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Dynamic Corporate Transformation

Vol. II, Ch. 2 · Part 1. Enterprise Optimization Foundations · Week 8

The General Enterprise Optimization Problem Revisited

Learning outcomes

After completing this chapter, the reader should be able to:

  1. Formulate the General Enterprise Optimization Problem in its canonical five-block form, discrete- and continuous-time.
  2. Distinguish objective functions from enterprise performance measures, and state what promotion from measure to objective adds and obliges.
  3. Identify enterprise decision variables and state variables, and keep the chosen/induced boundary exact.
  4. Construct enterprise constraint systems across the five mathematical forms and six architectural provenances.
  5. Differentiate equality and inequality constraints, and path, boundary, and decision-set constraints, with the multiplier previews each carries.
  6. Define enterprise feasibility for policies and trajectories, and certify it rather than assume it.
  7. Interpret enterprise solution spaces: feasible sets, optimal sets, and the value order that ranks only inside feasibility.
  8. Explain enterprise optimization architectures: the five-layer computational stack from declaration to implementation.
  9. Recognize the classes of enterprise optimization problems and place classical formulations as GEOP special cases.
  10. Prepare GEOP formulations for computational solution: standard forms, scaling, sparsity, and export.

Reading guide

Work through the chapter in section order; the full development, proofs, and worked examples are in the book — this page indexes them and does not replace them.

  1. Review of GEOP from Volume I

    Review of GEOP from Volume I

  2. Enterprise Decision Variables

    Enterprise Decision Variables

  3. Enterprise State Variables

    Enterprise State Variables

  4. Objective Functions

    Objective Functions

  5. Constraint Systems

    Constraint Systems

  6. Enterprise Feasible Regions

    Enterprise Feasible Regions

  7. Enterprise Optimization Architectures

    Enterprise Optimization Architectures

  8. Classes of Enterprise Optimization Problems

    Classes of Enterprise Optimization Problems

  9. Canonical GEOP Formulation

    Canonical GEOP Formulation

  10. Computational Preparation

    Computational Preparation

  11. Transition to Convex Optimization

    Transition to Convex Optimization

  12. Worked Examples

    Worked Examples

  13. Chapter Summary

    Chapter Summary

  14. Exercises

    Exercises

  15. Notes and Sources

    Notes and Sources

AXIOM

This chapter is instrumented by:

Launch the module, load the chapter model, modify inputs, run the optimization, and compare against the worked examples in the book.

Exercises

12 exercises, grouped A concept checks · B mathematical · C computational · D enterprise applications. Starred (★) exercises are on the advanced track. Full solutions appear in the Instructor's Manual, Chapter 2.

A. Concept checks

  1. 2.1
    Name the five constraint blocks of the canonical GEOP and give one Meridian instance of each, citing the block's architectural provenance.
  2. 2.2
    Explain why the Bolza/Mayer/Lagrange choice is representational while the LL-versus-ΦT\Phi_T content split is governance, citing Theorem (see book) and the window doctrine.

B. Mathematical exercises

  1. 2.3
    Write Example (see book)'s instance fully in the notation of Table (see book): every block, every symbol bound, empty blocks declared empty.
  2. 2.4
    Carry out Mayer-to-Lagrange for ΦT(x)=c⊤x\Phi_T(\x) = \mathbf{c}^{\top} \x on linear dynamics: compute the induced running term ℓk\ell_k explicitly and identify the policy-independent constant.
  3. 2.5
    Show that exponential discounting is a declared weight inside LL (define Lk=βkℓ(xk,uk)L_k = \beta^k \ell(\x_k, \uc_k)), and that re-expressing it as an augmented-state multiplicative factor yields the same optimal set; where does Theorem (see book)(iii) enter?
  4. 2.6
    Show that rescaling constraint gi↦γigig_i \mapsto \gamma_i g_i (γi>0\gamma_i > 0) and variables u↦Du\uc \mapsto \mathbf{D}\uc (diagonal, positive) preserves feasible and optimal sets; identify what the rescaling changes (conditioning, multiplier units) and record the multiplier transformation rule.

C. Computational exercises

  1. 2.7
    Prove Proposition (see book)'s open-loop clause in detail: the constant-rule subclass is a declared policy class, its induced problem has the same data as the sequence-decision problem, and feasible/optimal correspondences are bijections.
  2. 2.8
    Prove that minimal infeasible subsystems need not be unique: construct an instance with two distinct minimal subsystems and show a repair relaxing a constraint in their intersection fixes both, while one outside some subsystem fixes neither.

D. Enterprise applications

  1. 2.9
    (With AXIOM-02 or the Chapter 2 notebook.
  2. 2.10
    (With AXIOM-02 or the Chapter 2 notebook.
  3. 2.11
    Write one live decision of an enterprise you know in the full five-block template, empty blocks declared empty, every symbol of Table (see book) bound or explicitly unused.
  4. 2.12 ★
    Minimal infeasible subsystems can be exponentially numerous.

Downloads

All three companions consume the same seeded engine (26202), so their numbers agree by construction — the MFMF convention, carried forward.