a nonlocal supercritical neumann problem abstract we establish existence of positive non decreasing radial solutions for a nonlocal nonlinear neumann problem both in the ball and in the annulus the nonlinearity that we consider is rather general allowing for supercritical growth in the sense of sobolev embedding the consequent lack of compactness can be overcome by working in the cone of non negative and non decreasing radial functions within this cone we establish some a priori estimates which allow via a truncation argument to use variational methods for proving existence of solutions as a side result we prove a strong maximum principle for nonlocal neumann problems which is of independent interest

an equilibrium existence theorem bewley 14 theorem l proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of arrow and debreu 11 the assumptions on technology and preferences are natural and applicable in a wide variety of cases the proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dimensional case this paper establishes the existence of equilibrium under assumptions which are essentially the same as those given by bewley with the additional assumption that the preference orderings of consumers are representable by real valued utility functions this approach is related to the welfare approach of negishi 9 and arrow and hahn 2 chap 51 in the finite dimensional case and simplifies the approach originally adopted by bewley 5 in addition to making clear the role played by each of the assumptions in establishing the existence of equilibrium this approach has the merit of constructing directly a certain real valued function that is maximised at an equilibrium a result that provides a powerful tool in the analysis of qualitative properties of an equilibrium a model of resource allocation in continuous time over an infinite horizon that may be viewed as an application of the model that follows is given in 181

on the use of the varignon frame for single facility weber problems in the presence of convex barriers abstract this paper presents a new experimental approach to the weber problem in the presence of convex barriers by using the varignon frame the varignon frame is a mechanical system of strings weights and a board with holes that has been used to identify an optimal location for the classical weber problem we show through analytical results that the same analog can also be used for some of the weber problems in the presence of barriers some examples from the literature are revisited through experiments findings are compared to those found in the literature practical use of the analog is discussed as it provides rapid solutions allows for flexibility and enables one to visualize the problem

an efficient solution method for weber problems with barriers based on genetic algorithms in this paper we consider the problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers it is assumed that a barrier is a region where neither facility location nor travelling are permitted the resulting non convex optimization problem can be reduced to a finite series of convex subproblems which can be solved by the weiszfeld algorithm in case of the weber objective function and euclidean distances a solution method is presented that by iteratively executing a genetic algorithm for the selection of subproblems quickly finds a solution of the global problem visibility arguments are used to reduce the number of subproblems that need to be considered and numerical examples are presented

finite dimensional approximation in infinite dimensional mathematical programming we consider the problem of approximating an optimal solution to a separable doubly infinite mathematical program p with lower staircase structure by solutions to the programs p n obtained by truncating after the first n variables and n constraints of p viewing the surplus vector variable associated with the nth constraint as a state and assuming that all feasible states are eventually reachable from any feasible state we show that the efficient set of all solutions optimal to all possible feasible surplus states for p n converges to the set of optimal solutions to p a tie breaking algorithm which selects a nearest point efficient solution for p n is shown for convex programs to converge to an optimal solution to p a stopping rule is provided for discovering a value of n sufficiently large to guarantee any prespecified level of accuracy the theory is illustrated by an application to production planning

facility location in the presence of forbidden regions i formulation and the case of euclidean distance with one forbidden circle abstract a constrained form of the weber problem is formulated in which no path is permitted to enter a prespecified forbidden region r of the plane using the calculus of variations the shortest path between two points x y r which does not intersect r is determined if d x y is unconstrained distance we denote the shortes distance along a feasible path by d x y the constrained weber problem is then given points x j r and positive weights wj j 1 2 n find a point x r such that f x n j 1 d x x j is a minimum an algorithm is formulated for the solution of this problem when d x y is euclidean distance and r is a single circular region numerical results are presented

aggregation and discretization in multistage stochastic programming multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events in order to ensure computational tractability continuous state stochastic programs are usually discretized and frequently the curse of dimensionality dictates that decision stages must be aggregated in this article we construct two discrete stage aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem the approximate problems involve finitely many decisions and constraints thus principally allowing for numerical solution

some clarifications on the transversality condition in this paper the authors study a general concave discrete time infinite horizon optimal control problem they establish necessary and sufficient conditions for optimality in the weak sense of w a brock and for optimality in the strong sense of d gale the corresponding transversality conditions are general exhaustion properties of the limit value of the optimal state variables these properties cover and extend the well known results obtained in special cases copyright 1990 by the econometric society

an optimal algorithm for intersecting line segments in the plane the main contribution of this work is an o n log n k time algorithm for computing all k intersections among n line segments in the plane this time complexity is easily shown to be optimal within the same asymptotic cost our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment if any lies right above or below each intersection and each endpoint the algorithm has been implemented and performs very well the storage requirement is on the order of n k in the worst case but it is considerably lower in practice to analyze the complexity of the algorithm an amortization argument based on a new combinatorial theorem on line arrangements is used

the pohozaev identity for the fractional laplacian in this paper we prove the pohozaev identity for the semilinear dirichlet problem delta s u f u in omega u equiv0 in mathbb r n backslash omega here s in 0 1 s is the fractional laplacian in mathbb r n and is a bounded c1 1 domain to establish the identity we use among other things that if u is a bounded solution then u delta s omega is c up to the boundary where x dist x in the fractional pohozaev identity the function u delta s partial omega plays the role that u plays in the classical one surprisingly from a nonlocal problem we obtain an identity with a boundary term an integral over which is completely local as an application of our identity we deduce the nonexistence of nontrivial solutions in star shaped domains for supercritical nonlinearities

transversality conditions for some infinite horizon discrete time optimization problems this paper considers a class of discrete time infinite horizon optimization problems arising in economics necessary optimality conditions for such problems consist of an euler equation and a transversality condition at infinity our concern here is with the latter condition previous results for this type of problem have required such restrictions as concavity and boundedness of the utility functions whose sum is being maximized here we derive a transversality condition that remains valid under considerably more general hypotheses in doing so we introduce methods that may also be applicable to other situations such as optimization in continuous time

convex infinite horizon programs we establish conditions under which a sequence of finite horizon convex programs monotonically increases in value to the value of the infinite program a subsequence of optimal solutions converges to the optimal solution of the infinite problem if the conditions we impose fail then roughtly the optimal value of the infinite horizon problem is an improper convex function under more restrictive conditions we establish the necessary and sufficient conditions for optimality this constructive procedure gives us a way to solve the infinite long range problem by solving a finite short range problem it appears to work well in practice

a bi objective median location problem with a line barrier the multiple objective median problem momp involves locating a new facility with respect to a given set of existing facilities so that a vector of performance criteria is optimized a variation of this problem is obtained if the existing facilities are situated on two sides of a linear barrier such barriers like rivers highways borders or mountain ranges are frequently encountered in practice in this paper theory of an momp with line barriers is developed as this problem is nonconvex but specially structured a reduction to a series of convex optimization problems is proposed the general results lead to a polynomial algorithm for finding the set of efficient solutions the algorithm is proposed for bicriteria problems with different measures of distance

proportional discounting of future costs and benefits social benefits in the distant future receive very little importance when a public policy analysis uses a constant discounting model with its geometric timing weights to compare future benefits with present costs this paper defines conditions on social preferences that imply a model having arbitrary timing weights we then introduce a condition on preferences that restricts this general model to one in which the timing weights are linear fractional this model is proposed as an alternative to the constant discounting model it accords far more importance to the distant future than does constant discounting and like constant discounting it depends on the assessment of a single parameter

an information based approximation scheme for stochastic optimization problems in continuous time dynamic stochastic optimization problems with a large possibly infinite number of decision stages and high dimensional state vectors are inherently difficult to solve in fact scenario tree based algorithms are unsuitable for problems with many stages while dynamic programming type techniques are unsuitable for problems with many state variables this paper proposes a stage aggregation scheme for stochastic optimization problems in continuous time thus having an extremely large i e uncountable number of decision stages by perturbing the underlying data and information processes we construct two approximate problems that provide bounds on the optimal value of the original problem moreover we prove that the gap between the bounds converges to zero as the stage aggregation is refined if massive aggregation of stages is possible without sacrificing too much accuracy the aggregate approximate problems can be addressed by means of scenario tree based methods the suggested approach applies to problems that exhibit randomness in the objective and the constraints while the constraint functions are required to be additively separable in the decision variables and random parameters

gbsss the generalized big square small square method for planar single facility location abstract big square small square is a geometrical branch and bound algorithm devised by p hansen et al or 33 1251 1265 for the solution of constrained planar minisum single facility location problems with l p norms and continuous non decreasing costfunctions the method basically works by splitting the studied planar region into squares and either rejecting or further processing these squares by the evaluation of a lower bound we present a modified version of this algorithm aimed at correcting a small failure to converge accelerating the calculations minimising the information to be stored and most importantly determining a region of near optimality furthermore the method is applicable to any planar single facility problem with distances measured by mixed norms and as an objective any continuous function of the distances this includes nearly all the models which have been proposed in the literature

a reduction result for location problems with polyhedral barriers abstract in this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facilities where a set of polyhedral barriers restricts traveling this non convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities like e g the median and center objective functions an exact algorithm and a heuristic solution procedure based on this reduction result are developed

discrete time finite horizon approximation of infinite horizon optimization problems with steady state invariance the computation of large scale nonlinear intertemporal optimization problems requires time aggregation a procedure generally adopted is shown to introduce a dependency of the solution steady state to a specific choice of sequence of time intervals the authors establish necessary and sufficient conditions to avoid this dependency the result is a considerable improvement in the numerical accuracy of the time aggregated approximation the conditions apply to a broad class of models and prove useful in large scale applied general equilibrium modeling this conclusion is highlighted by a comparison with a spectral projection method using optimal orthogonal collocation as in k l judd 1992 copyright 1994 by the econometric society

algebraic properties of location problems with one circular barrier abstract the consideration of barriers to travel plays an increasingly important role in the transportation and location literature in one of the classical papers on location problems with barriers katz and cooper eur j oper res 6 1981 166 considered the weber problem often also referred to as median problem with one circular barrier region considering the same problem we develop new structural results showing that the set of feasible solutions can be subdivided into a polynomial number of cells of algebraic invariance on every convex subset of which the generally non convex objective function is convex these results imply improved exact and heuristic solution procedures based on convex optimization methods

an efficient algorithm for facility location in the presence of forbidden regions abstract this paper investigates a constrained form of the classical weber problem specifically we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized it is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the euclidean distance metric a solution procedure for this nonconvex programming problem is presented it is shown that by iteratively solving a series of unconstrained problems this procedure terminates at a local optimum to the original constrained problem numerical examples are presented

discounted value representations of temporal preferences this paper examines the proposition that temporal preferences may be represented in terms of discount functions an analytical framework is developed within which preferences are defined over a set of programs or prospects over time necessary and sufficient conditions for generalized discounting discounting as well as exponential discounting are established for a class of program sets over discrete and continuous time

the minisum and minimax location problems revisited the minisum minimax problem consists of locating a single facility in the plane with the aim of minimizing the sum of the weighted distances the maximum weighted distance to m given points we present two solution methods for generalized versions of these problems in which i location is restricted to the union of a finite number of convex polygons ii distances are approximated by norms that may differ with the given points and iii transportation costs are increasing and continuous functions of distance computational experience is described

existence results and finite horizon approximates for infinite horizon optimization problems the paper deals with infinite horizon optimization problems the existence of optimal solutions is obtained as a consequence of an asymptotic growth condition we also exhibit finite horizon approximates that yield upper and lower bounds for the optimal values and whose optimal solutions converge to the long term optimal trajectories

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

strong maximum principles for fractional laplacians we give a unified approach to strong maximum principles for a large class of nonlocal operators of order s 0 1 that includes the dirichlet the neumann restricted or regional and the neumann semirestricted laplacians

unbounded utility for savage s foundations of statistics and other models a general procedure for extending finite dimensional additive like representations for binary relations to infinite dimensional integral like representations is developed by means of a condition called truncation continuity the restriction of boundedness of utility met throughout the literature can now be dispensed with and for instance normal distributions or any other distribution with finite first moment can be incorporated classical representation results of expected utility such as savage 1954 von neumann and morgenstern 1944 anscombe and aumann 1963 de finetti 1937 and many others can now be extended the results are generalized to schmeidler s 1989 approach with nonadditive measures and choquet integrals and quiggin s 1982 rank dependent utility the different approaches have been brought together in this paper to bring to the fore the unity in the extension process

optimal control problems involving optimal economic growth with infinite horizon abstract we examine the existence of optimal solutions for a very general optimal economic growth problem with unbounded planning time horizon a convexity hypothesis about a certain orientor field plays a crucial role in the proof then we conduct sensitivity analysis and determine the relation between optimal pairs and the initial capital accumulation finally we see what happens if the convexity hypothesis we have made no longer exists this leads us to the introduction of the relaxed system and its comparison with the original one

an existence result for optimal economic growth problems abstract an existence result for optimal control problems of lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain this subsumes the main existence result of r f baum j optim theory appl 19 1976 89 116 and has the existence results for optimal economic growth problems of s i takekuma j math econom 7 1980 193 208 and m j p magill econometrica 49 1981 679 711 j math anal appl 82 1981 66 74 as simple corollaries in addition a new notion of uniform integrability is used which coincides with the classical notion if the time domain is bounded