On minimization of a convex quadratic function subjected to a system of linear equations

Abstract

In this paper a general solution of a problem of the minimization of the convex quadratic function f(χ) = (χ -μ)' B(χ - μ) + d subject to the system of equations Rχ = s, where the matrix B is nonnegative definite and the matrix R is not of full row rank, is presented. For the derived formula a necessary and sufficient condition for the optimal solution of the considered problem is formulated. The obtained formulae are recurrent with respect to additional constraints imposed on the vector χ. To derive these formulae the Lagrange multiplier method and generalized inverse of matrices were used.