Accurate values for the nonrelativistic energies of the lowest singlet and triplet S-states of the4He-Isotop

Accurate lower and upper bounds for the nonrelativistic lowest energies¹ E ₀ and³ E ₀ of the singlet and triplet-system of the⁴He-Isotop are calculated with the linearized method of variance minimization. The same was done for¹ E ₁ the energy of the first excitedS-state 2¹ S. The results especially for¹ E ₀ and³ E ₀ in a.u. are −2.9033076997₅ ≤¹ E ₀ ≤ −2.9033076921₈ −2.1749324263₇ ≤³ E ₀ ≤ −2.1749324245₉ i.e. the values are determined with an absolute error smaller than 0.00167 cm⁻¹ for¹ E ₀ and 0.00039 cm⁻¹ for³ E ₀.     

A nonadiabatic lower bound calculation of H 2 + and D 2 +

Accurate nonadiabatic lower and upper bounds for groundstate energies of H ₂ ⁺ and D ₂ ⁺ are calculated with the linearized method of variance minimization. The results in a.u. are –0.59713906₃<E ₀(H ₂ ⁺ )<–0.59713899₄ –0.59878877₅<E ₀(D ₂ ⁺ )<–0.59877873₈ i.e. the values are determined with an absolute error smaller than 0.02 cm^–1 for H ₂ ⁺ and 0.01 cm^–1 for D ₂ ⁺ .     

An open optimization problem in statistical quality control

We ask the experts in global optimization if there is an efficient solution to an optimization problem in acceptance sampling: Here, one often has incomplete prior information about the quality of incoming lots. Given a cost model, a decision rule for the inspection of a lot may then be designed that minimizes the maximum loss compatible with the available information. The resulting minimax problem is sometimes hard to solve, as the loss functions may have several local maxima which vary in an unpredictable way with the parameters of the decision rule.     

On affine scaling algorithms for nonconvex quadratic programming

We investigate the use of interior algorithms, especially the affine-scaling algorithm, to solve nonconvex — indefinite or negative definite — quadratic programming (QP) problems. Although the nonconvex QP with a polytope constraint is a hard problem, we show that the problem with an ellipsoidal constraint is easy. When the hard QP is solved by successively solving the easy QP, the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions.Research supported in part by NSF Grant DDM-8922636 and the College Summer Grant, College of Business Administration, The University of Iowa.     

ε-optimality criteria for convex programming problems via exact penalty functions

In this paper, we present-optimality criteria for convex programming problems associated with exact penalty functions. Several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved. We assume the solvability of neither the convex problem nor the dual problem. To derive our criteria, we estimate the size of the penalty parameter in terms of an-solution for the dual problem.     

Convexlike and concavelike conditions in alternative,minimax, and minimization theorems

Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.In this paper, we study various known convexlike conditions for vector-valued functions on a set and investigate convexlike and concavelike conditions for real-valued functions on a product setC×D, where we are mainly interested in the relationships between these conditions. At the end of the paper, we point out several conclusions from our results for the above-mentioned mathematical fields.The author is indebted to Dr. R. Reemtsen and Dr. V. Jeyakumar for their helpful comments during the preparations of this paper.     

Numerical solution of minimax problems of optimal control,part 1

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

An investigation of glasses for capillary chromatography

An investigation was conducted of various glasses, other than soda lime or borosilicate, for use in glass capillary gas chromatography. The work has uncovered some unique chromatographic qualities in the use of potash soda lead and fused silica glasses as materials for making glass capillary columns. The fused silica proved to be an ideal material for capillary column construction, being inherently more inert than glass containing metal oxides. It has been shown that through the use of thin wall capillary tubing of high flexibility many of the mechanical problems associated with glass capillary columns, such as fragility and column straightening, can be avoided.     

Stability theorems for infinitely constrained mathematical programs

The primary concern of this paper is to investigate stability conditions for the mathematical program: findx E ⁿ that maximizesf(x):g ^j(x)0 for somej J, wheref is a real scalarvalued function and eachg is a real vector-valued function of possibly infinite dimension. It should be noted that we allow, possibly infinitely many, disjunctive forms. In an earlier work, Evans and Gould established stability theorems wheng is a continuous finite-dimensional real-vector function andJ=1. It is pointed out that the results of this paper reduce to the Evans-Gould results under their assumptions. Furthermore, since we use a slightly more general definition of lower and upper semicontinuous point-to-set mappings, we can dispense with the continuity ofg (except in a few instances where it is implied by convexity assumptions).     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.