Math. Program. 85: 193–206 (1999) / DOI 10.1007/s10107980031a  Springer-Verlag 1999 N.N. Tam · N.D. Yen Continuity properties of the Karush-Kuhn-Tucker point set in quadratic programming problems Received March 18, 1996 / Revised version received August 8, 1997 Published online November 24, 1998 Abstract. We obtain necessary and sufficient conditions for the set of the Karush-Kuhn-Tucker points in a canonical quadratic programming problem to be upper semicontinuous or lower semicontinuous with respect to the problem parameters. Key words. quadratic programming problem – Karush-Kuhn-Tucker point set – upper semicontinuity – lower semicontinuity – R 0 -matrix 1. Introduction A quadratic programming (QP, for brevity) problem is the one of minimizing (or maxi- mizing) a quadratic function over a polyhedral set. Studying various stability aspects of quadratic programs is an interesting topic. Although the general stability theory in non- linear mathematical programming is applicable to convex and nonconvex QP problems, the specific structure of the latter allows one to have more complete results. Several authors have made efforts in studying stability properties of the QP problems. Daniel [7] established some basic facts about the solution stability of a QP problem whose objective function is a positive definite quadratic form. Guddat [12] studied continuity properties of the solution set of a convex QP problem. Robinson [21, Theorem 2] obtain- ed a fundamental result on the stable behavior of the solution set of a monotone affine generalized equation (an affine variational inequality in the terminology of [11]), which yields a fact on the Lipschitz continuity of the solution set of a convex QP problem. Best and Chakravarti [3] obtained some results on the continuity and differentiability of the optimal value function in a perturbed convex QP problem. By using the lin- ear complementarity theory, in [6, Chapter 7] Cottle, Pang and Stone studied in detail the stability of convex QP problems. Recently, Best and Ding [4] proved a result on the continuity of the optimal value function in a convex QP problem. Auslender and Coutat [2] established some results on stability and differential stability of generalized linear-quadratic programs, which include convex QP problems as a special case. Some attempts have been made to study the stability of nonconvex QP problems: Nhan [17] gave a sufficient condition for the upper semicontinuity of the solution set in a general QP problem, Tam [25] developed furthermore the result of Nhan and obtained a com- plete characterization for the lower semicontinuity of the solution set in a general QP problem. In this paper, which is a revised version of [26], we will discuss the upper N.N. Tam: Department of Mathematics, Hanoi Pedagogical Institute 2, Melinh, Vinhphuc, Vietnam N.D. Yen: Institute of Mathematics, P.O. Box 631 Bo Ho, Hanoi, Vietnam