Pointwise versions of the maximum theorem with applications in optimization

AbstractWe establish a sequential version of the Maximum Theorem which is suitable for solving general optimization problems by successive approximation, e.g. finite truncation of an ”infinite” optimization problem. This can then be used to obtain convergence of optimal values and (partial) convergence of optimal solutions. In particular, we do this for general problems in infinite horizon optimization and semi-infinite programming

Optimal solution characterization for infinite positive semi-definite programming

We give a set-theoretic description of the set of optimal solutions to a general positive semi-definite quadratic programming problem over an affine set. We also show that the solution space is again an affine set, thus offering the opportunity to find an optimal solution by solving a corresponding operator equation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31452/1/0000373.pd

Convergence of selections with applications in optimization

We consider the problem of finding an easily implemented tie-breaking rule for a convergent set-valued algorithm, i.e., a sequence of compact, non-empty subsets of a metric space converging in the Hausdorff metric. Our tie-breaking rule is determined by nearest-point selections defined by "uniqueness" points in the space, i.e., points having a unique best approximation in the limit set of the convergent algorithm. Convergence of the algorithm is shown to be equivalent to convergence of all such nearest-point selections. Under reasonable additional hypotheses, all points in the metric space have the uniqueness property. Consequently, all points yield convergent nearest-point selections, i.e., tie-breaking rules, for a convergent algorithm.We then show how to apply these results to approximate solutions for the following types of problems: infinite systems of inequalities, semi-infinite mathematical programming, non-convex optimization, and infinite horizon optimization.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29485/1/0000571.pd

Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales

We obtain Euler-Lagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and discrete settings: our results seem new and interesting even in the particular cases when the time scale is the set of real numbers or the set of integers.Comment: This is a preprint of a paper whose final and definite form will appear in Journal of Optimization Theory and Applications (JOTA). Paper submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for publication 15-April-201

Transversality Conditions for Infinite Horizon Variational Problems on Time Scales

We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler-Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.Comment: Submitted 6-October-2009; Accepted 19-March-2010 in revised form; for publication in "Optimization Letters"

On the Minors of an Incidence Matrix and its Smith Normal Form

Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertex-disjoint odd cycles (i.e., the odd tulgeity) of the graph. The problem of determining this number is shown to be NP-hard. Turning to maximal minors, we determine the rank of the incidence matrix. This depends on the number of components of the graph containing no odd cycle. We then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix, as well as its Smith normal form. These results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs. Finally, we give a sufficient condition for a system of equations (whose coefficient matrix is an incidence matrix) to admit an integer solution