A simple heuristic approach to simplex efficiency

Consider the standard linear program:

$\text{Minimize}\text{cx}\text{subject to}\text{Ax=b, x?0}$

where A is an m × n matrix. The simplex algorithm solves this linear program by moving from an extreme point of the feasibility region to a better (in terms of the objective function cx) extreme point (via the pivot operation) until the optimal is reached. In order to obtain a feel for the number of necessary iterations, we consider a simple probabilistic (Markov chain) model as to how the algorithm moves along the extreme points. At first we suppose that if at any time the algorithm is at the jth best extreme point then after the next pivot the resulting extreme point is equally likely to be any of the j?1 best. Under this assumption, we show that the time to get from the Nth best to the best extreme point has approximately, for large N, a Poisson distribution with mean equal to the algorithm (base e) of N. We also consider a more general probabilistic model in which we drop the uniformity assumption and suppose that when at the jth best the next one is chosen probabilistically according to weights w_i, i = 1, …, j?1.     

Brechungsverm?gen

Ohne Zusammenfassung     

Preparation,Reactions, and Stereochemistry of 4-Tert-Butyl-1-Chlorophosphorinane 1-Sulfide and Derivatives

Abstract

Nucleophilic substitution at tetracoordinated phosphorus centers has been extensively investigated.¹ Transesterification of acyclic phosphinates proceeds with inversion of configuration,² whereas four-membered phosphinates react with retention.³ In previous work⁴ we suggested that transesterification in six-membered phosphinates occurs via an S_N2 mechanism. The title compound was synthesized as a model structure to study the stereochemical and mechanistic aspects of nucleophlic substitution at phosphorus in a six-membered ring. A mixture of diastereomers was produced and separated by column chromatography. Both the mixture and separate isomers were used for an investigation of substitution with methanol, methoxide ions, and phenoxide ions; the reactions proceeded with inversion. Transesterification reactions of the resulting thiophosphinates were studied and found to proceed with inversion. Reactions of these with phenyllithium, benzyllithium, lithium phenylacetylide, and the corresponding organomagnesium halides were also carried out. The stereochemical assignments for the title compound (cis and trans) and two derivatives have been firmly anchored by X-ray studies;⁵ assignments for others are tentative and based on spectroscopic measurements, including ¹H, ¹³C, and ³¹P NMR data.