Many fixed-order suboptimal control problems with stability, performance and robustness specifications can be reduced to a search for a matrix X > 0 satisfying a linear matrix inequality (LMI) while X(-1) satisfies another LMI. This paper defines a certain class of these problems we shall call the ’dual LMI problem’, and a computational algorithm to solve our dual LMI problem is given. Properties and limitations of the algorithm are discussed in comparison with the existing algorithm (the min/max algorithm). An extension to optimal control problems is provided. Numerical examples for the fixed-order stabilization problem and the static output feedback linear quadratic optimal control problem demonstrate the applicability of the proposed algorithm.