Let (Formula presented.) be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace (Formula presented.) we partially solve the operator version of Birkhoff–James orthogonality problem, if (Formula presented.) is orthogonal to (Formula presented.) when does there exist a unit vector x 0 such that (Formula presented.) and (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) is a L 1-predual space as well as a M-ideal in Y, we show that if (Formula presented.) is orthogonal to (Formula presented.) then there is a unit vector x 0 with (Formula presented.) and (Formula presented.) is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved. © 2021 Taylor & Francis Group, LLC.