Categorization and Selection of Suitable Solution Approaches

All-At-Once Problem Formulation and Decomposition

To categorize solution approaches, a common starting point is the All-At-Once (AAO) optimization problem. It aggregates the design variables, constraints, and objectives of all subsystems \(i \in \{0,\dots,m\} =:\) \(M\) into a single problem. Shared design variables \({}^{i}_{j}\)\(z\) and \({}^{j}_{i}\)\(z\) between coupled subsystems \(i\) and \(j\) are condensed into a single variable \(z_{\{i,j\}}\), yielding:

\[\begin{aligned} \left\{{}^{i}x^{*}\right\}_{i\in M},\;\left\{z_{\{i,j\}}^{*}\right\}_{\substack{j\in{}^{i}N \\ i\in M}} := \; & \argmin_{\substack{{}^{i}x \;\in\; {}^{i}\mathcal{X} \;\forall\; i\in M \\ z_{\{i,j\}} \;\in\; {}^{i}_{j}\mathcal{Z} \cap {}^{j}_{i}\mathcal{Z} \;\forall\; j\in{}^{i}N,\; i\in M}} \quad \sum_{i\in M} {}^{i}v_f\!\left({}^{i}r\right) \\ & \text{s.t.} \quad {}^{i}v_g\!\left({}^{i}r\right) \leq 0 \quad \forall\; i\in M, \\ & \phantom{\text{s.t.}} \quad {}^{i}v_h\!\left({}^{i}r\right) = 0 \quad \forall\; i\in M, \\ & \text{where} \quad {}^{i}r := {}^{i}r\!\left({}^{i}x,\;\left\{z_{\{i,j\}},\;{}^{j}_{i}h\right\}_{j\in{}^{i}N}\right) \quad \forall\; i\in M, \\ & \phantom{\text{where}} \quad {}^{j}_{i}h := {}^{j}_{i}H\!\left({}^{j}r\right) \quad \forall\; j\in{}^{i}N,\; i\in M. \end{aligned}\]

Here, \({}^{i}\)\(r\) depends on both \({}^{i}\)\(x\) and neighboring subsystems' design variables through the coupling mapping identity \({}^{j}_{i}h := {}^{j}_{i}H\!\left({}^{j}r\right)\).

To address these inter-subsystem dependencies, many approaches apply a decomposition ^[1, 2]: \({}^{j}_{i}\)\(h\) and \({}^{i}_{j}\)\(z\) are treated as additional design variables of subsystem \(i\), and pairwise coupling constraints are introduced:

\[{}^{i}_{j}c := \begin{bmatrix} {}^{i}_{j}c_{h} \\ {}^{i}_{j}c_{z} \end{bmatrix} := \begin{bmatrix} {}^{i}_{j}H\!\left({}^{i}r\right) - {}^{i}_{j}h \\ {}^{i}_{j}z - {}^{j}_{i}z \end{bmatrix} = 0 \quad \forall\; j\in{}^{i}N,\; i\in M.\]

Redundant shared-variable constraints are omitted (only one of \({}^{i}_{j}c_{z}=0\) and \({}^{j}_{i}c_{z}=0\) is imposed). Each component is only introduced if the corresponding quantities exist. This yields the decomposed AAO formulation ^[1, 2]:

\[\begin{aligned} \left\{{}^{i}x^{*}\right\}_{i\in M},\;\left\{{}^{i}_{j}z^{*},\;{}^{j}_{i}h^{*}\right\}_{\substack{j\in{}^{i}N \\ i\in M}} := \; & \argmin_{\substack{{}^{i}x \;\in\; {}^{i}\mathcal{X} \;\forall\; i\in M \\ {}^{i}_{j}z \;\in\; {}^{i}_{j}\mathcal{Z} \;\forall\; j\in{}^{i}N,\; i\in M \\ {}^{j}_{i}h \;\in\; {}^{j}_{i}\mathcal{H} \;\forall\; j\in{}^{i}N,\; i\in M}} \quad \sum_{i\in M} {}^{i}v_f\!\left({}^{i}r\right) \\ & \text{s.t.} \quad {}^{i}v_g\!\left({}^{i}r\right) \leq 0 \quad \forall\; i\in M, \\ & \phantom{\text{s.t.}} \quad {}^{i}v_h\!\left({}^{i}r\right) = 0 \quad \forall\; i\in M, \\ & \phantom{\text{s.t.}} \quad {}^{i}_{j}c = 0 \quad \forall\; j\in{}^{i}N,\; i\in M, \\ & \text{where} \quad {}^{i}r := {}^{i}r\!\left({}^{i}x,\;\left\{{}^{i}_{j}z,\;{}^{j}_{i}h\right\}_{j\in{}^{i}N}\right) \quad \forall\; i\in M. \end{aligned}\]

The bound set \({}^{j}_{i}\mathcal{H}\) must be chosen to preserve equivalence with the original AAO problem (achievable by selecting an unbounded set). Unlike the original formulation, the decomposed version makes response \({}^{i}\)\(r\) depend solely on the design variables \({}^{i}\)\(x\), \({}^{i}_{j}\)\(z\), \({}^{j}_{i}\)\(h\) of subsystem \(i\).

Monolithic and Distributed Approaches

Solution approaches are first categorized by whether a single or multiple optimization routines determine design variables ^[2]. Single-optimizer approaches are called monolithic ^[3, 4]. Individual Discipline Feasible (IDF) operates on the decomposed formulation. A single optimizer determines \({}^{i}\)\(x\), \({}^{i}_{j}\)\(z\), \({}^{j}_{i}\)\(h\) \(\;\forall\; j\in{}^{i}\)\(N\), \(i\in\)\(M\) and enforces all constraints (local and coupling) via its built-in constraint handling ^{[5, 4, 6]}. Multidisciplinary Feasible (MDF) operates on the original AAO problem directly. For each candidate set of design variables, a convergence driver (typically a Fixed-Point Iteration (FPI) scheme such as Gauss–Seidel or Jacobi) iterates the coupled response functions to mutual consistency before the optimizer proposes new variables ^{[3, 5, 4]}:

\[\begin{aligned} {}^{0}r &:= {}^{0}r\!\left({}^{0}x,\;\left\{z_{\{0,j\}},\;{}^{j}_{0}H\!\left({}^{j}r\right)\right\}_{j\in{}^{0}N}\right) \\ &\;\;\vdots \\ {}^{m}r &:= {}^{m}r\!\left({}^{m}x,\;\left\{z_{\{m,j\}},\;{}^{j}_{m}H\!\left({}^{j}r\right)\right\}_{j\in{}^{m}N}\right) \end{aligned}\]

Software frameworks such as OpenMDAO and GEMSEO support MDF/IDF approaches. Crucially, while monolithic methods allow distributed evaluation of local responses \({}^{i}\)\(r\), the decision-making (i.e., determining optimal design variables) remains centralized ^{[7, 6, 2]}. This framework provides distributed approaches instead.

Distributed solution approaches decompose the AAO problem into an equivalent set of coupled optimization subproblems, each solved by its own coordinated optimizer ^{[3, 4, 2]}. Prominent methods include Collaborative Optimization (CO), Concurrent SubSpace Optimization (CSSO), BLISS, BLISS-2000, Analytical Target Cascading (ATC), and Augmented Lagrangian Coordination (ALC) ^{[3, 5, 2, 8]}.

ALC belongs to the class of primal-dual approaches ^[9]. Pure primal methods also exist, but most cannot handle nonconvex local constraints ^[9]. A notable exception is Sensitivity-Based Distributed Programming (SBDP) ^[10].

Our previous publication [insert link to review paper] identified relaxation-based primal-dual approaches (such as ALC) and SBDP — which grant subsystem autonomy over local and shared design variables and support bidirectional coupling — as the most suitable approaches for distributed design optimization.

The following sections apply these most promising approaches to the decomposed problem formulation above. First, representative distributed primal-dual methods based on Lagrangian relaxation of coupling constraints are derived: ALC and ALADIN. Then, SBDP is considered for the same formulation. Alternatively, these approaches can also be applied to a consensus reformulation. Finally, a unified algorithmic structure is identified.

Primal Problem (Compact Notation)

Before detailing algorithmic structures, the decomposed problem is rewritten in compact form. The local, shared, and coupling design variables of subsystem \(i\) are stacked into a combined design variable vector \({}^{i}\)\(d\):

\[{}^{i}d := \begin{bmatrix} {}^{i}x \\ \left\{{}^{i}_{j}z\right\}_{j\in{}^{i}N} \\ \left\{{}^{j}_{i}h\right\}_{j\in{}^{i}N} \end{bmatrix} \in\; {}^{i}\mathcal{D} := {}^{i}\mathcal{X} \times \prod_{j\in{}^{i}N} {}^{i}_{j}\mathcal{Z} \times \prod_{j\in{}^{i}N} {}^{j}_{i}\mathcal{H}.\]

The bound constraint \({}^{i}d \in {}^{i}\mathcal{D}\) can equivalently be expressed as an inequality:

\[{}^{i}v_{\mathcal{D}}\!\left({}^{i}d\right) := \begin{bmatrix} {}^{i}d_{1} - {}^{i}\bar{d}_{1} \\ {}^{i}\underline{d}_{1} - {}^{i}d_{1} \\ \vdots \\ {}^{i}d_{n_d} - {}^{i}\bar{d}_{n_d} \\ {}^{i}\underline{d}_{n_d} - {}^{i}d_{n_d} \end{bmatrix} \leq 0.\]

Individual components of \({}^{i}\)\(d\) are retrieved via selector matrices:

\[{}^{i}_{j}S_{z}\;{}^{i}d = {}^{i}_{j}z, \qquad {}^{j}_{i}S_{h}\;{}^{i}d = {}^{j}_{i}h.\]

Coupling constraints now depend on \({}^{i}\)\(d\) and \({}^{j}\)\(d\):

\[{}^{i}_{j}c\!\left({}^{i}d,\;{}^{j}d\right) := \begin{bmatrix} {}^{i}_{j}H\!\left({}^{i}r\right) - {}^{i}_{j}h \\ {}^{i}_{j}z - {}^{j}_{i}z \end{bmatrix} = 0.\]

Using this notation, the decomposed problem is rewritten as the primal problem:

\[\begin{aligned} \left\{{}^{i}d^{*}\right\}_{i\in M} := \; & \argmin_{{}^{i}d \;\in\; {}^{i}\mathcal{D} \;\forall\; i\in M} \quad \sum_{i\in M} {}^{i}v_f\!\left({}^{i}r\right) \\ & \text{s.t.} \quad {}^{i}v_g\!\left({}^{i}r\right) \leq 0 \quad \forall\; i\in M, \\ & \phantom{\text{s.t.}} \quad {}^{i}v_h\!\left({}^{i}r\right) = 0 \quad \forall\; i\in M, \\ & \phantom{\text{s.t.}} \quad {}^{i}_{j}c\!\left({}^{i}d,\;{}^{j}d\right) = 0 \quad \forall\; j\in{}^{i}N,\; i\in M, \\ & \text{where} \quad {}^{i}r := {}^{i}r\!\left({}^{i}d\right) \quad \forall\; i\in M. \end{aligned}\]

For readability, the response identity \({}^{i}r := {}^{i}r\!\left({}^{i}d\right)\) is omitted in the remainder of this work.