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Dynamic Corporate Transformation

Vol. II, Ch. 11 · Part 3. Enterprise Optimization Under Uncertainty · Week 12

Distributionally Robust Enterprise Optimization

Learning outcomes

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

  1. Distinguish stochastic, robust, neuro-fuzzy robust, and distributionally robust optimization.
  2. Define ambiguity sets for enterprise uncertainty.
  3. Formulate distributionally robust enterprise optimization problems.
  4. Construct moment-based ambiguity sets.
  5. Construct Wasserstein ambiguity sets.
  6. Construct φ\varphi-divergence ambiguity sets.
  7. Interpret worst-case probability distributions.
  8. Analyze enterprise decisions under distributional ambiguity.
  9. Implement DREO computationally with calibrated radii and honest certificates.
  10. Prepare enterprise models for multi-objective optimization.

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. Motivation for Distributionally Robust Enterprise Optimization

    Motivation for Distributionally Robust Enterprise Optimization

  2. Enterprise Distributional Ambiguity

    Enterprise Distributional Ambiguity

  3. Ambiguity Sets

    Ambiguity Sets

  4. Moment-Based Ambiguity Sets

    Moment-Based Ambiguity Sets

  5. Wasserstein Ambiguity Sets

    Wasserstein Ambiguity Sets

  6. \texorpdfstring{$\varphi$

    \texorpdfstring{φ\varphi

  7. Worst-Case Distribution Theory

    Worst-Case Distribution Theory

  8. Distributionally Robust Enterprise Policies

    Distributionally Robust Enterprise Policies

  9. Computational Algorithms

    Computational Algorithms

  10. Enterprise Applications

    Enterprise Applications

  11. Comparison with NF-REO

    Comparison with NF-REO

  12. Preparation for Multi-Objective Enterprise Optimization

    Preparation for Multi-Objective Enterprise Optimization

  13. Chapter Summary

    Chapter Summary

  14. Worked Examples

    Worked Examples

  15. Exercises

    Exercises

  16. 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 11.

A. Concept checks

  1. 11.1
    For each source row of Table (see book), justify the recommended set-design response and exhibit an enterprise decision where the wrong family (e.
  2. 11.2
    Explain, using the saddle point of Theorem (see book), why publishing the worst-case distribution alongside a DR decision is the correct governance disclosure, and what a board should do when that law is (a) absurd, (b) plausible.

B. Mathematical exercises

  1. 11.3
    Derive Scarf's bound of Example (see book) from the dual (see book) with ψ=(ξ,ξ2)\psi = (\xi, \xi^{2}): exhibit the optimal quadratic majorant of (ξ−d)+(\xi - d)^{+} and the two-point worst-case law, verifying Theorem (see book)'s k+1=3k + 1 = 3 bound is not tight here.
  2. 11.4
    Prove the KL closed form (see book) on finite support: Lagrangian, exponential-tilt stationarity, and the one-dimensional dual; verify the Example (see book) weights.
  3. 11.5
    Prove the d=1d = 1 concentration inequality behind Proposition (see book) using W(P,Q)=∫∣FP−FQ∣\Wass(P, Q) = \int\abs{F_P - F_Q} and Dvoretzky–Kiefer–Wolfowitz; calibrate c1,c2c_1, c_2 explicitly for compactly supported laws.
  4. 11.6
    Show the worst-case expectation of Definition (see book) is a coherent risk measure, and identify the ambiguity sets representing (a) CVaRα_{\alpha}, (b) mean plus θ×\theta \times standard deviation (state precisely when the latter fails coherence).

C. Computational exercises

  1. 11.7
    Prove strong duality in Theorem (see book) directly for finite Ξ\Xi via LP duality, and exhibit a two-point Ξ\Xi instance with the Slater condition violated where a duality gap opens.
  2. 11.8
    Prove that for ε>0\varepsilon > 0 fixed and N→∞N \to \infty, the Wasserstein DREO decision converges to the ε\varepsilon-robust decision around PtrueP_{\mathrm{true}}, and combine with Proposition (see book) to obtain the full consistency diagram of data-driven DREO.

D. Enterprise applications

  1. 11.9
    Reproduce Example (see book); then implement the Wasserstein-DR newsvendor on support [0,∞)[0, \infty) via Theorem (see book)(ii) and report whether transport-based sets repair the disappointment gap at matched regret.
  2. 11.10
    Build the cyber layer schedule of Example (see book) across retentions d∈[3,10]d \in [3, 10] under (a) Scarf, (b) Scarf plus unimodality (Gauss-type bound; Notes), (c) fitted lognormal; plot the three curves and locate where structure assumptions matter most.
  3. 11.11
    Meridian growth-market entry: quarterly demand history (N=16N = 16, provided in the AXIOM-11 module) with suspected survivorship bias.
  4. 11.12 ★
    Develop dynamic DREO for the Chapter 9 controlled diffusion: define rectangular Wasserstein ambiguity per time step, derive the distributionally robust HJB equation, prove a verification theorem, and quantify on the liquidity instance how the Chapter 9 gain stiffens with the per-step radius; discuss time-consistency failures for non-rectangular sets.

Downloads

All three companions consume the same seeded engine (26211), so their numbers agree by construction — the MFMF convention, carried forward.