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.     

A general saddle point result for constrained optimization

A saddle point theory in terms of extended Lagrangian functions is presented for nonconvex programs. The results parallel those for convex programs conjoined with the usual Lagrangian formulation.Sponsored in part by the Office of Naval Research under Grant No. N00014-67A-0298-0019 (NR047-004).Sponsored in part by the Office of Naval Research under Grant No. N00014-67A-0321-0003 (NR047-096).     

The synthesis of optimal controls for linear,time-optimal problems with retarded controls

Optimization problems involving linear systems with retardations in the controls are studied in a systematic way. Some physical motivation for the problems is discussed. The topics covered are: controllability, existence and uniqueness of the optimal control, sufficient conditions, techniques of synthesis, and dynamic programming. A number of solved examples are presented.This research was supported in part by the National Aeronautics and Space Administration under Grant No. NGL-40-002-015, in part by the Air Force Office of Scientific Research under Grant No. AF-AFOSR-693-67B, and in part by the National Science Foundation under Grant No. GP-15132.     

Metric spaces of semicontinuous functions

This work was supported by the Russian Foundation of Fundamental Research under Grant No. 93-011-236.     

A density version of the Hales-Jewett theorem

Research supported by the National Science Foundation under Grant No. DMS86-05098.     

Weakly symmetric spaces and spherical varieties

Weakly symmetric homogeneous spaces were introduced by A. Selberg in 1956. We prove that, for a real reductive algebraic group, they can be characterized as the spaces of real points of affine spherical homogeneous varieties of the complexified group. As an application, under the same assumption on the transitive group, we show that weakly symmetric spaces are precisely the homogeneous Riemannian manifolds with commutative algebra of invariant differential operators.Supported by the Alexander von Humboldt Foundation and Russian Foundation for Basic Research, Grant No. 95-01-01263.Supported by the U. S. Civilian Research and Development Foundation, Award No. 206, Russian Foundation for Basic Research, Grant No. 98-01-00598, and the Alexander von Humboldt Foundation.     

Streams of a M/M/1 feedback queue in statistical equilibrium

Summary It is shown that, under statistical equilibrium, no stream in the M/M/1 feedback queue of Jackson [7], besides the exogenous input and the final output, is poissonian. The proof relies on martingale methods, especially the filtering formulas of [1].Work supported in part by Army Research Office Grant No. DAAG29-75-G-0189 and National Science Foundation Grant No. SOC76-80333, while the author was visiting the Dept. of Electrical Engineering and Computer Science at U.C. Berkeley     

Adaptive control of discounted Markov decision chains

In this paper, we consider discounted-reward finite-state Markov decision processes which depend on unknown parameters. An adaptive policy inspired by the nonstationary value iteration scheme of Federgruen and Schweitzer (Ref. 1) is proposed. This policy is briefly compared with the principle of estimation and control recently obtained by Schäl (Ref. 4).This research was supported in part by the Consejo Nacional de Ciencia y Tecnología under Grant No. PCCBBNA-005008, in part by a grant from the IBM Corporation, in part by the Air Force Office of Scientific Research under Grant No. AFOSR-79-0025, in part by the National Science Foundation under Grant No. ECS-0822033, and in part by the Joint Services Electronics Program under Contract No. F49620-77-C-0101.     

Asymptotic power comparison of the chi-square and likelihood ratio tests

Summary A Hoeffding-type power comparison is made between the likelihood ratio and chi-square tests for a simple hypothesis in a multinomial. The power comparison is based on the fact that under an alternative hypothesis the distribution of the test statistic can be approximated by a normal distribution. The theory of large deviations is used to match the significance levels. This research was partially supported by the National Science Foundation Grant No. GP-33697X2, U.S. Energy Research and Development Agency Grant 7064100, Environmental Protection Agency Grant R805379-01-0 and U.S. Public Health Service Grant USPHS ES01299-15, at the University of California-Berkeley.     

An efficient one—dimensional search procedure for barrier functions

In minimizing interior penalty functions, most of the computational time is spent on the one-dimensional search. This paper presents a method for performing this search on barrier functions which is significantly faster than current techniques. The method exploits the special structure of barrier functions. Comparative computational results are given for a set of six test problems.This research was partially supported by the National Aeronautics and Space Administration under Research Grant NSG 110-61 and by the Office of Naval Research under Grant No. N00014-67-A-0404-0010.