This paper deals with the existence of solutions and the conditions for the strong convergence of minimizing sequences towards the set of solutions of the quadratic function minimization problem on the intersection of two ellipsoids in Hilbert space. | Yugoslav Journal of Operations Research 12 (2002), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS* M. JA]IMOVI], I. KRNI] Department of Mathematics University of Montenegro, Podgorica Abstract: This paper deals with the existence of solutions and the conditions for the strong convergence of minimizing sequences towards the set of solutions of the quadratic function minimization problem on the intersection of two ellipsoids in Hilbert space. Keywords: Quadratic functional, minimization, wellposedness. 1. INTRODUCTION Suppose that H , F , G1 and G2 are Hilbert spaces; A : H → F , H → G1 and C : H → G2 - bounded linear operators; f ∈ F a fixed element; r1 > 0 and r2 > 0 are given real numbers; U1 and U2 ellipsoids in the space H defined by operators B and C: U1 = {u ∈ H : || Bu || ≤ r1}, U 2 = {u ∈ H : || Cu || ≤ r2} . This paper deals with the extremal problem: J (u) = || Au − f ||2 → inf, u ∈ U = U1 ∩ U 2 . (1) We study the existence of solutions and the wellposedness of the problem in the Tikhonov sense. As an example of the problem of this type, we can quote the problem of minimization of the function * This research is supported by the Yugoslav Ministry of Sciences and Ecology, Grant OSI263. 50 M. Ja}imovi}, I. Krni} / On Wellposedness Quadratic Function Minimization Problem J (u) = || x (T , u) − z ||2Rn , where z ∈ Rn and x(t , u) is a solution of the system of differential equation x′( t ) = B(t ) x( t ) + D( t )u(t ), t ∈ (0, T ), x(0) = 0 ∈ Rn , T T || u(t ) ||Lr := ∫ | u(t ) |R2 dt ≤ r1 , || x(t, u) || = ∫ | x(t, u(t )) |2 dt ≤ r2 2 0 n 0 with given matrices B(⋅) = (bij (⋅))n×n and D(⋅) = ( dij (⋅)) n×r . These conditions guarantee the existence of the solution x(t , u) ∈ Hn1 [0, T ] of the previous system for each u ∈ Lr2 [0, T ] . The same problem with different set of constraints U , was considered in [1], [2] and [3]. In [3], the set of constraints U was a ball. In [2]
