This paper considers the design of a variable structure observer for discrete-time systems. The output of the system is assumed to be multidimensional, which extends existing work in which unidimensional outputs are assumed. The system is affected by both matched and unmatched disturbances. An ultimately bounded stable variable structure observer is proposed. The main novelty of the approach is that the design of the observer is written as an optimization problem with linear constraints. The objective is the minimization of the ultimate boundedness of the response. Using Linear Matrix Inequalities, the degrees of freedom of the observer are constructively exploited, which provides advantages when compared with existing frameworks.