Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming

We study the preconditioning of symmetric indefinite linear systems of equations that arise in interior point solution of linear optimization problems. The preconditioning method that we study exploits the block structure of the augmented matrix to design a similar block structure preconditioner to improve the spectral properties of the resulting preconditioned matrix so as to improve the convergence rate of the iterative solution of the system. We also propose a two-phase algorithm that takes advantage of the spectral properties of the transformed matrix to solve for the Newton directions in the interior-point method. Numerical experiments have been performed on some LP test problems in the NETLIB suite to demonstrate the potential of the preconditioning method discussed.

An Extension to the Tactical Planning Model for a Job Shop: Continuous-Time Control

We develop an extension to the tactical planning model (TPM) for a job shop by the third author. The TPM is a discrete-time model in which all transitions occur at the start of each time period. The time period must be defined appropriately in order for the model to be meaningful. Each period must be short enough so that a job is unlikely to travel through more than one station in one period. At the same time, the time period needs to be long enough to justify the assumptions of continuous workflow and Markovian job movements. We build an extension to the TPM that overcomes this restriction of period sizing by permitting production control over shorter time intervals. We achieve this by deriving a continuous-time linear control rule for a single station. We then determine the first two moments of the production level and queue length for the workstation.

Type-alpha DPLs

This paper introduces Denotational Proof Languages (DPLs). DPLs are languages for presenting, discovering, and checking formal proofs. In particular, in this paper we discus type-alpha DPLs---a simple class of DPLs for which termination is guaranteed and proof checking can be performed in time linear in the size of the proof. Type-alpha DPLs allow for lucid proof presentation and for efficient proof checking, but not for proof search. Type-omega DPLs allow for search as well as simple presentation and checking, but termination is no longer guaranteed and proof checking may diverge. We do not study type-omega DPLs here. We start by listing some common characteristics of DPLs. We then illustrate with a particularly simple example: a toy type-alpha DPL called PAR, for deducing parities. We present the abstract syntax of PAR, followed by two different kinds of formal semantics: evaluation and denotational. We then relate the two semantics and show how proof checking becomes tantamount to evaluation. We proceed to develop the proof theory of PAR, formulating and studying certain key notions such as observational equivalence that pervade all DPLs. We then present NDL, a type-alpha DPL for classical zero-order natural deduction. Our presentation of NDL mirrors that of PAR, showing how every basic concept that was introduced in PAR resurfaces in NDL. We present sample proofs of several well-known tautologies of propositional logic that demonstrate our thesis that DPL proofs are readable, writable, and concise. Next we contrast DPLs to typed logics based on the Curry-Howard isomorphism, and discuss the distinction between pure and augmented DPLs. Finally we consider the issue of implementing DPLs, presenting an implementation of PAR in SML and one in Athena, and end with some concluding remarks.

Structure and Interpretation of Computer Programs

"The Structure and Interpretation of Computer Programs" is the entry-level subject in Computer Science at the Massachusetts Institute of Technology. It is required of all students at MIT who major in Electrical Engineering or in Computer Science, as one fourth of the "common core curriculum," which also includes two subjects on circuits and linear systems and a subject on the design of digital systems. We have been involved in the development of this subject since 1978, and we have taught this material in its present form since the fall of 1980 to approximately 600 students each year. Most of these students have had little or no prior formal training in computation, although most have played with computers a bit and a few have had extensive programming or hardware design experience. Our design of this introductory Computer Science subject reflects two major concerns. First we want to establish the idea that a computer language is not just a way of getting a computer to perform operations, but rather that it is a novel formal medium for expressing ideas about methodology. Thus, programs must be written for people to read, and only incidentally for machines to execute. Secondly, we believe that the essential material to be addressed by a subject at this level, is not the syntax of particular programming language constructs, nor clever algorithms for computing particular functions of efficiently, not even the mathematical analysis of algorithms and the foundations of computing, but rather the techniques used to control the intellectual complexity of large software systems.