5.5 Shared Formulation and Practical Differences

What can be equivalent?¶

Nested and simultaneous CCD can represent the same joint optimization problem:

min_{p,z} J_{sys}(p,z), R(p,z)=0, g(p,z) ≤ 0.

The nested form eliminates $z$ through an exact inner optimum; the simultaneous form retains $z$ . Under compatible models, objectives, constraints, discretization, regularity assumptions, and sufficiently accurate solves, both should reach comparable local solutions from comparable basins.

The sequential baselines are generally not mathematically equivalent to this problem. They freeze variables or use discipline-specific objectives and therefore search a restricted or different problem.

Workflow	One system objective?	All coupled constraints?	Plant revisited using closed-loop value?	Classification
One-pass sequential	No	No	No	Non-CCD baseline
Iterated sequential baseline	No	Partially	Through handoffs only	Non-CCD baseline
Nested	Yes	Yes	Yes	CCD
Simultaneous	Yes	Yes	Yes	CCD

Fair comparison¶

Agreement is assessed after independent simulation using the same mesh-refinement tolerance and validation model. Objective proximity without feasibility is not equivalence.

Activity 5.5: test nested–simultaneous agreement¶

Activity 5.5: test nested–simultaneous agreement

Two CCD implementations return:

Result	$J_{sys}$	$k_s$ (kN/m)	gain $K$	max constraint	validation $J$
Nested	0.7180	20.8	1.24	0.998	0.726
Simultaneous	0.7165	20.6	1.27	1.003	0.724

Compute the relative objective difference using the nested value as reference.
Determine which reported solution is feasible before validation.
If re-solving the simultaneous result gives max constraint 0.999 and $J=0.7184$ , recompute the relative difference.
Compute relative differences in $k_s$ and $K$ after the re-solve.
State whether the evidence supports practical agreement within 0.5% objective tolerance and explain what it does not prove about global optimality.