Large-scale optimization plays important role in many control and learning problems. Sequential subspace optimization is a novel approach particularly suitable for large-scale optimization problems. It is based on sequential reduction of the initial optimization problem to optimization problems in a low-dimensional space. In this paper we consider a problem of multidimensional convex real-valued function optimization. In a framework of sequential subspace optimization we develop a new method based on a combination of quasi-Newton and conjugate gradient method steps. We provide its formal justification and derive several of its theoretical properties. In particular, for quadratic programming problem we prove linear convergence in a finite number of steps. We demonstrate superiority of the proposed algorithm over common state of the art methods by carrying out comparative analysis on both modelled and real-world optimization problems.