Irwin E. Schochetman

Algebraic graph theory without orientation

Abstract Let G be an undirected graph with vertices {v 1 ,v 2 ,…,>;v ⋎ } and edges {e1,e2, …,eϵ}. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if ej is a link incident with vi, 2 if ej is a loop at vi, and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1s to a − 1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from Rϵ to R⋎ defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MMt, to count certain spanning substructures of G. These formulas may be viewed as generalizations of the matrix tree theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges of G analogous to the cycle matroid and its dual. In this setting, the analogue of a spanning forest can have components with one odd cycle, and the analogue of an edge cut has the property that its removal creates a new bipartite component.

Infinite horizon optimization

We consider the general problem of choosing a discounted cost minimizing infinite sequence of decisions from a closed subset of the product space formed by a sequence of arbitrary compact metric spaces. Examples include equipment replacement, production planning and, more generally, infinite stage mathematical programs. It is shown that the optimal costs for finite horizon approximating problems converge to the optimal infinite horizon cost as the horizons diverge to infinity. Moreover, the existence of a unique algorithmically optimal i.e. accumulation point solution is shown to be a necessary and sufficient condition for convergence in the product topology i.e. policy convergence of all finite horizon optima. Under the weaker condition of Hausdorff convergence of the sets of finite horizon optima to the set of infinite horizon optima, we show how to force policy convergence through a natural tie-breaking rule. Finally, a forward algorithm is presented which, in the presence of a unique infinite horizon optimum, is guaranteed to converge.

On the minors of an incidence matrix and its Smith normal form

Abstract 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.

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 firstN variables andN constraints of (P). Viewing the surplus vector variable associated with theNth 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 ofN sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.

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 detined 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.

Infinite horizon optimality criteria for equipment replacement under technological change

We consider the problem of optimally acquiring and retiring assets that are undergoing technological change over an unbounded horizon. Assuming discounted future per-period acquisition and maintenance costs go to zero, while total discounted costs diverge to infinity, we show that efficient solutions exist and are overtaking optimal.

Solution and Forecast Horizons for Infinite-Horizon Nonhomogeneous Markov Decision Processes

We consider a nonhomogeneous infinite-horizon Markov Decision Process (MDP) problem with multiple optimal first-period policies. We seek an algorithm that, given finite data, delivers an optimal first-period policy. Such an algorithm can thus recursively generate, within a rolling-horizon procedure, an infinite-horizon optimal solution to the original problem. However, it can happen that no such algorithm exists, i.e., the MDP is not well posed. Equivalently, it is impossible to solve the problem with a finite amount of data. Assuming increasing marginal returns in actions (with respect to states) and stochastically increasing state transitions (with respect to actions), we provide an algorithm that is guaranteed to solve the given MDP whenever it is well posed. This algorithm determines, in finite time, a forecast horizon for which an optimal solution delivers an optimal first-period policy. As an application, we solve all well-posed instances of the time-varying version of the classic asset-selling problem.

Existence and Discovery of Average Optimal Solutions in Deterministic Infinite Horizon Optimization

We consider the problem of making a sequence of decisions, each chosen from a finite action set over an infinite horizon, so as to minimize its associated average cost. Both the feasibility and cost of a decision are allowed to depend upon all of the decisions made prior to that decision; moreover, time-varying costs and constraints are allowed. A feasible solution is said to be efficient if it reaches each of the states through which it passes at minimum cost. We show that efficient solutions exist and that, under a state reachability condition, efficient solutions are also average optimal. Exploiting the characterization of efficiency via a solutions short-run as opposed to long-run behavior, a forward algorithm is constructed which recursively discovers the first, second, and subsequent decisions of an efficient, and hence average optimal, infinite horizon solution.

Pointwise versions of the maximum theorem with applications in optimization

Abstract We 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.

ON THE CLOSURE OF THE SUM OF CLOSED SUBSPACES

We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators.