We consider a multidimensional singular stochastic control problem with state-dependent diffusion matrix and drift vector and control cost depending on the position and direction of displacement of the controlled process. The objective is to minimize the total expected discounted cost. We write an equivalent infinite-dimensional linear programming problem on a subspace of the space conjugate to C(R(n)) x C(R(n) x B), where B is the unit sphere in R(n). We write a dual linear program and prove absence of duality gap. The dual program characterizes the optimal cost function as a maximal solution to the variational inequality with gradient constraints.