relativistic chern simons higgs vortex equations an existence theorem is established for the solutions to the non abelian relativistic chern simons higgs vortex equations over a doubly periodic domain when the gauge group g assumes the most general and important prototype form g su n mathematics subject classification 2010 35c08 35j50 35q70 81e13

a variational analysis of the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface we will give existence results by using variational methods in a noncoercive case a key tool in our analysis is a new moser trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u1 and u2 2012 wiley periodicals inc

an ellipsoid trust region bundle method for nonsmooth convex minimization this paper presents a bundle method of descent for minimizing a convex possibly nonsmooth function f of several variables at each iteration the algorithm finds a trial point by minimizing a polyhedral model of f subject to an ellipsoid trust region constraint the quadratic matrix of the constraint which is updated as in the ellipsoid method is intended to serve as a generalized hessian to account for second order effects thus enabling faster convergence the interpretation of generalized hessians is largely heuristic since so far this notion has been made precise by j l goffin only in the solution of linear inequalities global convergence of the method is established and numerical results are given

a redistributed proximal bundle method for nonconvex optimization proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives this naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting this work proposes an approach based on generating cutting planes models not of the objective function as most bundle methods do but of a local convexification of the objective function the corresponding convexification parameter is calculated on the fly in such a way that the algorithm can inform the user as to what proximal parameters are sufficiently large that the objective function is likely to have well defined proximal points this novel approach shown to be sound from both the objective function and subdifferential modelling perspectives opens the way to create workable nonconvex algorithms based on nonconvex mathcal vu theory both theoretical convergence analysis and some encouraging preliminary numerical experience are provided

solving mixed integer nonlinear programs by outer approximation a wide range of optimization problems arising from engineering applications can be formulated as mixed integer nonlinear programming problems minlps duran and grossmann 1986 suggest an outer approximation scheme for solving a class of minlps that are linear in the integer variables by a finite sequence of relaxed milp master programs and nlp subproblems

robust optimization for unconstrained simulation based problems in engineering design an optimized solution often turns out to be suboptimal when errors are encountered although the theory of robust convex optimization has taken significant strides over the past decade all approaches fail if the underlying cost function is not explicitly given it is even worse if the cost function is nonconvex in this work we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large partial differential equations pde solver response surface and kriging metamodels moreover this technique can be employed for most real world problems because it operates directly on the response surface and does not assume any specific structure of the problem we present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics relevant to nanophotonic design the corresponding objective function is highly nonconvex and resides in a 100 dimensional design space starting from an optimized design we report a robust solution with a significantly lower worst case cost while maintaining optimality we further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties

su 3 chern simons vortex theory and toda systems motivated by the study of the asymptotic properties of non topological condensates in the non abelian chern simons vortex theory see 26 we analyze the su 3 toda system p 1 2 zj j e 2i 1kijzi me i 12kijzi 1 on m mzj 0j 1 2 where m r2 z2 k kij 2 1 12 is the su 3 cartan matrix and j are positive parameters we study the variational problem associated with the system p 1 2 in a range of parameters where the trivial solution is a strict local minimum and the corresponding sobolev type inequality fails to apply in this situation a lack of compactness may occur due to concentration phenomena nonetheless we are able to establish the existence of a non trivial solution for p 1 2 which is not a minimizer

forced oscillations in quadratically damped systems bayliss 1975 has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations in the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear the properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution by using the lipschitz continuity of the mapping and the contracting mapping principle almost periodic solutions can be found for perturbations of the considered equation attention is also given to the lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory

oracle based robust optimization via online learning robust optimization is a common optimization framework under uncertainty when problem parameters are unknown but it is known that they belong to some given uncertainty set in the robust optimization framework a min max problem is solved wherein a solution is evaluated according to its performance on the worst possible realization of the parameters in many cases a straightforward solution to a robust optimization problem of a certain type requires solving an optimization problem of a more complicated type which might be np hard in some cases for example solving a robust conic quadratic program such as those arising in a robust support vector machine svm with an ellipsoidal uncertainty set leads in general to a semidefinite program in this paper we develop a method for approximately solving a robust optimization problem using tools from online convex optimization where at every stage a standard nonrobust optimization program is solved our algorithms find an approximate robust solution usin

existence and multiplicity result for the singular toda system abstract we consider the toda system on a compact surface g u 1 2 1 h 1 e u 1 h 1 e u 1 d v g 1 2 h 2 e u 2 h 2 e u 2 d v g 1 4 j 1 j 1 j p j 1 u 2 2 2 h 2 e u 2 h 2 e u 2 d v g 1 1 h 1 e u 1 h 1 e u 1 d v g 1 4 j 1 j 2 j p j 1 where h i are smooth positive functions i are positive real parameters p j are given points on and i j are numbers greater than 1 we give existence and multiplicity results using variational and morse theoretical methods it is the first existence result when some of the i j s are allowed to be negative

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

a topological degree counting for some liouville systems of mean field type let a a ij n n be an invertible matrix and a 1 a ij n n be the inverse of a in this paper we consider the generalized liouville system n 0 1 g u i n a ij j h j e u j h j e u j 1 0 in m j 1 where 0 h j c 1 m and j and prove that under the assumptions of h 1 and h 2 see introduction the leray schauder degree of 0 1 is equal to m 1 m n n if 1 n satisfies 8 n n n i 1 i 1 i j n a ij i j 8 n 1 n n i 1 i equation 0 1 is a natural generalization of the classic liouville equation and is the euler lagrangian equation of the nonlinear function u 1 2 m 1 i j n a ij g u i g u j n n i 1 m i u i n n i 1 i log m h i e u i the liouville system 0 1 has arisen in many different research areas in mathematics and physics our counting formulas are the first result in degree theory for liouville systems

an outer approximation algorithm for a class of mixed integer nonlinear programs an outer approximation algorithm is presented for solving mixed integer nonlinear programming problems of a particular class linearity of the integer or discrete variables and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure based on principles of decomposition outer approximation and relaxation the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed integer linear master program convergence and optimality properties of the algorithm are presented as well as a general discussion on its implementation numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper finally a theoretical comparison with generalized benders decomposition is presented on the lower bounds predicted by the relaxed master programs

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

existence and non existence results for the su 3 singular toda system on compact surfaces we consider the su 3 singular toda system on a compact surface g u1 2 1 h1eu1 h1eu1dvg 1 2 h2eu2 h2eu2dvg 1 4 m 1m 1m pm 1 u2 2 2 h2eu2 h2eu2dvg 1 1 h1eu1 h1eu1dvg 1 4 m 1m 2m pm 1 where hi are smooth positive functions on i r pm and im 1 we give both existence and non existence results under some conditions on the parameters i and im existence results are obtained using variational methods which involve a geometric inequality of new type non existence results are obtained using blow up analysis and localized poho aev type identities

orbit optimization by radial basis function interpolation in trust regions we present a new derivative free algorithm orbit for unconstrained local optimization of computationally expensive functions a trust region framework using interpolating radial basis function rbf models is employed the rbf models considered often allow orbit to interpolate nonlinear functions using fewer function evaluations than the polynomial models considered by present techniques approximation guarantees are obtained by ensuring that a subset of the interpolation points is sufficiently poised for linear interpolation the rbf property of conditional positive definiteness yields a natural method for adding additional points we present numerical results on test problems to motivate the use of orbit when only a relatively small number of expensive function evaluations are available results on two very different application problems calibration of a watershed model and optimization of a pde based bioremediation plan are also encouraging and support orbit s effectiveness on blackbox functions for which no special mathematical structure is known or available

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

a general existence result for the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface u1 2 1 h1eu1 h1eu1dvg 1 2 h2eu2 h2eu2dvg 1 u1 4 j 1m 1 j pj 1 u2 2 2 h2eu2 h2eu2dvg 1 1 h1eu1 h1eu1dvg 1 u2 4 j 1m 2 j pj 1 which is motivated by the study of models in non abelian chern simons theory here h1 h2h1 h2 are smooth positive functions 1 2 1 2 two positive parameters pipi points of the surface and 1 i 2 j 1 i 2 j non negative numbers we prove a general existence result using variational methods the same analysis applies to the following mean field equation u 1 heu heudvg 1 2 he u he udvg 1 which arises in fluid dynamics

new concentration phenomena for su 3 toda system in this paper we consider the following su 3 toda system 0 1 u1 2 2h1eu1 2h2eu2 0in b1 0 u2 2 2h2eu2 2h1eu1 0in b1 0 u1 u2 0on b1 0 where is a positive parameter which will go to zero h1 x h2 x h x is a positive smooth function and b1 0 is a unit ball in r2 in 29 the authors construct a family of solutions u1 u2 of 0 1 for h 1 such that the solutions blow up at the origin with mass 8 4 after some scaling with the limiting profile wi x i 2ewi in r2 r2 x i 2ewi where i 23 i for i 1 2 in that paper the authors ask whether there exist blow up solutions of mass 8 4 with the limiting profile w 2ew 0in r2 r2ew such that u1 is the sum of two bubbles and u2 has one bubble in this paper we prove that such solution exists

composition of pseudo almost periodic functions and semilinear differential equations under the assumption of uniformly continuity we establish a composition theorem of pseudo almost periodic functions in general banach spaces as an application we give some existence theorems of pseudo almost periodic solutions and mild pseudo almost periodic solutions for some differential equations

asymmetric blow up for the su 3 toda system abstract we consider the so called toda system in a smooth planar domain under homogeneous dirichlet boundary conditions we prove the existence of a continuum of solutions for which both components blow up at the same point this blow up behavior is asymmetric and moreover one component includes also a certain global mass the proof uses singular perturbation methods

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

the existence of pseudo almost periodic solutions for some nonlinear differential equations in banach spaces abstract let e be a real banach where x h denotes the limit of the quotient x x t h t as 0 t 0 consider the differential equation d x d t a t x f t where we assume amongst other properties that a is strongly dissipative in the following sense x y a t x a t y p t x y x y where is a function of the type u 0 and p is a function majorized by an almost periodic function with negative mean value first we show that this equation has a unique pseudo almost periodic solution then we apply this result to the class of the following equations d x d t q t x x f t 0 where q is a pseudo almost periodic function with positive mean value

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

trust region methods without using derivatives worst case complexity and the nonsmooth case trust region methods are a broad class of methods for continuous optimization that found application in a variety of problems and contexts in particular they have been studied and applied for problems without using derivatives the analysis of trust region derivative free methods has focused on global convergence and they have been proven to generate a sequence of iterates converging to stationarity independently of the starting point most of such an analysis is carried out in the smooth case and moreover little is known about the complexity or global rate of these methods in this paper we start by analyzing the worst case complexity of general trust region derivative free methods for smooth functions for the nonsmooth case we propose a smoothing approach for which we prove global convergence and bound the worst case complexity effort for the special case of nonsmooth functions that result from the composition of smooth and nonsmooth convex components we show how to improve the existing resu

existence uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in banach spaces abstract we give sufficient conditions ensuring the existence uniqueness and global attractiveness of a pseudo compact almost automorphic solution of the following differential equation x t f t x t in a banach space e where f r e e is a pseudo almost automorphic function with respect to the first argument we essentially assume that the function f is dissipative then we apply the main results to the following equation x t q t x t x t f t 0

degree counting and shadow system for toda system of rank two one bubbling abstract we initiate the program for computing the leray schauder topological degree for toda systems of rank two this program still contains a lot of challenging problems for analysts as the first step we prove that if a sequence of solutions u 1 k u 2 k blows up then one of h j e u j k m h j e u j k d v g j 1 2 tends to a sum of dirac measures this is so called the phenomena of weak concentration our purposes in this article are i to introduce the shadow system due to the bubbling phenomena when one of parameters i crosses 4 and j 4 n where 1 i j 2 ii to show how to calculate the topological degree of toda systems by computing the topological degree of the general shadow systems iii to calculate the topological degree of the shadow system for one point blow up we believe that the degree counting formula for the shadow system would be useful in other problems

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

pseudo almost periodic solutions of some differential equations ii we first present some results about pseudo almost periodic functions then we use these results to show some theorems about existence and uniqueness of the pseudo almost periodic solutions of the linear and the quasi linear nonautonomous equations finally we set up some theorems of the existence of pseudo almost periodic solutions implying the existence of almost periodic solutions of general ordinary differential equation systems

global convergence of radial basis function trust region algorithms for derivative free optimization we analyze globally convergent derivative free trust region algorithms relying on radial basis function interpolation models our results extend the recent work of conn scheinberg and vicente siam j optim 20 2009 pp 387 415 to fully linear models that have a nonlinear term we characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first order critical points for the orbit algorithm of wild regis and shoemaker siam j sci comput 30 2008 pp 3197 3219 using orbit we present numerical results for different types of radial basis functions on a series of test problems we also demonstrate the use of orbit in finding local minima on a computationally expensive environmental engineering problem involving remediation of contaminated groundwater

degree counting for toda system with simple singularity one point blow up abstract in this paper we study the degree counting formula of the rank two toda system with simple singular source when 1 0 4 4 8 and 2 4 n the key step is to derive the degree formula of the shadow system which arises from the bubbling solutions as 1 tends to 4 in order to compute the topological degree of the shadow system we need to find some suitable deformation during this deformation we shall deal with new difficulty arising from the phenomenon blow up does not necessarily imply concentration of mass this phenomenon occurs due to the collapsing of singularities this is a continuation of the previous work 25

benchmarking derivative free optimization algorithms we propose data profiles as a tool for analyzing the performance of derivative free optimization solvers when there are constraints on the computational budget we use performance and data profiles together with a convergence test that measures the decrease in function value to analyze the performance of three solvers on sets of smooth noisy and piecewise smooth problems our results provide estimates for the performance difference between these solvers and show that on these problems the model based solver tested performs better than the two direct search solvers tested

derivative free robust optimization by outer approximations we develop an algorithm for minimax problems that arise in robust optimization in the absence of objective function derivatives the algorithm utilizes an extension of methods for inexact outer approximation in sampling a potentially infinite cardinality uncertainty set clarke stationarity of the algorithm output is established alongside desirable features of the model based trust region subproblems encountered we demonstrate the practical benefits of the algorithm on a new class of test problems

a double mean field equation related to a curvature prescription problem abstract we study a double mean field type pde related to a prescribed curvature problem on compacts surfaces with boundary 0 1 u 2 k e u k e u 1 in u 2 h e u 2 h e u 2 1 on here and are real parameters k h are smooth positive functions on and respectively and is the outward unit normal vector to we provide a general blow up analysis then a moser trudinger inequality which gives energy minimizing solutions for some range of parameters finally we provide existence of min max solutions for a wider range of parameters which is dense in the plane if is not simply connected

an approximation technique for robust nonlinear optimization nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst case formulation that is however difficult to treat computationally in this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set in case of any norm bounded parameter uncertainty this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints the main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems one similar to the forward mode the other similar to the reverse mode of automatic differentiation we show how to generalize the techniques to optimal control problems and discuss how even infinite dimensional uncertainties can be treated efficiently finally we present optimization results for an example from process engineering a batch distillation

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

manifold sampling for optimization of nonconvex functions that are piecewise linear compositions of smooth components we develop a manifold sampling algorithm for the minimization of a nonsmooth composite function f triangleq psi h circ f when psi is smooth with known derivatives h is a known nonsmoo

blow up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities abstract a class of equations with exponential nonlinearities on a compact riemannian surface is considered more precisely we study an asymmetric sinh gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities we start by performing a blow up analysis in order to derive some information on the local blow up masses as a consequence we get a compactness property in a supercritical range we next introduce a variational argument based on improved moser trudinger inequalities which yields existence of solutions for any choice of the underlying surface

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