OR-Spektrum 1, 51-55 (1979) ORSpektrum 9 by Springer-Verlag 1979 Semi-Infinite Quadratic Programming U. Eckhardt Institut fiir Angewandte Mathematik, Universit/it Hamburg, Bundesstrage 55, D-2000 Hamburg 13 Received March 23, 1979 j Accepted June 1, 1979 Summary. A method is presented for minimizing a defi- nite quadratic function under an infinite number of linear inequality restrictions. Special features of the method are that it generates a sequence of feasible solutions and a sequence of basic solutions simultaneously and that it has very favourable properties concerning numerical sta- bility. Zusammenfassung. Eine Methode zur Minimierung einer positiv definiten quadratischen Funktion unter unend- lich vielen linearen Nebenbedingungen wird vorgestellt. Diese Methode hat die Eigenschaft, dafo eine Folge von zul~issigen L6sungen des Problems sowie eine Folge von Basisl6sungen gleichzeitig erzeugt wird. Auf diese Weise erh~lt man Einschliet~ungen des Zielfunktions- wertes der L6sung des Problems. Zudem weist die Methode hervorragende Stabilit/itseigenschaften auf. 1. Introduction The topic of this paper is to present a method for minimizing a positive definite quadratic form q(x) = 89 xTCx - cTx, (1) with positive definite symmetric (d, d)-matrix C and d-vector c, under the conditions aTx <~ bt, t e T, (2) a t e IN d, b t e AN for all t, and T is a finite or infinite index set. For a finite set T there exist efficfent methods for solving the problem stated. The method ofC. E. Lemke [11] is widely used for this purpose and the HarwelI Subroutine VE02A [7] is a useful code for many applica- tions. The method presented here is adapted to situations which have the following properties: The number d of variables is relatively small (some hundreds at most), - The number of inequalities is large, even an infinite number of restrictions is allowed, - The matrix C, although being positive definite, is poorly conditioned (i. e., the ratio of the largest to the smallest eigenvalue of C is very large), - A point 2- 0 e/R d fulfdling (2) is known. 2. Practical Applications The situation described so far is typical for certain important classes of practical applications. Consider the situation that a set of observed data Yl, ..., Ym has to be approximated by a linear combination of given func- tions ~q(t), . ...... ~(t) in the least squares sense: m n j=l xi.~i(ti)_ yj]2 _~ Minimum (3) and assume further that some conditions are imposed on the approximating function such as monotonicity, convexity, or some explicit bounds for the function and some of its derivatives are known. Such conditions can be expressed by an infinite number - or at least by a large number - of linear inequalities. It is well known that in many cases the matrix of the quadratic form belonging to (3) is ill-conditioned. Usually it is not difficult to find a function fulfdling the extra conditions if one has a rough idea of the solution of the problem. A similar situation is present when a system is modelled by an integral operator and one wants to 0171-6468/79/0001/0051/tg01.00