A powertrain system includes an internal combustion engine, a multi-mode transmission having a plurality of torque machines, and a driveline. A method to determine extrema for an objective function employed to control operation of the powertrain system includes establishing an objective component equation related to an objective function and corresponding to an object component of interest. A plurality of linear constraints and a non-linear constraint are imposed on the objective component equation. The objective component equation is solved in relation to the plurality of linear constraints and the non-linear constraint to determine the extrema for the objective function. The extrema for the objective function are employed to control operation of the powertrain system.