Function initialize_optimization_state 

pub fn initialize_optimization_state(
    problem: &Problem,
    initial_params: &HashMap<String, (ManifoldType, DVector<f64>)>,
) -> Result<InitializedState, ApexSolverError>

Initialize optimization state from problem and initial parameters.

This is the common initialization sequence used by all optimizers:

1. Create variables from initial values
2. Build variable-to-column index mapping
3. Build symbolic sparsity structure for Jacobian
4. Compute initial cost