RABBIT: A Compiler for SCHEME

We have developed a compiler for the lexically-scoped dialect of LISP known as SCHEME. The compiler knows relatively little about specific data manipulation primitives such as arithmetic operators, but concentrates on general issues of environment and control. Rather than having specialized knowledge about a large variety of control and environment constructs, the compiler handles only a small basis set which reflects the semantics of lambda-calculus. All of the traditional imperative constructs, such as sequencing, assignment, looping, GOTO, as well as many standard LISP constructs such as AND, OR, and COND, are expressed in macros in terms of the applicative basis set. A small number of optimization techniques, coupled with the treatment of function calls as GOTO statements, serve to produce code as good as that produced by more traditional compilers. The macro approach enables speedy implementation of new constructs as desired without sacrificing efficiency in the generated code. A fair amount of analysis is devoted to determining whether environments may be stack-allocated or must be heap-allocated. Heap-allocated environments are necessary in general because SCHEME (unlike Algol 60 and Algol 68, for example) allows procedures with free lexically scoped variables to be returned as the values of other procedures; the Algol stack-allocation environment strategy does not suffice. The methods used here indicate that a heap-allocating generalization of the "display" technique leads to an efficient implementation of such "upward funargs". Moreover, compile-time optimization and analysis can eliminate many "funargs" entirely, and so far fewer environment structures need be allocated at run time than might be expected. A subset of SCHEME (rather than triples, for example) serves as the representation intermediate between the optimized SCHEME code and the final output code; code is expressed in this subset in the so-called continuation-passing style. As a subset of SCHEME, it enjoys the same theoretical properties; one could even apply the same optimizer used on the input code to the intermediate code. However, the subset is so chosen that all temporary quantities are made manifest as variables, and no control stack is needed to evaluate it. As a result, this apparently applicative representation admits an imperative interpretation which permits easy transcription to final imperative machine code. These qualities suggest that an applicative language like SCHEME is a better candidate for an UNCOL than the more imperative candidates proposed to date.

On the Behavior of the Homogeneous Self-Dual Model for Conic Convex Optimization

There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of ε-optimal solutions, and (ii) the maximum distance of ε-optimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stopping-rule theory for HSD-based interior-point methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the ε-optimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous self-dual model that might improve the resulting solution time in practice