Let be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace we partially solve the operator version of Birkhoff-James orthogonality problem, if is orthogonal to when does there exist a unit vector x 0 such that and is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for is a L 1 -predual space as well as a M-ideal in Y, we show that if is orthogonal to then there is a unit vector x 0 with and is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved.