Tools from control and dynamical systems have proven valuable for analyzing and developing optimization methods. In this paper, we establish rigorous theoretical foundations for using feedback linearization—a well-established nonlinear control technique—to solve constrained optimization problems. For equality-constrained optimization, we establish global convergence rates to first-order Karush-Kuhn-Tucker (KKT) points and uncover the close connection between the FL method and the Sequential Quadratic Programming (SQP) algorithm. Building on this relationship, we extend the FL approach to handle inequality-constrained problems. Furthermore, we introduce a momentum-accelerated feedback linearization algorithm and provide a rigorous convergence guarantee.

A single physical computer memory may be used for more than one type of page, with the amount of memory devoted to each type under software control.For example, the effective size of memory can be increased by storing some pages in compressed form and decompressing on demand.This introduces another level to the memory hierarchy, and the sizes of two adjacent levels may be varied dynamically.We present an algorithm for varying the mixture of pages in a two-type cache, and then analyze the "competitiveness" of our algorithm.We also report progress in partially automating the design and analysis of on-line algorithms.

This paper attaches a name to a natural class of combinatorial problems and points out that the class includes many important special cases. One special case, a simple problem of placing nonoverlapping labels on a rectangular map, is shown to be NP-complete.

The preemptive scheduling of sporadic tasks on a uniprocessor is considered. A task may arrive at any time, and is characterized by a value that reflects its importance, an execution time that is the amount of processor time needed to completely execute the task, and a deadline by which the task is to complete execution. The goal is to maximize the sum of the values of the completed tasks. An online scheduling algorithm that achieves optimal performance when the system is underloaded and provides a nontrivial performance guarantee when the system is overloaded is designed. The algorithm is implemented using simple data structures to run at a cost of O(log n) time per task, where n bounds the number of tasks in the system at any instant. Upper bounds on the best performance guarantee obtainable by an online algorithm in a variety of settings are derived.>

The authors study the performance of online algorithms in environments where no value is obtained for the partial execution of a request. They prove that no online scheduling algorithm can have a competitive factor greater than 0.25 times the optimal. They further refine this bound by considering the effect of the loading factor. Other models of task systems (for example, tasks systems consisting of many types of task requests), are considered. Similar upper bounds on the competitive factor that can be made by online scheduling algorithms in these environments are proved. It is shown that the performance bound of 0.25 is tight by means of a simple online uniprocessor scheduling algorithm has a competitive factor of 1/4. The authors extend the discussion to systems with dual processors. They show that the upper bound for the dual-processor online scheduling problem is 1/2 if all tasks have the same value density. This bound is tight if the tasks all also have zero laxity.>

The technique of speeding up access into search structures by maintaining fingers that point to various locations of the search structure is considered. The problem of choosing, in a large search structure, locations at which to maintain fingers is treated. In particular, a server problem in which k servers move along a line segment of length m, where m is the number of keys in the search structure, is addressed. Since fingers may be arbitrarily copied, a server is allowed to jump, or fork, to a location currently occupied by another server. Online algorithms are presented and their competitiveness analyzed. It is shown that the case in which k=2 behaves differently from the case in which k>or=3, by showing that there is a four-competitive algorithm for k=2 that never forks its fingers. For k>or=3, it is shown that any online algorithm that does not fork its fingers can be at most Omega (m/sup 1/2/)-competitive. The main result is that for k=3 there is an online algorithm that forks and is constant competitive (independent of m, the size of the search structure). The algorithm is simple and implementable.>

Motivated by applications in data compression, debugging, and physical simulation, we consider the problem of adaptively choosing locations in a long computation at which to save intermediate results.Such checkpoints allow faster recomputation of arbitrary requested points within the computation.We abstract the problem to a server problem in which k servers move along a line in a single direction, modeling the fact that most computations are not reversible.Since checkpoints may be arbitrarily copied, we allow a server to jump to any location currently occupied by another server.We present online algorithms and analyze their competitiveness.We give lower bounds on the competitiveness of any online algorithm and show that our algorithms achieve these bounds within relatively small factors.

The combinatorial structure of visibility in simple orthogonal polygons is studied. It is shown that the visibility graph of a horizontally or vertically convex polygon is a permutation graph. In general, orthogonal polygons can have concavities (dents) with four possible orientations. In the case where the polygon has three dent orientations, it is shown that the visibility graph is weakly triangulated. Since weakly triangulated graphs are perfect, a polynomial algorithm for this polygon covering problem is obtained. Furthermore, the following duality relationship is obtained. The minimum number of orthogonally convex polygons needed to cover an orthogonal polygon P with at most three dent orientations is equal to the maximum number of points of P, no two of which can be contained together in an orthogonally convex covering polygon. Finally, it shown that in the case of orthogonal polygons with all four dent orientations, the above duality relationship fails to hold.

We consider the problem of covering simple orthogonal polygons with star polygons. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q.

A formal study of three-dimensional topological graph theory is initiated. The problem of deciding whether a graph can be embedded in 3-space so that no collection of vertex-disjoint cycles is topologically linked is considered first. The Robertson-Seymour Theory of Graph Minors is applicable to this problem and guarantees the existence of an O(n/sup 3/) algorithm for the decision problem. However, not even a finite-time decision procedure was known for this problem. A small set of forbidden minors for linkless embeddable graphs is exhibited, and it is shown that any graph with these minors cannot be embedded without linked cycles. It is further established that any graph that does not contain these minors is embeddable without linked cycles. Thus, an O(n/sup 3/) algorithm for the decision problem is demonstrated. It is believed that the proof technique will lead to an algorithm for actually embedding a graph, provided it does not contain the forbidden minors.>