A characterization of open mapping in terms of convergent sequences

It is certainly well known that a mapping between metric spaces is continuous if and only if it preserves convergent sequences. Does there exist a comparable characterization for the mapping to be open? Of course, the inverse mapping is set-valued, in general. In this research/expository note, we show that a mapping is open if and only if the set-valued inverse mapping preserves convergent sequences in an appropriate set-theoretic sense

A characterization of open mapping in terms of convergent sequences

It is certainly well known that a mapping between metric spaces is continuous if and only if it preserves convergent sequences. Does there exist a comparable characterization for the mapping to be open? Of course, the inverse mapping is set-valued, in general. In this research/expository note, we show that a mapping is open if and only if the set-valued inverse mapping preserves convergent sequences in an appropriate set-theoretic sense

A characterization of open mapping in terms of convergent sequences

It is certainly well known that a mapping between metric spaces is continuous if and only if it preserves convergent sequences. Does there exist a comparable characterization for the mapping to be open? Of course, the inverse mapping is set-valued, in general. In this research/expository note, we show that a mapping is open if and only if the set-valued inverse mapping preserves convergent sequences in an appropriate set-theoretic sense

OPTIMALITY CRITERIA FOR DETERMINISTIC DISCRETE-TIME INFINITE HORIZON OPTIMIZATION

We consider the problem of selecting an optimality criterion, when total costs diverge, in deterministic infinite horizon optimization over discrete time. Our formulation allows for both discrete and continuous state and action spaces, as well as time-varying, that is, nonstationary, data. The task is to choose a criterion that is neither too overselective, so that no policy is optimal, nor too underselective, so that most policies are optimal. We contrast and compare the following optimality criteria: strong, overtaking, weakly overtaking, efficient, and average. However, our focus is on the optimality criterion of efficiency. (A solution is efficient if it is optimal to each of the states through which it passes.) Under mild regularity conditions, we show that efficient solutions always exist and thus are not overselective. As to underselectivity, we provide weak state reachability conditions which assure that every efficient solution is also average optimal, thus providing a sufficient condition for average optima to exist. Our main result concerns the case where the discounted per-period costs converge to zero, while the discounted total costs diverge to infinity. Under the assumption that we can reach from any feasible state any feasible sequence of states in bounded time, we show that every efficient solution is also overtaking, thus providing a sufficient condition for overtaking optima to exist. 1

Convergence of best approximations from unbounded sets

Given a metric space whose bounded sets are relatively compact (i.e., have compact closures), we show that a nearest point selection from a sequence of Kuratowski converging sets converges to the nearest point in the limit set whenever the latter point is unique. The result is extended to Kuratowski limits of linear varieties in infinite dimensional Hilbert spaces where this nearest point (relative to the origin) is necessarily unique. Finally, we show that the Kuratowski limit of hyperplanes must itself be a hyperplane and that a necessary and sufficient condition for the associated nearest points to the origin to converge as above is that the canonial points parametrizing the hyperplanes converge.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30064/1/0000434.pd

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 N th 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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47924/1/10107_2005_Article_BF01586057.pd

A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem ( P ) of selecting a continuous function x over a σ-compact Hausdorff space T to a metric space A , from a feasible region X of such functions, so as to minimize a functional c on X . We require that X consist of a closed equicontinuous family of functions lying in the product (over T ) of compact subsets Y t of A . (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c ( x ) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C ( T , A ) of continuous functions from T to A , the feasible region X is compact. Thus optimal solutions x * to ( P ) exist under the assumption that c is continuous. We wish to approximate such an x * by optimal solutions to a net { P i }, i ∈ I , of approximating problems of the form min   x ∈ X i c i ( x ) for each i ∈ I , where (1) the net of sets { X i } I converges to X in the sense of Kuratowski and (2) the net { c i } I of functions converges to c uniformly on X . We show that for large i , any optimal solution x * i to the approximating problem ( P i ) arbitrarily well approximates some optimal solution x * to ( P ). It follows that if ( P ) is well-posed, i.e., lim sup  X i * is a singleton { x * }, then any net { x i * } I of ( P i )-optimal solutions converges in C ( T , A ) to x * . For this case, we construct a finite algorithm with the following property: given any prespecified error δ and any compact subset Q of T , our algorithm computes an i in I and an associated x i * in X i * which is within δ of x * on Q . We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44092/1/10479_2004_Article_351310.pd

Convergence of nearest-point selections from unbounded sets

http://deepblue.lib.umich.edu/bitstream/2027.42/7452/5/bal9403.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/7452/4/bal9403.0001.001.tx

A finite algorithm for solving optimization problems

http://deepblue.lib.umich.edu/bitstream/2027.42/7454/5/ban0108.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/7454/4/ban0108.0001.001.tx