Least-index resolution of degeneracy in quadratic programming

In this study, we combine least-index pivot selection rules with Keller's algorithm for quadratic programming to obtain a finite method for processing degenerate problems.Research and reproduction of this report were partially supported by National Science Foundation Grant MCS76-81259; and the Office of Naval Research Contract N00014-75-C-0267.     

A generalization of the Frank—Wolfe theorem

The Frank—Wolfe theorem states that a quadratic function, bounded below on a nonempty polyhedral convex set, attains its infimum there. This paper gives sufficient conditions under which a function either attains its infimum on a nonempty polyhedral convex set or is unbounded below on some halfline of that set. Quadratic functions are shown to satisfy these sufficient conditions.Research and reproduction of this report were partially supported by the National Science Foundation Grant MCS76-81259; and the Office of Naval Research Contract N00014-75-C-0267.     

A note on a sufficient-decrease criterion for a non-derivative step-length procedure

A step-length algorithm is an essential part of many descent methods for unconstrained and constrained optimization. In this note we present a criterion that defines an acceptable step length when only function values are available at trial step lengths.This research was supported by the U.S. Department of Energy Contract DE-AC03-76SF00326, PA No. DE-AT03-76ER72018; National Science Foundation Grants MCS-7926009 and ECS-8012974; the Office of Naval Research Contract N00014-75-C-0267; and the U.S. Army Research Office Contract DAAG29-79-C-0110.     

Distributed computing viewed as a team problem with constraints

The optimal distribution of the workload in a system of interconnected computer units is considered. Formulated as a team decision problem with a singular cost criterion and with equality and inequality constraints, it is shown that the problem admits always a unique piecewise linear strategy which is globally optimal. Some interesting particular cases are studied.The research reported in this paper was made possible through support from the Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-C-0648 and Contract No. N00014-77-C-0531 and by the National Science Foundation, Grant No. ENG-76-11824.     

Solving singularly constrained generalized network problems

The singularly constrained generalized network problem represents a large class of capacitated linear programming (LP) problems. This class includes any LP problem whose coefficient matrix, ignoring single upper bound constraints, containsm + 1 rows which may be ordered such that each column has at most two non-zero entries in the firstm rows. The paper describes efficient procedures for solving such problems and presents computational results which indicate that, on large problems, these procedures are at least twenty-five times more efficient than the state of the art LP systemapex-iii.This research was partly supported by ONR Contract N00014-76-C-0383 with Decision Analysis and Research Institute and by Project NR047-021, ONR Contracts N00014-75-C-0616 and N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Solving geometric programs using GRG: Results and comparisons

This paper describes the performance of a general-purpose GRG code for nonlinear programming in solving geometric programs. The main conclusions drawn from the experiments reported are: (i) GRG competes well with special-purpose geometric programming codes in solving geometric programs; and (ii) standard time, as defined by Colville, is an inadequate means of compensating for different computing environments while comparing optimization algorithms.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865, the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18, and the National Science Foundation, Grant No. DCR75-04544 at Stanford University; and by the Office of Naval Research under Contract No. N00014-75-C-0240, and the National Science Foundation, Grant No. SOC74-23808, at Case Western Reserve University.     

The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.This paper is an extension of part of the author's dissertation in the Department of Operations Research, Stanford University, Stanford, California. The research was supported at Stanford by the Department of Energy under Contract DE-AC03-76SF00326, PA#DE-AT03-76ER72018, Office of Naval Research under Contract N00014-75-C-0267 and the National Science Foundation under Grants MCS76-81259, MCS-7926009 and ECS-8012974 (formerly ENG77-06761).     

On sending one bit of information over an AGWN channel with noiseless feedback

We study the problem described by the title under the real-time constraint that no block coding is allowed. Three separate schemes, including the well known center-of-gravity scheme, and their performance are compared.The research reported in this paper was made possible through support from the Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-C-0648 and Contract No. N00014-77-C-0531, and by the National Science Foundation, Grant No. ENG-76-11824.The formulation of this problem took place when the first author was SRC Senior Fellow at Imperial College during the summer of 1977. He appreciates discussions with M. Davis and D. Mayne of Imperial College.     

Optimal approximation of sparse hessians and its equivalence to a graph coloring problem

We consider the problem of approximating the Hessian matrix of a smooth non-linear function using a minimum number of gradient evaluations, particularly in the case that the Hessian has a known, fixed sparsity pattern. We study the class of Direct Methods for this problem, and propose two new ways of classifying Direct Methods. Examples are given that show the relationships among optimal methods from each class. The problem of finding a non-overlapping direct cover is shown to be equivalent to a generalized graph coloring problem—the distance-2 graph coloring problem. A theorem is proved showing that the general distance-k graph coloring problem is NP-Complete for all fixedk≥2, and hence that the optimal non-overlapping direct cover problem is also NP-Complete. Some worst-case bounds on the performance of a simple coloring heuristic are given. An appendix proves a well-known folklore result, which gives lower bounds on the number of gradient evaluations needed in any possible approximation method. This research was partially supported by the Department of Energy Contract AM03-76SF00326. PA#DE-AT03-76ER72018; Army Research Office Contract DAA29-79-C-0110; Office of Naval Research Contract N00014-74-C-0267; National Science Foundation Grants MCS76-81259, MCS-79260099 and ECS-8012974.     

Low order polynomial bounds on the expected performance of local improvement algorithms

We present a general abstract model of local improvement, applicable to such diverse cases as principal pivoting methods for the linear complementarity problem and hill climbing in artificial intelligence. The model accurately predicts the behavior of the algorithms, and allows for a variety of probabilistic assumptions that permit degeneracy. Simulation indicates an approximately linear average number of iterations under a variety of probability assumptions. We derive theoretical bounds of 2en logn and en ² for different distributions, respectively, as well as polynomial bounds for a broad class of probability distributions. We conclude with a discussion of the applications of the model to LCP and linear programming.The author was supported by the New Faculty Research Development Program of the Georgia Institute of Technology. This work is based on the author's Ph.D. thesis, performed under George Dantzig at Stanford 1978–81, at the Systems Optimization Laboratory. While at Stanford, research was supported in part by Department of Energy Contract AM03-76SF00326, PA #DE-AT03-76ER72018; Office of Naval Research Contract N00014-75-C-0267; National Science Foundation Grants MCS76-81259, MCS-7926009 and ECS-8012974; and Army Research Office Contract DAA29-79-C-0110. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Large-scale linearly constrained optimization

An algorithm for solving large-scale nonlinear programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.This research was supported by the U.S. Office of Naval Research (Contract N00014-75-C-0267), the National Science Foundation (Grants MCS71-03341 A04, DCR75-04544), the U.S. Energy Research and Development Administration (Contract E(04-3)-326 PA #18), the Victoria University of Wellington, New Zealand, and the Department of Scientific and Industrial Research Wellington, New Zealand.     

Optimal design of pitched laminated wood beams

The optimal design of a pitched laminated wood beam is considered. An engineering formulation is given in which the volume of the beam is minimized. The problem is then reformulated and solved as a generalized geometric (signomial) program. Sample designs are presented.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865; by the US Energy Research and Development Administration Contract No. E(04-3)-326 PA-18; and by the National Science Foundation, Grant No. DCR75-04544 at Stanford University. This work was carried out during the first author's stay at the Management Science Division of the University of British Columbia and the Systems Optimization Laboratory of Stanford University. The authors are indebted to Mr. S. Liu and Mrs. M. Ratner for their assistance in performing the computations.     

Brownian approximations to first passage probabilities

Summary By direct probabilistic argument one term of an Edgeworth type asymptotic expansion is obtained for certain first passage distributions for random walks. These results provide partial justification for and extensions of approximations suggested earlier as a heuristic consequence of Laplace transform calculations.Research supported by ONR Contract N00014-77-C-0306, NSF Grant MCS77-16974, and by the Humboldt Stiftung     

Error estimates for the finite element solution of variational inequalities

Summary We study the mixed finite element approximation of variational inequalities, taking as model problems the so called obstacle problem and unilateral problem. Optimal error bounds are obtained in both cases.Supported in part by National Science Foundation grant MCS 75-09457, and by Office of Naval Research grant N00014-76-C-0369     

The behavior of a multiplicity independent root-finding scheme in the presence of error

King and Kioustelidis independently proposed a derivative free scheme that permits root finding methods, such as the secant method to preserve high order convergence when the root in question is multiple. In this note it is shown that the scheme can fail to achieve the maximum accuracy that is attainable at a fixed precision of computation.This work was supported in part by the Office of Naval Research under contract No. N00014-76-C-0391.     

A note on a theorem of Yamamuro

Conditions are presented which are necessary for the existence of a regular fixed point of aC ¹ map.Work supported in part by NSF Grant No. MCS 77-15509.Work supported in part by ONR Grant No. N0014-75-C-0495 and NSF Grant No. Eng. 76-81058.     

A unification of two classes ofQ-matrices

This note presents a class ofQ-matrices which includes Saigal's classN ofQ-matrices with negative principal minors and the classE of strictly semi-monotoneQ-matrices.Research supported by the Office of Naval Research under Contract N00014-75-C-0621 NR 047-048.     

Optimal radius of convergence of interpolatory iterations for operator equations

The convergence of the class of direct interpolatory iterationsI _n for a simple zero of a non-linear operatorF in a Banach space of finite or infinite dimension is studied.A general convergence result is established and used to show that ifF is entire the radius of convergence goes to infinity withn while ifF is analytic in a ball of radiusR the radius of convergence increases to at leastR/2 withn.The research was supported in part by the National Science Foundation under Grant MCS 75-222-55 and the office of Naval Research under Contract N00014-76-C-0370, NR 044-422.     

On the perturbation analysis of discrete-event dynamic systems

The paper describes a new approach to the analysis and optimization of discrete-event dynamic systems, such as queueing networks.Dedicated to G. LeitmannThe work reported in this paper was supported in part by ONR Contracts Nos. N00014-75-C-0648 and N00014-79-C-0776 and in part by NSF Grant No. NSF-ECS-82-13680.     

A weighted gram-schmidt method for convex quadratic programming

Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number ofgeneral constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases. This research was supported by the U.S. Department of Energy Contract DE-AC03-76SF00326, PA No. DE-AT03-76ER72018; National Science Foundation Grants MCS-7926009 and ECS-8012974; the Office of Naval Research Contract N00014-75-C-0267; and the U.S. Army Research Office Contract DAAG29-79-C-0110. The work of Nicholas Gould was supported by the Science and Engineering Research Council of Great Britain.