An algorithm of successive minimization in convex programming

A gênerai exchange algorithm is gwen for the mimmizatwn ofa convex function with equahty and inequahty constraintsAt is a generahzation of the Cheney-Goldstein algorithm, but following an idea gwen by Topfer, afimte séquence ofsub-problems the dimension ofwhich is decreasing, is considered at each itération Gwen a positive number 8, under very gênerai conditions, it is proved that the method, after afimte number of itérations, leads to an "e-solutwn" In 1959, Cheney and Goldstein [6] (see also Goldstein [7]) proposed an algorithm for solving the problem of minimizing a convex function: « \ ( £ u / * j(x)=max( iJbl(t)xl — c\ teS \ i = l under the constramts: £ bt{t)xt^c(t) for ail tel/, i = i where S and U are two disjoint compact sets and blt. . ., bn, c are continuous real functions defîned on S u U. At each itération v of this algorithm, a polyhedral approximation of the problem is associated to a suitable subset A consisting of n + 1 points of S u U. Using the exchange theorem (Stiefel [11, 12, 13]; see also [8, 9]) a new element t e S u U is introduced: A x =(A\t0) u t\ We propose hère a new algorithm which is an extension of the CheneyGoldstein algorithm for solving the same problem but under much weaker assumptions: the sets S and U are arbitrary and the mappings blt. . ., bn)c are (*) Rtui décembre 1977 () Mathématiques appliquées I M A G Université Scientifique et Médicale de Grenoble () U E R de Sciences, Université de Saiat-Étienne, Saint-Étienne R A I R O Analyse numénque/Numencal Analysis, vol 12, n° 4, 1978 378 P. J. LAURENT, C. CARASSO only supposed to be bounded. Moreover, no Haar condition is introduced. At each itération, we consider a séquence of nested minimization problems. The algorithm is based on an extension of the exchange theorem in which the exchanged quantities are not just a single point {see [3, 4]). The idea of the algorithm is similar to the recursive method introduced by Töpfer [14], [15] {see also [3]) for problems of Tchebycheff best approximation. In the case of a best approximation problem the algorithm becomes an extension of the Rémès aigorithm {see [5]). For other applications, see [2]. 1. PROBLEM AND ASSUMPTIONS We dénote by E the n-dimensional Euclidean space and by < x, x' > the usual inner-product of x and x' in E. 1 .1 . The minimization problem We dénote by L a finite set with l éléments (/ < n) and by S and U two arbitrary sets. Suppose that L, S and U have no common point and let T = S u U. Let b and c be two bounded mappings from L u Tinto E and R respectively (i. e., b{T) and c{T) are bounded). We define the functionals ƒ and g by: / (x)=Sup«x,6( t )>-c( t ) ) . teS