Sulla convergenza di un algoritmo di direzioni ammissibili per problemi di ottimo a vincoli lineari con funzione oggetto convessa non differenziabile

Abstract

Se, essendof la funzione obiettivo del problema, {x ^k} e {f(x ^k)} sono le successioni delle approssimazioni rispettivamente di una soluzione ottimax ^* e dell' ottimof(x ^*) generate da un noto algoritmo di direzioni ammissibili a parametri antizigzag ε _k , mostriamo che per avere (a) lim _k→∞ f(x ^*)=f(x ^*) basta assumere lim _k→∞ ε _k =0. Inoltre, ove si assuma in più la stretta convessità dif, si ha anche (b) lim _k→∞ x ^k=x ^*. Da quest'ultima condizione deriviamo infine specifiche ipotesi, in ordine alla (b), per il caso particolare del problema di trasporto stocastico.

Summary

The aim of the present paper is to analyze, without differentiability of the objective functionf, the convergence of a known «feasible directions» algorithm for constrained optimization problems having the constraints linear [8], 6.5.2.

In these circumstances (i.e. iff is not differentiable) one must, almost in general, verify some preliminary conditions to obtain convergence [4]. Nevertheless, this work is not always easy to accomplish particularly in absence of differentiability.

Here, we establish that under the convexity assumption forf, the only condition lim _k→∞ ε _k =0, where the ε _k are the antizigzag parameters, suffices to obtain the convergence of the algorithm, i.e. lim _k→∞ f(x ^k)=opt., thex ^k being the approximate solutions to problem. The proof is obtained by application of the Th. 24.5, [6]. Successively, we consider the question if one has also the convergence of {x ^k} to optimal solution. By using now the Cor. 27.2.2, [6], we establish, for this purpose, that under an additional general qualification forf — precisely the strict convexity — the convergence of {x ^k} is also stated. Finally, we examine the above property for the stochastic transportation problem [1] for which we indicate special conditions in order to verify the latter convergence property.