On the equivalence between some discrete and continuous optimization problems

The simplex algorithm for linear programming is based on the well-known equivalence between the problem of maximizing a linear functionf on a polyhedronP and the problem of maximizingf over the setV _P of all vertices ofP. The equivalence between these two problems is also exploited by some methods for maximizing a convex or quasi-convex function on a polyhedron.In this paper we determine some very general conditions under which the problem of maximizingf overP is equivalent, in some sense, to the problem of maximizingf overV _P . In particular, we show that these two problems are equivalent whenf is convex or quasi-convex on all the line segments contained inP and parallel to some edge ofP.In the case whereP is a box our results extend a well-known result of Rosenberg for 0–1 problems. Furthermore, whenP is a box or a simplex, we determine some classes of functions that can be maximized in polynomial time overP.This paper has been partially written while the author was visiting the Rutgers Center for Operations Research (RUTCOR). The support of the Air Force grants AFORS-89-0512 and AFORS-90-0008 is gratefully acknowledged.     

One-pot synthesis of polyfunctionalized cyclic urea derivatives

[reaction: see text] Unexpectedly, imidazolidin-2-ones are easily obtained from beta-keto esters upon treatment with tert-butyl nosyloxycarbamate and calcium oxide under mild conditions. The structure of the product was established by X-ray crystallography.     

On the image of a constrained extremum problem and some applications to the existence of a minimum

A new approach to constrained optimization, which has appeared recently under various forms and in several contexts, is presented in a general and unifying setting. This approach is then employed to establish some new conditions for the existence of the minimum of a constrained minimum problem.     

Discrete 2-convex functions

Mathematical Programming - In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form $$x \in \bigcup _{i=1}^m P_i$$ , where the...     

Facile and highly stereoselective one-pot synthesis of either (E)- or (Z)-nitro alkenes

Aliphatic aldehydes were reacted with nitro alkanes in the presence of catalytic amounts of piperidine over 4 A molecular sieves. Simply by changing reaction conditions (solvent and temperature) it is possible to control the stereochemical outcome of the reactions, obtaining pure (E)- and (Z)-nitro alkenes in high to excellent yields. The role of molecular sieves on the stereochemical control seems crucial in addition to that of piperidine, especially for the synthesis of the Z isomer.     

Solution-phase synthesis of 2-cyano and 2-amido aziridinyl peptides

Starting from a library of 2-l-α-amino acyl (E)-acrylonitriles, different short 2-cyano and 2-amido aziridinyl peptides, potential protease inhibitors, were obtained under parallel solution-phase conditions. The transformations include careful selection of conditions for aziridine deprotection and cyano group partial hydrolysis.     

Synthesis of optically active trifluoromethyl substituted diaziridines and oxaziridines

(E)-Trifluoromethyl imines were considered as ideal substrates to be transformed into the corresponding diaziridines by a direct amination reaction with nosyloxycarbamates. The diastereoselective induction was strongly controlled by the N-substituent. Similar results were obtained in the epoxidation reactions performed on the same substrates using m-CPBA as oxidant. Starting from enantiopure imines chiral diaziridines or oxaziridines were obtained with very high enantiopurity.