Peter A. Beling

Using fast matrix multiplication to find basic solutions

Abstract We consider the problem of finding a basic solution to a system of linear constraints (in standard form) given a non-basic solution to the system. We show that the known arithmetic complexity bounds for this problem admit considerable improvement. Our technique, which is similar in spirit to that used by Vaidya to find the best complexity bounds for linear programming, is based on reducing much of the computation involved to matrix multiplication. Consequently, our complexity bounds in their most general form are a function of the complexity of matrix multiplication. Using the best known algorithm for matrix multiplication, we achieve a running time of O(m1.594 n) arithmetic operations for an m × n problem in standard form. Previously, the best bound was O(m2 n) arithmetic operations.

Polynomial algorithms for linear programming over the algebraic numbers

We derive a bound on the computational complexity of linear programs whose coefficients are real algebraic numbers. Key to this result is a notion of problem size that is analogous in function to the binary size of a rational-number problem. We also view the coefficients of a linear program as members of a finite algebraic extension of the rational numbers. The degree of this extension is an upper bound on the degree of any algebraic number that can occur during the course of the algorithm, and in this sense can be viewed as a supplementary measure of problem dimension. Working under an arithmetic model of computation, and making use of a tool for obtaining upper and lower bounds on polynomial functions of algebraic numbers, we derive an algorithm based on the ellipsoid method that runs in time bounded by a polynomial in the dimension, degree, and size of the linear program. Similar results hold under a rational number model of computation, given a suitable binary encoding of the problem input.

Hybrid evolutionary algorithms for a multiobjective financial problem

We examine the use of numeric score functions that allow one to rank order a universe of stocks based on profitability. We use a genetic algorithm to evolve sets of implicit-positive binary classification rules. Using each rule set, we induce a scoring model by weighting the individual terms in a representation of the rule in terms of binary variables. We report on the empirical performance of the proposed family of scoring algorithms on several large historical stock data sets. We also compare our approach with a polynomial network technique.

Polynomial algorithms for LP over a subring of the algebraic integers with applications to LP with circulant matrices

We show that a modified variant of the interior point method can solve linear programs (LPs) whose coefficients are real numbers from a subring of the algebraic integers. By defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring, we prove the modified algorithm runs in time polynomial in the encoding size of the input coefficients, the dimension of the problem, and the order of the subring. We then extend the Tardos scheme to our case, obtaining a running time which is independent of the objective and right-hand side data. As a consequence of these results, we are able to show that LPs with real circulant coefficient matrices can be solved in strongly polynomial time. Finally, we show how the algorithm can be applied to LPs whose coefficients belong to the extension of the integers by a fixed set of square roots.

Induction of rule-based scoring functions

We consider the problem that many portfolio managers face of selecting, on a regular basis, stocks for investment and recommendation to clients. In typical solution strategies, binary rules are developed to classify stocks as strong or weak performers based on technical indicators. Strategies based on binary classification rules have been shown to be very effective at maximizing the total profitability of the stocks that are selected. Having a fixed number of target stocks is important for portfolio maintenance and for client choice, however, and so the selection problem also engenders the additional constraint of limiting the total number of stocks selected. Binary classification rule strategies do not address this constraint. In this paper we investigate the use of scoring functions, which have the advantage of allowing one to rank order the population based on profitability, as an alternative to binary classification rules. A key feature of this work is that we develop the scoring functions by incorporating binary classification rules. In particular, we induce the score model by assigning optimal weights to sets of implicit positive binary classification rules. We use a genetic algorithm with supervised batch learning to evolve classification rules. Fitness of a rule set is evaluated based on the success of the scoring function that it induces. We report on the relative empirical performance of this method on several large historical data sets.

A heuristic for the topological design of two-tiered networks

A basic hierarchical network design problem is that of selecting access area and backbone designs that minimize the sum total cost of the network. Because of its computational difficulty, network designers typically segment the hierarchical design problem, first solving the access area problem to obtain a set of backbone nodes and then solving the backbone design problem on the subgraph induced by these nodes. Each individual problem is far easier to solve than the complete network design problem, but in general the procedure gives a poor overall solution. In this paper, we describe a technique for integrating the access area and backbone design problems into a single mathematical program. The fundamental idea of this approach is to incorporate backbone network cost information into the access area problem without increasing the computational difficulty of the resulting problem significantly beyond that of the access area problem.

A fast symmetric penalty algorithm for the linear complementarity problem

In an earlier paper, the authors presented a new parameterization algorithm for the the linear complementarity problem. The trajectory associated with this parameterization is distinguished by a naturally defined starting point and by a piecewise characterization as a fractional polynomial function of a single parameter. In order to follow this trajectory, however one needs to isolate roots of polynomials-a computationally expensive operation. In this paper, we present an algorithm in which we parameterize one dimension at a time. This results in polynomials of degree one, which allows trivial root isolation. The algorithm stays unaffected in other key respects. For example, the average number of pieces in the trajectory is still O(n/sup 2/), where n is the dimension of the problem space. This implies that our algorithm is competitive, in an average sense, to Lemkes method.

Tensored nearest-neighbor classifiers

Among the various parametric and nonparametric techniques available for classification, k-nearest neighbor is a well known nonparametric method. This paper describes some experiments with modification of a k-nearest neighbor method to use the neighbor information as attributes to a second k-nearest neighbor classifier.