purpose we introduce a generator and a new subdifferential for quasiconvex functions by using penot and volle s theorem

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

on quasi convex duality the familiar fenchel moreau rockafellar duality scheme deduced from a conjugation is shown to be applicable to quasi convex problems here quasi affine functions take the place of affine functions the links with other quasi convex dualities are examined

sandwich theorem for quasiconvex functions and its applications abstract in convex programming sandwich theorem is very important because it is equivalent to fenchel duality theorem in this paper we investigate a sandwich theorem for quasiconvex functions also we consider some applications for quasiconvex programming

isometries on spaces of absolutely continuous vector valued functions abstract this paper investigates the surjective not necessarily linear isometries between spaces of absolutely continuous vector valued functions with respect to the norm max v where and v denote the supremum norm and the total variation of a function respectively and gives an absolutely continuous version of a celebrated theorem by jerison as a consequence in the scalar valued case we obtain generalizations of all known results concerning such isometries by a different approach

regularity of lorentzian busemann functions a general theory of regularity for lorentzian busemann functions in future timelike geodesically complete spacetimes is presented this treatment simplifies and extends the local regularity developed by eschenburg galloway and newman to prove the lorentzian splitting theorem criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of bartnik

necessary and sufficient conditions for some constraint qualifications in quasiconvex programming abstract in this paper we investigate relations between constraint qualifications in quasiconvex programming at first we show a necessary and sufficient condition for the closed cone constraint qualification for quasiconvex programming q cccq and investigate some sufficient conditions for the q cccq also we consider a relation between the q cccq and the basic constraint qualification for quasiconvex programming q bcq and we compare the q bcq with some constraint qualifications

the lower semi continuous quasi convex hull as a normalized second conjugate let e be a real locally convex space and e its conjugate i e the linear space of all continuous linear functionals e i cq it is well known that if f e r u m or iffz a thenf e i qz 3 1 the lower semi continuous convex hull of f i e the greatest lower semi continuous convex functional majorized by f coincides with f the second convex conjugate off since this fact has important applications in optimization theory several attempts have been made to generalize it to f4 the lower semi continuous quasi convex hull off the greatest lower semi continuous quasi convex functional majorized by f i e to represent f as a second conjugate off in some sense thus alternatively to the a quasi conjugate f of greenberg and pierskalla 5 w ic we shall recall below crouzeix h h has introduced in 2 chapter ii for any f e r and a e r another a quasi conjugate of f as a certain functional f f e i and has shown that f sup f f and fq inf f f

global minimization of large scale constrained concave quadratic problems by separable programming the global minimization of a large scale linearly constrained concave quadratic problem is considered the concave quadratic part of the objective function is given in terms of the nonlinear variablesx r n while the linear part is in terms ofy r k for large scale problems we may havek much larger thann the original problem is reduced to an equivalent separable problem by solving a multiple cost row linear program with 2n cost rows the solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum and also gives a bound on the relative error in the function value of this incumbent ana priori bound on this relative error is obtained which is shown to be 0 25 in important cases if the incumbent is not a satisfactory approximation to the global minimum a guaranteede approximate solution is obtained by solving a single liner zero one mixed integer programming problem this integer problem is formulated by a simple piecewise linear underestimation of the separable problem

bicircular projections on some matrix and operator spaces a projection on a complex banach space is bicircular if its linear combinations with the complementary projection having coefficients of modulus one are isometries such projections are always bicontractive in this paper we prove structure theorems for bicircular projections acting on the spaces of the full operator algebra symmetric operators and antisymmetric operators

bicircular projections on some banach spaces in this paper we show that the bicircular projections are precisely the hermitian projections and prove some immediate consequences of this result

decomposition theorems for lorentzian manifolds with nonpositive curvature soit m g un espace temps globalement hyperbolique de dimension 2 avec partout des courbures sectionnelles du genre temps non positives k 0 qui contient une ligne du genre temps complete m g alors m g est isometrique a un produit r h dt 2 h ou h h est une variete de riemann complete le facteur r dt 2 est represente par et h h est represente par un ensemble de niveau d une fonction de busemann associee a

cutting planes for low rank like concave minimization problems concavity cuts play an important role in several algorithms for concave minimization such as pure cutting plane algorithms conical algorithms and branch and bound algorithms for concave quadratic minimization problems konno et al 1998 have demonstrated that the lower the rank of the problem i e the smaller the number of nonlinear variables the deeper the concavity cuts usually turn out to be in this paper we examine the case where the number of nonlinear variables of a concave minimization problem is large but most of the objective value of a good solution is determined by a small number of variables only we will discuss ways to exploit such a situation to derive deep cutting planes to this end we apply concepts usually applied for efficiently solving low rank concave minimization problems

a general theory of dual optimization problems we construct a unified theory of dual optimization problems which encompasses as particular cases the known dual problems for each concept of dual problem we define first an unperturbational version from which we deduce via a certain scheme the corresponding perturbational version we generalize simultaneously the lagrangian and surrogate cases using coupling functionals we study the connections between dual problems and define lagrangian functionals for them we study the class of perturbation functionals which can be written as the upper sum of the primal objective functional h and a functional with values not depending on h

a conical algorithm for globally minimizing a concave function over a closed convex set in this paper we are concerned with the problem of finding the global minimum of a concave function over a closed convex possibly unbounded set in rn the intrinsic difficulty of this problem is due to the fact that a local minimum of the objective function may fail to be a global one which makes the conventional methods of local optimization almost useless

on the degree and separability of nonconvexity and applications to optimization problems we study qualitative indications for d c representations of closed sets in and functions on hilbert spaces the first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity we show that a closed set is weakly closed if this indication is finite using this result we can prove the solvability of nonconvex minimization problems by duality a minimization problem on a feasible set in which this indication is low can be reduced to a quasi concave minimization over a convex set in a low dimensional space the second indication is the separability which can be incorporated in solving dual problems both the index of nonconvexity and the separability can be characteristics to good d c representations for practical computation we present a notion of clouds which enables us to obtain a good d c representation for a class of nonconvex sets using a generalized caratheodory s theorem we present various applications of clouds

generalized bi circular projections abstract recall that a projection p on a complex banach space x is a generalized bi circular projection if p i p is a surjective isometry for some such that 1 and 1 it is easy to see that every hermitian projection is generalized bi circular a generalized bi circular projection is said to be nontrivial if it is not hermitian botelho and jamison showed that a projection p on c 0 1 is a nontrivial generalized bi circular projection if and only if p i p is a surjective isometry in this article we prove that if p is a projection such that p i p is a surjective isometry for some then either p is hermitian or is an n th unit root of unity we also show that for any n th unit root of unity there are a complex banach space x and a nontrivial generalized bi circular projection p on x such that p i p is an isometry

a convex like duality scheme for quasi convex programs this paper describes a symmetric duality relation for quasi convex programs we are able to strengthen previous results and to define necessary and sufficient conditions for the absence of duality gap in the present scheme one can generate quasi convex quasi concave lagrangians and discuss the correspondence between saddle points of the lagrangians and the solutions to the dual and primal programs the present scheme is very similar to rockafellar s scheme for convex programs and in this sense it may be viewed as a unified approach several examples are also given

duality in reverse convex optimization a duality theorem for the general problem of minimizing an extended real valued convex function on a locally convex linear space under a reverse convex constraint is considered in the particular case of the distance to a reverse convex subset in a normed linear space we recover as a corollary a duality theorem due to c franchetti and i singer boll un mat ital b 5 17 1980 pp 33 43 similar to the one known for the distance to a convex subset the general theorem also contains the duality principle of toland singer for d c optimization