Chapter 11: Solving Nested and Simultaneous CCD Problems

Derivatives, sparsity, scaling, and computational efficiency¶

Many reported CCD failures are implementation failures: inaccurate derivatives, poor scaling, inconsistent inner convergence, insufficient discretization, hidden nonsmoothness, or unfair comparisons.

Learning objectives¶

After completing this chapter, you should be able to:

1. explain and apply outer plant loop;

2. explain and apply inner control solve;

3. explain and apply sparse simultaneous NLP;

4. explain and apply derivatives and scaling;

5. formulate and verify the chapter methods on nested and simultaneous suspension formulations using identical meshes, derivatives, tolerances, and starts.

Mathematical lens¶

The recurring quantities are outer $p$, trajectory $w=[X,U]$, and sparse residuals $R(p,w)$:

Running example¶

The recurring example is nested and simultaneous suspension formulations using identical meshes, derivatives, tolerances, and starts. Retaining one system prevents apparent improvements from being caused by changed physics, information, loads, or metrics.

Recommended workflow¶

1. scale formulation.

2. verify derivatives.

3. warm start.

4. match tolerances.

5. compare solutions.

Chapter map¶

1. Structure of the Nested Problem

2. Structure of the Simultaneous Problem

3. Gradient-Based Algorithms

4. Finite Differences and Complex-Step Derivatives

5. Algorithmic Differentiation

6. Direct and Adjoint Sensitivities

7. Sparse Jacobians and Hessians

8. Variable and Constraint Scaling

9. Initialization and Warm Starting

10. Parallel Computation

11. Local Versus Global Optimization

12. Bilevel Sensitivities and Inner-Loop Convergence