a note on two convergence acceleration methods for ordinary continued fractions abstract in this note we compare two recent methods of convergence acceleration for ordinary continued fractions the first one introduced by thron and waadeland 13 and further developed by brezinski 1 the second one by jacobsen and waadeland 4 5

weak efficiency for multiobjective variational problems abstract sometimes to locate efficient solutions for multiobjective variational problems mvps is quite costly so in this paper we tackle the study of weakly efficient solutions for mvps a new concept of weak vector critical point which generalizes other ones already existent and a new class of pseudoinvex functions are introduced we will apply a new approach to prove that the new class of pseudoinvex functions is equivalent to the class of functions whose weak vector critical points are weakly efficient solution for mvps

optimality and duality for nondifferentiable multiobjective variational problems abstract the concept of efficiency is used to formulate duality for nondifferentiable multiobjective variational problems wolfe and mond weir type vector dual problems are formulated by using the generalized schwarz inequality and a characterization of efficient solution we established the weak strong and converse duality theorems under generalized f convexity assumptions

on efficiency and duality for multiobjective variational control problems with f ρ convexity abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational control problems wolfe and mond weir type duals are formulated under the generalized f convexity on the functions involved weak and strong duality theorems are proved

unbalanced optimal transport dynamic and kantorovich formulations abstract this article presents a new class of distances between arbitrary nonnegative radon measures inspired by optimal transport these distances are defined by two equivalent alternative formulations i a dynamic formulation defining the distance as a geodesic distance over the space of measures ii a static kantorovich formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer transport creation and destruction of mass between two measures both formulations are convex optimization problems and the ability to switch from one to the other depending on the targeted application is a crucial property of our models of particular interest is the wasserstein fisher rao metric recently introduced independently by 7 15 defined initially through a dynamic formulation it belongs to this class of metrics and hence automatically benefits from a static kantorovich formulation

mixed type duality for multiobjective variational problems with generalized f ρ convexity a mixed type dual for multiobjective variational problems is formulated several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized f convexity static mixed type dual multiobjective problems are particular cases of these problems

duality for nondifferentiable static multiobjective variational problems involving generalized f ϱ convex functions using the generalized schwarz inequality we establish some weak duality theorems for nondifferentiable static multiobjective variational problems involving generalized f convex functions later in the sequel we introduce three dual models for the nondifferentiable static multiobjective fractional variational problems and derive weak duality results for such programs

geometric properties of cones with applications on the hellinger kantorovich space and a new distance on the space of probability measures abstract we study general geometric properties of cone spaces and we apply them on the hellinger kantorovich space m x hk we exploit a two parameter scaling property of the hellinger kantorovich metric hk and we prove the existence of a distance shk on the space of probability measures that turns the hellinger kantorovich space m x hk into a cone space over the space of probabilities measures p x shk we provide a two parameter rescaling of geodesics in m x hk and for p x shk we obtain a full characterization of the geodesics we finally prove finer geometric properties including local angle condition and partial k semiconcavity of the squared distances that will be used in a future paper to prove existence of gradient flows on both spaces

symmetric duality for multiobjective fractional variational problems with generalized invexity the concept of symmetric duality for multiobjective fractional problems has been extended to the class of multiobjective variational problems weak strong and converse duality theorems are proved under generalized invexity assumptions a close relationship between these problems and multiobjective fractional symmetric dual problems is also presented

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

symmetric duality for multiobjective variational problems abstract a mond weir type symmetric dual for a multiobjective variational problem is formulated weak and strong duality theorems under generalized convexity assumptions are proved for properly efficient solutions under an additional condition on the kernel function that occurs in the formulation of the problems a self duality theorem is proved a close relationship between these variational problems and symmetric dual nonlinear multiobjective programming problems is also incorporated

generalized compressible flows and solutions of the h div geodesic problem we study the geodesic problem on the group of diffeomorphism of a domain m rd equipped with the h div metric the geodesic equations coincide with the camassa holm equation when d 1 and represent one of its possible multi dimensional generalizations when d 1 we propose a relaxation a la brenier of this problem in which solutions are represented as probability measures on the space of continuous paths on the cone over m we use this relaxation to prove that smooth h div geodesics are globally length minimizing for short times we also prove that there exists a unique pressure field associated to solutions of our relaxation finally we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized camassa holm and incompressible euler solutions

convex sobolev inequalities related to unbalanced optimal transport abstract we study the behavior of various lyapunov functionals relative entropies along the solutions of a family of nonlinear drift diffusion reaction equations coming from statistical mechanics and population dynamics these equations can be viewed as gradient flows over the space of radon measures equipped with the hellinger kantorovich distance the driving functionals of the gradient flows are not assumed to be geodesically convex or semi convex we prove new isoperimetric type functional inequalities allowing us to control the relative entropies by their productions which yields the exponential decay of the relative entropies

on variational problems characterization of solutions and duality in this paper we introduce a new class of pseudoinvex functions for variational problems using this new concept we obtain a necessary and sufficient condition for a critical point of the variational problem to be an optimal solution illustrated with an example also weak strong and converse duality are established

a class of nondifferentiable continuous programming problems abstract optimality conditions duality and converse duality results are obtained for a class of continuous programming problems with a nondifferentiable term in the integrand of the objective function the proofs are based on a fritz john theorem for constrained optimization in abstract spaces the results generalize various well known results in variational problems with differentiable functions and also give a dynamic analogue of certain nondifferentiable programming problems

duality with bf type i functions for a class of nondifferentiable multiobjective variational problems a class of bf type i functions and its extensions are introduced in the continuous case an example is presented in support utilizing these new concepts sufficient optimality conditions and duality results are presented for multiobjective variational problems involving arbitrary norms

a variational approach to the mean field planning problem abstract we investigate a first order mean field planning problem of the form t u h x d u f x m in 0 t r d t m m h p x d u 0 in 0 t r d m 0 m 0 m t m t in r d associated to a convex hamiltonian h with quadratic growth and a monotone interaction term f with polynomial growth we exploit the variational structure of the system which encodes the first order optimality condition of a convex dynamic optimal entropy transport problem with respect to the unknown density m and of its dual involving the maximization of an integral functional among all the subsolutions u of an hamilton jacobi equation combining ideas from optimal transport convex analysis and renormalized solutions to the continuity equation we will prove existence and at least partial uniqueness of a weak solution m u a crucial step of our approach relies on a careful analysis of distributional subsolutions to hamilton jacobi equations of the form t u h x d u under minimal summability conditions on and to a measure theoretic description of the optimality via a suitable contact defect measure finally using the superposition principle we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players

on properties of the generalized wasserstein distance the wasserstein distances wp p geqq 1 defined in terms of a solution to the monge kantorovich problem are known to be a useful tool to investigate transport equations in particular the benamou brenier formula characterizes the square of the wasserstein distance w2 as the infimum of the kinetic energy or action functional of all vector fields transporting one measure to the other another important property of the wasserstein distances is the kantorovich rubinstein duality stating the equality between the distance w1 of two probability measures and the supremum of the integrals in d of lipschitz continuous functions with lipschitz constant bounded by one an intrinsic limitation of wasserstein distances is the fact that they are defined only between measures having the same mass to overcome such a limitation we recently introduced the generalized wasserstein distances w p a b defined in terms of both the classical wasserstein distance wp and the total variation or l1 distance see piccoli and rossi in archive for rational mechanics and analysis 211 1 335 358 2014 here p plays the same role as for the classic wasserstein distance while a and b are weights for the transport and the total variation term in this paper we prove two important properties of the generalized wasserstein distances 1 a generalized benamou brenier formula providing the equality between w 2 a b and the supremum of an action functional which includes a transport term kinetic energy and a source term 2 a duality a la kantorovich rubinstein establishing the equality between w 1 1 1 and the flat metric

optimality conditions and duality for multiobjective variational problems with generalized b invexity we extend the concepts of b type i and generalized b type i functions to the continuous case and we use these concepts to establish sufficient optimality conditions and duality results for multiobjective variational programming problems

generalized wasserstein distance and its application to transport equations with source in this article we generalize the wasserstein distance to measures with different masses we study the properties of this distance in particular we show that it metrizes weak convergence for tight sequences we use this generalized wasserstein distance to study a transport equation with a source in which both the vector field and the source depend on the measure itself we prove the existence and uniqueness of the solution to the cauchy problem when the vector field and the source are lipschitzian with respect to the generalized wasserstein distance

a necessary and sufficient condition for duality in multiobjective variational problems abstract in this paper we move forward in the study of duality and efficiency in multiobjective variational problems we introduce new classes of pseudoinvex functions and prove that not only it is a sufficient condition to establish duality results but it is also necessary moreover these functions are characterized in order that all kuhn tucker or fritz john points are efficient solutions recent papers are improved we provide an example to show this improvement and illustrate these classes of functions and results

symmetric duality for multiobjective fractional variational problems involving cones in this paper a pair of multiobjective fractional variational symmetric dual problems over cones is formulated weak strong and converse duality theorems are established under generalized f convexity assumptions moreover self duality theorem is also discussed

the optimal partial transport problem given two densities f and g we consider the problem of transporting a fraction m in 0 min f l 1 g l 1 of the mass of f onto g minimizing a transportation cost if the cost per unit of mass is given by x y 2 we will see that uniqueness of solutions holds for m in f wedge g l 1 min f l 1 g l 1 this extends the result of caffarelli and mccann in ann math in print where the authors consider two densities with disjoint supports the free boundaries of the active regions are shown to be n 1 rectifiable provided the supports of f and g have lipschitz boundaries and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex moreover assuming f and g supported on two bounded strictly convex sets omega lambda subset mathbb r n and bounded away from zero and infinity on their respective supports c 0 alpha rm loc regularity of the optimal transport map and local c1 regularity of the free boundaries away from omega cap lambda are shown finally the optimal transport map extends to a global homeomorphism between the active regions

duality for a class of multiobjective control problems with generalized invexity abstract b mond and i smart j math anal appl 136 1988 325 333 defined a kind of invexity and discussed the duality and sufficiency in scalar control problems with such invexity d bhatia and p kumar j math anal appl 189 1995 676 692 defined another kind of invexity corresponding generalized invexity and discussed the duality for multiobjective control problems with such generalized invexity in this paper the duality results for multiobjective control problems with mond and smart s generalized invexity are discussed

convergence acceleration of limit periodic continued fractions under asymptotic side conditions we present a method of convergence acceleration for limitk periodic continued fractionsk an 1 ork 1 bn satisfying certain asymptotic side conditions the method represents an improvement of the fixed point modification considered by thron and waadeland 8 under these conditions the regularc fraction expansions of hypergeometric functions2f1 a 1 c z and2f1 a b c z 2f1 a b 1 c 1 z are examples of continued fractions satisfying these conditions

a note on the relation between two convergence acceleration methods for ordinary continued fractions in this note we relate two methods of convergence acceleration for ordinary continued fractions the first one is due to lorentzen and waadeland jacobsen and waadeland 1980 1998 the second one to waadeland 1986 1987 1987 1987 1988

on the convergence of augmented lagrangian method for optimal transport between nonnegative densities the dynamical formulation of the optimal transport problem introduced by j d benamou and y brenier corresponds to the time space search of a density and a momentum minimizing a transport energy between two densities in order to solve this problem an algorithm has been proposed to estimate a saddle point of a lagrangian we will study the convergence of this algorithm to a saddle point of the lagrangian in the most general conditions particularly in cases where initial and final densities vanish on some areas of the transportation domain the principal difficulty of our study will consist in the proof under these conditions of the existence of a saddle point and especially in the uniqueness of the density momentum component indeed these conditions imply to have to deal with non regular optimal transportation maps for these reasons a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

sufficient optimality criteria and duality for multiobjective variational control problems with v invexity abstract the sufficient optimality criteria for a class of multiobjective variational control problems under the v invexity assumption are presented duality results are proved under a variety of v invexity assumptions multiobjective fractional variational control problems are also studied

integral representation of convex functionals on a space of measures abstract in duality pairs such as m b c 0 and w 1 p w 0 1 p a convex integral functional on the space of functions has a polar which admits an integral representation this representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part the duality is useful in plasticity theory in the sobolev case the study of non parametric integrands is new a description of the sub differential is obtained

generalized gradients of lipschitz functionals abstract the purpose of this article is threefold i to present in a unified fashion the theory of generalized gradients whose elements are at present scattered in various sources ii to give an account of the ways in which the theory has been applied iii to prove new results concerning generalized gradients of summation functionals pointwise maxima and integral functionals on subspaces of l these last mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by rockafellar ioffe and levin valadier and others

symmetric duality in continuous time programming via minmax duality abstract in this paper we present continuous time generalizations of the symmetric duality formulation of dantzig eisenberg and cottle and self duality formulation of mond and cottle and establish appropriate duality relations we also show that continuous time analogues of several pairs of symmetric dual problems which are well known in the area of finite dimensional nonlinear programming can be obtained as special cases of our general symmetric programs

generalized type i invexity and duality in multiobjective variational problems abstract in this paper we introduce new classes of generalized v type i invex functions for variational problems and consider a multiobjective variational problem vp a number of sufficiency results are established using lagrange multiplier conditions under various types of generalized v type i invexity requirements duality results are obtained for mond weir type duals under the above generalized v type i invexity assumptions and their generalizations

improving a method for computing non dominant solutions of certain second order recurrence relations of poincare type in this paper we present a method of convergence acceleration for the calculation of non dominant solutions of second order linear recurrence relations for which the coefficients satisfy certain asymptotic conditions it represents an improvement of the method recently proposed by jacobsen and waadeland 3 4 for limit periodic continued fractions for continued fractions the method corresponds to a repeated application of the bauer muir transformation some examples and a generalization to non homogeneous recurrence relations are given

nonlinear fokker planck equations with reaction as gradient flows of the free energy abstract we interpret a class of nonlinear fokker planck equations with reaction as gradient flows over the space of radon measures equipped with the recently introduced hellinger kantorovich distance the driving entropy of the gradient flow is not assumed to be geodesically convex or semi convex we prove new generalized dissipation inequalities which allow us to control the relative entropy by its production we establish the entropic exponential convergence of the trajectories of the flow to the equilibrium along with other applications this result has an ecological interpretation as a trend to the ideal free distribution for a class of fitness driven models of population dynamics our existence theorem for weak solutions under mild assumptions on the nonlinearity is new even in the absence of the reaction term