Richard V. Helgason

A polynomially bounded algorithm for a singly constrained quadratic program

This paper presents a characterization of the solutions of a singly constrained quadratic program. This characterization is then used in the development of a polynomially bounded algorithm for this class of problems.

Improved efficiency of the Frank-Wolfe algorithm for convex network programs

We discuss methods for speeding up convergence of the Frank-Wolfe algorithm for solving nonlinear convex programs. Models involving hydraulic networks, road networks and factory-warehouse networks are described. The PARTAN technique and heuristic variations of the Frank-Wolfe algorithm are described which serve to significantly improve the convergence rate with no significant increase in memory requirements. Computational results for large-scale models are reported.

A matrix method for ordinary differential eigenvalue problems

This paper describes a method for solving ordinary differential eigenvalue problems of the form N(u) + λM(u) = 0, where N and M are linear differential operators and u(x) is a scaler variable. The boundary conditions are independent of λ. The problem is transformed into a matrix problem ∥ A + λB ∥ = 0. This is reduced to the standard eigenvalue problem ∥A + λI ∥ = 0 which is then solved by the Q - R algorithm. The computer program is organized so that it can solve a wide range of problems with minimal effort on the users part. The method is applied to a hydrodynamic stability problem and compared to the shooting method.

A new procedure for identifying the frame of the convex hull of a finite collection of points in multidimensional space

Abstract Consider a set, A of n points in m-dimensional space. The convex hull of these points is a polytope, P , in R m . The frame, F , of these points in the set of extreme points of teh polytope P with F ⊆ A . The problem of identifying the frame plays a central role in optimization theory (redundancy in linear programming and stochastic programming), economics (data envelopment analysis), computational geometry (facial decomposition of polytopes) and statistics (Gastwirth estimators). The standard approach for finding the elements of F consists of solving linear programs with m rows and n − 1 columns; one for each element of A , differing only in the right-hand side vectors. Although enhancements to reduce the total number of linear programs which must ultimately be solved as well as to reduce the number of columns in the technology matrix are known, the utility of this approach is severely limited by its laboriousness and computational demands. We introduce a new procedure also based on solving linear programs but with an important and distinguishing difference. The linear programs begin small and grow larger, but never have more columns than the number of extreme points of P . Experimental results indicate that the time to find the frame using the new procedure is between about one-third and two-thirds that of an enhanced implementation of the established method currently in use.

Primal simplex network codes: State-of-the-art implementation technology

In recent years there have been several extremely successful specialization of the primal simplex method for solving network flow problems. Much of this success is due to the development of highly efficient computational techniques for implementing the primal simplex algorithm. We view these efficient techniques as a new body of knowledge which we call implementation technology. This exposition presents the state-of-the-art of implementation technology.

A generalization of Polyak's convergence result for subgradient optimization

This paper generalizes a practical convergence result first presented by Polyak. This new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach.

The one-to-one shortest-path problem: an empirical analysis with the two-tree Dijkstra algorithm

Four new shortest-path algorithms, two sequential and two parallel, for the source-to-sink shortest-path problem are presented and empirically compared with five algorithms previously discussed in the literature. The new algorithm, S22, combines the highly effective data structure of the S2 algorithm of Dial et al., with the idea of simultaneously building shortest-path trees from both source and sink nodes, and was found to be the fastest sequential shortest-path algorithm. The new parallel algorithm, PS22, is based on S22 and is the best of the parallel algorithms. We also present results for three new S22-type shortest-path heuristics. These heuristics find very good (often optimal) paths much faster than the best shortest-path algorithm.

Cruise Missile Mission Planning: A Heuristic Algorithm for Automatic Path Generation

This manuscript presents a heuristic algorithm based on geometric concepts for the problem of finding a path composed of line segments from a given origin to a given destination in the presence of polygonal obstacles. The basic idea involves constructing circumscribing triangles around the obstacles to be avoided. Our heuristic algorithm considers paths composed primarily of line segments corresponding to partial edges of these circumscribing triangles, and uses a simple branch-and-bound procedure to find a relatively short path of this type. This work was motivated by the military planning problem of developing mission plans for cruise missiles, but is applicable to any two-dimensional path-planning problem involving obstacles.

Technical Note-Computational Comparison among Three Multicommodity Network Flow Algorithms

This note presents our computational experience using specialized techniques for solving multicommodity network flow problems. The algorithms investigated include a price-directive decomposition procedure, a resource-directive decomposition procedure using subgradient optimization, and a primal partitioning procedure.

An Algorithm for Identifying the Frame of a Pointed Finite Conical Hull

We present an algorithm for identifying the extreme rays of the conical hull of a finite set of vectors whose generated cone is pointed. This problem appears in diverse areas including sto-chastic programming, computational geometry, and non-parametric efficiency measurement. The standard approach consists of solving a linear program for every element of the set of vectors. The new algorithm differs in that it solves fewer and substantially smaller LPs. Extensive computational testing vali-dates the algorithm and demonstrates that for a wide range of problems it is computationally superior to the standard approach.