Optimization Methods and Software Vol. 22, No. 3, June 2007, 453–467 Minimizing quadratic functions with separable quadratic constraints R. KU ˇ CERA* VŠB-Technical University, Ostrava, Czech Republic (Received 31 October 2004; in final form 1 February 2006) This article deals with minimizing quadratic functions with a special form of quadratic constraints that arise in 3D contact problems of linear elasticity with isotropic friction [Haslinger, J., Kuˇ cera, R. and Dostál, Z., 2004, An algorithm for the numerical realization of 3D contact problems with Coulomb friction. Journal of Computational and Applied Mathematics, 164/165, 387–408.]. The proposed algorithm combines the active set method with the conjugate gradient method. Its general scheme is similar to the algorithms of Polyak’s type that solve the quadratic programming problems with simple bounds. As the algorithm does not terminate in a finite number of steps, the convergence is proved. The implementation uses an adaptive precision control of the conjugate gradient loops. Numerical experiments demonstrate the computational efficiency of the method. Keywords: Quadratic function; Separable quadratic constraints; Active set; Convergence; Conjugate gradient method; Adaptive precision control 2000 Mathematics Subject Classification: 65K05, 90C20 1. Introduction We shall be concerned with solving min x∈ f(x) (1) where f(x) = (1/2)x ⊤ Ax − x ⊤ b,  ={x ∈ R 2m : x 2 i + x 2 2i ≤ g 2 i , i = 1,...,m}, A ∈ R 2m×2m is a symmetric positive definite matrix, b ∈ R 2m , g i are positive values and x i denotes the i th entry of a vector x ∈ R 2m . Let us point out that the quadratic constraints x 2 i + x 2 2i ≤ g 2 i (2) are separable with respect to the pairs (x i ,x 2i ) ⊤ ∈ R 2 and can be interpreted so that the i th pair lies in the circle with the centre in the origin of R 2 and with the radius g i . Problem (1) arises if we want to solve 3D contact problems of the linear elasticity with isotropic friction. In our previous paper [1], we have shown how to approximate the circles by *Email: radek.kucera@vsb.cz Optimization Methods and Software ISSN 1055-6788 print/ISSN 1029-4937 online © 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10556780600609246