Further reading¶

This book teaches with synthetic data and with models small enough to solve by hand or in a few seconds. The methods themselves are not ours. This page lists the primary sources behind them: the textbooks to learn a topic properly and the papers where an idea first appeared or was made practical.

How to use it:

Start with the core shelf. Ten books cover most of what the chapters use.
Then go chapter by chapter. Each entry says in one line why it matters to that chapter.
Each entry is tagged Textbook, Paper or Standard. Papers carry a DOI or a stable publisher link. Books carry the publisher, edition and year, with a DOI where the publisher gives one.
The list is short on purpose. It names sources we have checked, not everything we have read. Where a topic has a hundred good references, we list the one or two that explain it best to someone who knows energy operations and is learning the method.

The chapter numbers follow the book, not the order of reading.

Core shelf¶

Read these first. Each one is described again under the chapter where it matters most.

Bertsimas and Tsitsiklis, Introduction to Linear Optimization (1997): linear programming and duality. See Chapter 1.
Boyd and Vandenberghe, Convex Optimization (2004): convexity, Lagrangian duality, conic and quadratic programs. See Chapter 7.
Wolsey, Integer Programming, 2nd edition (2021): MILP formulation strength. See Chapter 5.
Postek, Zocca, Gromicho and Kantor, Hands-On Mathematical Optimization with Python (2025): worked Python models in the style this book follows. See T1.
Nocedal and Wright, Numerical Optimization, 2nd edition (2006): gradient methods and the theory behind solver behaviour. See T3.
Conejo, Carrión and Morales, Decision Making Under Uncertainty in Electricity Markets (2010): the energy-market counterpart to all of the above. See Chapter 2.
Kirschen and Strbac, Fundamentals of Power System Economics, 2nd edition (2019): why prices, constraints and shadow prices look the way they do. See Chapter 1.
Rausand, Barros and Høyland, System Reliability Theory, 3rd edition (2021): reliability, availability and maintenance models. See Chapter 3.
McNeil, Frey and Embrechts, Quantitative Risk Management, revised edition (2015): VaR, expected shortfall, coherence and risk allocation. See Chapter 13.
Birge and Louveaux, Introduction to Stochastic Programming, 2nd edition (2011): decisions under uncertainty with recourse. See Chapter 14.

Chapter 1: Two wind farms, one connection (LP)¶

Chapter 1: a linear programme, its dual and the shadow price of a shared limit.

Bertsimas, D. and Tsitsiklis, J. N. (1997). Introduction to Linear Optimization. Athena Scientific, Belmont, MA. ISBN 978-1-886529-19-9. Textbook. The clearest treatment of the geometry of LP, duality and sensitivity; our graphical feasible region is its chapter 1 and 2 in energy dress.
Kirschen, D. S. and Strbac, G. (2019). Fundamentals of Power System Economics, 2nd edition. Wiley, Hoboken, NJ. ISBN 978-1-119-21324-6. Textbook. Explains marginal pricing and why a binding network limit separates prices, which is what the shadow price in the chapter measures.
Schweppe, F. C., Caramanis, M. C., Tabors, R. D. and Bohn, R. E. (1988). Spot Pricing of Electricity. Kluwer Academic, Boston. ISBN 978-0-89838-260-0. Textbook. The origin of the idea that the price at a node is the dual variable of the dispatch problem.
Wood, A. J., Wollenberg, B. F. and Sheblé, G. B. (2014). Power Generation, Operation, and Control, 3rd edition. Wiley, Hoboken, NJ. Textbook. The standard engineering reference for economic dispatch and constrained dispatch, the problem NEMDE approximates in far more detail than we do.

Chapter 2: A battery and a day of prices (LP to MILP)¶

Chapter 2: arbitrage and joint energy and FCAS dispatch for a battery.

Conejo, A. J., Carrión, M. and Morales, J. M. (2010). Decision Making Under Uncertainty in Electricity Markets. International Series in Operations Research and Management Science, vol. 153. Springer, New York. doi:10.1007/978-1-4419-7421-1. Textbook. Offering and scheduling models for market participants, written as optimisation problems in the same spirit as this book.
Morales, J. M., Conejo, A. J., Madsen, H., Pinson, P. and Zugno, M. (2014). Integrating Renewables in Electricity Markets: Operational Problems. International Series in Operations Research and Management Science, vol. 205. Springer, New York. doi:10.1007/978-1-4614-9411-9. Textbook. Forecasting, market clearing and trading with stochastic renewables, including storage and balancing.
Williams, H. P. (2013). Model Building in Mathematical Programming, 5th edition. Wiley, Chichester. ISBN 978-1-118-44333-0. Textbook. How to turn a verbal operating rule, such as no simultaneous charge and discharge, into constraints; the binary variable in this chapter is one of its standard devices.

Chapter 3: Reliability, availability and when to replace¶

Chapter 3: MTBF and MTTR, Poisson intervals, censored Weibull fits, age replacement.

Rausand, M., Barros, A. and Høyland, A. (2021). System Reliability Theory: Models, Statistical Methods, and Applications, 3rd edition. Wiley, Hoboken, NJ. ISBN 978-1-119-37352-0. Textbook. The broadest single reference for failure rates, availability, maintenance models and life-data analysis.
Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. Wiley, New York. A second edition with F. G. Pascual appeared in 2022. Textbook. The reference for censored likelihoods and Weibull and lognormal fits, which is how the chapter treats units still running.
Billinton, R. and Allan, R. N. (1992). Reliability Evaluation of Engineering Systems: Concepts and Techniques, 2nd edition. Plenum Press, New York. ISBN 978-0-306-44063-2. Textbook. Written for power systems; its treatment of repairable components and availability is the one most operators recognise.
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Textbook. The classical account of failure-rate classes and renewal arguments behind replacement policies.
Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293-297. Paper. The paper that introduced the distribution we fit to failure times.
Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481. doi:10.1080/01621459.1958.10501452. Paper. Shows how to use units that have not yet failed instead of discarding them.
Garwood, F. (1936). Fiducial limits for the Poisson distribution. Biometrika, 28(3-4), 437-442. doi:10.1093/biomet/28.3-4.437. Paper. The exact interval for a count of failures, used for the Poisson confidence intervals.
Barlow, R. E. and Hunter, L. (1960). Optimum preventive maintenance policies. Operations Research, 8(1), 90-100. doi:10.1287/opre.8.1.90. Paper. The age-replacement result: replace at a fixed age when the failure rate is rising and a failure costs more than a planned replacement.

Chapter 4: Choosing a maintenance window¶

Chapter 4: a chance constraint on safe wind hours using ensemble forecasts, and a backtest.

Charnes, A. and Cooper, W. W. (1959). Chance-constrained programming. Management Science, 6(1), 73-79. doi:10.1287/mnsc.6.1.73. Paper. The origin of requiring a constraint to hold with a stated probability instead of always.
Prékopa, A. (1995). Stochastic Programming. Mathematics and Its Applications, vol. 324. Kluwer Academic, Dordrecht. doi:10.1007/978-94-017-3087-7. Textbook. The mathematical theory of probabilistic constraints, including when they stay convex.
Calafiore, G. C. and Campi, M. C. (2006). The scenario approach to robust control design. IEEE Transactions on Automatic Control, 51(5), 742-753. doi:10.1109/TAC.2006.875041. Paper. Says how many forecast scenarios you need before a constraint enforced on them holds for the real weather with a chosen confidence.
Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477), 359-378. doi:10.1198/016214506000001437. Paper. Why a probabilistic forecast should be judged with a proper score and not by whichever number flatters it.
Hyndman, R. J. and Athanasopoulos, G. (2021). Forecasting: Principles and Practice, 3rd edition. OTexts, Melbourne. Free online at otexts.com/fpp3. Textbook. A practical introduction to forecasting, prediction intervals and evaluating forecasts out of sample.

Chapter 5: A replacement campaign: crews, crane and spares¶

Chapter 5: a time-indexed scheduling MILP with a greedy baseline.

Wolsey, L. A. (2021). Integer Programming, 2nd edition. Wiley, Hoboken, NJ. ISBN 978-1-119-60653-6. Textbook. The best short account of why one formulation solves in seconds and an equivalent one does not, which is the point of the time-indexed form.

Chapter 6: Two farms, two PPAs, one constraint¶

Chapter 6: contract value, the price cliff and offer clearing behind a shared limit.

Stoft, S. (2002). Power System Economics: Designing Markets for Electricity. Wiley-IEEE Press, New York. ISBN 978-0-471-15040-4. Textbook. Clear on offers, price caps and floors, and what a generator's offer does to the clearing price.
Biggar, D. R. and Hesamzadeh, M. R. (2014). The Economics of Electricity Markets. Wiley-IEEE Press, Chichester. ISBN 978-1-118-77575-2. Textbook. Written by authors with Australian regulatory experience; strong on congestion, constrained-off generation and incentives under a pay-as-bid or pay-as-cleared design.

Chapter 7: Lenders, covenants and the value of a dollar¶

Chapter 7: a covenant as a constraint, and its Lagrange multiplier as the value of a dollar.

Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press, Cambridge. ISBN 978-0-521-83378-3. Free PDF at web.stanford.edu/~boyd/cvxbook. Textbook. Chapter 5 (duality) and section 5.6 (sensitivity analysis) explain why a multiplier is a price.

Chapter 8: Ten price bands¶

Chapter 8: turning a target quantity at each price into a bid, with a band-design MILP.

Conejo, Carrión and Morales (2010), listed under Chapter 2. Textbook. Its offering-strategy chapters treat a price-taker's stepwise offer curve as the output of an optimisation.
Pinson, P., Chevallier, C. and Kariniotakis, G. N. (2007). Trading wind generation from short-term probabilistic forecasts of wind power. IEEE Transactions on Power Systems, 22(3), 1148-1156. doi:10.1109/TPWRS.2007.901117. Paper. Shows the optimal bid for a wind producer is a quantile of its forecast, set by the imbalance prices.
Morales, J. M., Conejo, A. J. and Pérez-Ruiz, J. (2010). Short-term trading for a wind power producer. IEEE Transactions on Power Systems, 25(1), 554-564. doi:10.1109/TPWRS.2009.2036810. Paper. A stochastic-programming bid for a wind producer across several markets.

Chapter 9: Was the battery well traded?¶

Chapter 9: backtests without look-ahead, rolling control, perfect-foresight benchmark and Shapley attribution.

Rawlings, J. B., Mayne, D. Q. and Diehl, M. M. (2017). Model Predictive Control: Theory, Computation, and Design, 2nd edition. Nob Hill Publishing, Madison, WI. ISBN 978-0-9759377-3-0. Textbook. The theory of the receding-horizon schedule our policies use: what is re-solved, what is committed, and when it is stable.
Shapley, L. S. (1953). A value for n-person games. In Kuhn, H. W. and Tucker, A. W. (eds), Contributions to the Theory of Games, Volume II, Annals of Mathematics Studies 28, pp. 307-317. Princeton University Press. Paper. The original definition of the fair share we use to split profit between decisions.
Hyndman and Athanasopoulos (2021), listed under Chapter 4. Textbook. Time-series cross-validation is the forecasting version of an honest, no-look-ahead backtest.

Chapter 10: A battery's life¶

Chapter 10: cycle counting, calendar and cycle fade, state-of-health filtering.

ASTM International. ASTM E1049-85 (Reapproved 2017): Standard Practices for Cycle Counting in Fatigue Analysis. store.astm.org/e1049-85r17.html. Standard. The definition of rainflow counting that the chapter follows.
Downing, S. D. and Socie, D. F. (1982). Simple rainflow counting algorithms. International Journal of Fatigue, 4(1), 31-40. doi:10.1016/0142-1123(82)90018-4. Paper. The compact algorithm behind most implementations.
Xu, B., Oudalov, A., Ulbig, A., Andersson, G. and Kirschen, D. S. (2018). Modeling of lithium-ion battery degradation for cell life assessment. IEEE Transactions on Smart Grid, 9(2), 1131-1140. doi:10.1109/TSG.2016.2578950. Paper. Puts rainflow-counted cycle depth into a degradation cost that an optimiser can use.
Schmalstieg, J., Käbitz, S., Ecker, M. and Sauer, D. U. (2014). A holistic aging model for Li(NiMnCo)O2 based 18650 lithium-ion batteries. Journal of Power Sources, 257, 325-334. doi:10.1016/j.jpowsour.2014.02.012. Paper. A published calendar-plus-cycle ageing model fitted to cell tests.
Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35-45. doi:10.1115/1.3662552. Paper. The filter that combines a fade model with noisy capacity tests.

Chapter 11: Cycle, augment or rest?¶

Chapter 11: an annual life-planning MILP, dynamic programming and augmentation.

Bertsekas, D. P. (2017). Dynamic Programming and Optimal Control, Volume I, 4th edition. Athena Scientific, Belmont, MA. ISBN 978-1-886529-43-4. Textbook. The reference for finite-horizon dynamic programming used to check the MILP.
Wolsey (2021), listed under Chapter 5. Textbook. Fixed-charge and indicator constraints for the augmentation decision.

Chapter 12: Two contracts, one line¶

Chapter 12: sharing a network limit as a cooperative game.

Shapley (1953), listed under Chapter 9. Paper. The axioms that single out one allocation of the gain.
Shapley, L. S. and Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48(3), 787-792. doi:10.2307/1951053. Paper. The first application of the Shapley value outside pure theory.
Peleg, B. and Sudhölter, P. (2007). Introduction to the Theory of Cooperative Games, 2nd edition. Theory and Decision Library C, vol. 34. Springer, Berlin. doi:10.1007/978-3-540-72945-7. Textbook. Covers the core, nucleolus and Shapley value in one place.
Simshauser, P. (2021). Renewable energy zones in Australia's National Electricity Market. Energy Economics, 101, 105446. doi:10.1016/j.eneco.2021.105446. Paper. Peer-reviewed account of connection lags, system strength and curtailment on shared NEM networks.

Chapter 13: How bad can a year get?¶

Chapter 13: VaR, expected shortfall, coherence, risk contributions, backtesting a risk number, and the shape of renewable revenue.

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools, revised edition. Princeton University Press, Princeton, NJ. ISBN 978-0-691-16627-8. Textbook. The standard reference for VaR, expected shortfall, coherence, Euler allocation and backtesting.
Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk, 3rd edition. McGraw-Hill, New York. ISBN 978-0-07-146495-6. Textbook. The practitioner's account of what a VaR number means and how it is validated.
Hull, J. C. (2018). Risk Management and Financial Institutions, 5th edition. Wiley, Hoboken, NJ. Textbook. A plain introduction to VaR, expected shortfall and their regulatory use.
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228. doi:10.1111/1467-9965.00068. Paper. The four axioms, and why VaR can fail the one that rewards diversification.
Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26(7), 1443-1471. doi:10.1016/S0378-4266(02)00271-6. Paper. Defines CVaR for discrete scenario sets and shows it is coherent.
Acerbi, C. and Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking and Finance, 26(7), 1487-1503. doi:10.1016/S0378-4266(02)00283-2. Paper. Pins down the definition of expected shortfall that stays coherent when losses have atoms.
Tasche, D. (2008). Capital allocation to business units and sub-portfolios: the Euler principle. In Resti, A. (ed.), Pillar II in the New Basel Accord: The Challenge of Economic Capital, pp. 423-453. Risk Books, London. Preprint: arXiv:0708.2542. Paper. Why splitting a portfolio risk number by derivatives is the one allocation that suits performance measurement.
Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73-84. doi:10.3905/jod.1995.407942. Paper. The proportion-of-failures test for the number of VaR breaches.
Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841-862. doi:10.2307/2527341. Paper. Adds a test that breaches are not clustered, which Kupiec's test cannot see.
Efron, B. (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics, 7(1), 1-26. doi:10.1214/aos/1176344552. Paper. The bootstrap, used to put an error bar on a tail estimate from a short history.
Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York. ISBN 978-0-412-04231-7. Textbook. The practical guide to resampling, including its limits for dependent data.
Hirth, L. (2013). The market value of variable renewables: the effect of solar wind power variability on their relative price. Energy Economics, 38, 218-236. doi:10.1016/j.eneco.2013.02.004. Paper. Shows the average price a wind farm receives falls below the time-average price as wind penetration rises, the cannibalisation effect.
Hirth, L., Ueckerdt, F. and Edenhofer, O. (2015). Integration costs revisited: an economic framework for wind and solar variability. Renewable Energy, 74, 925-939. doi:10.1016/j.renene.2014.08.065. Paper. Frames profile, balancing and grid costs as parts of the same value gap.
Weron, R. (2014). Electricity price forecasting: a review of the state-of-the-art with a look into the future. International Journal of Forecasting, 30(4), 1030-1081. doi:10.1016/j.ijforecast.2014.08.008. Paper. The survey of price models, and of how to test them, behind any scenario generator.

Chapter 14: Hedging the portfolio¶

Chapter 14: mean-CVaR and mean-variance hedging, volume risk, and why an optimised hedge disappoints out of sample.

Rockafellar, R. T. and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21-41. risk.net/journal-risk/2161159. Paper. The linear-programming formulation of CVaR minimisation that the chapter solves.
Krokhmal, P., Palmquist, J. and Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4(2), 43-68. risk.net/journal-of-risk/2161203. Paper. Puts CVaR in the constraints as well as the objective, so a return target and a risk limit can sit in one LP.
Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77-91. doi:10.1111/j.1540-6261.1952.tb01525.x. Paper. The mean-variance trade-off, and the minimum-variance portfolio as its simplest case.
Bessembinder, H. and Lemmon, M. L. (2002). Equilibrium pricing and optimal hedging in electricity forward markets. Journal of Finance, 57(3), 1347-1382. doi:10.1111/1540-6261.00463. Paper. Shows why electricity forwards carry a premium or discount to expected spot, and how demand risk shapes the optimal hedge.
DeMiguel, V., Garlappi, L. and Uppal, R. (2009). Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Review of Financial Studies, 22(5), 1915-1953. doi:10.1093/rfs/hhm075. Paper. Estimation error can outweigh the gain from optimising, so a simple split is a fair benchmark.
Smith, J. E. and Winkler, R. L. (2006). The optimizer's curse: skepticism and postdecision surprise in decision analysis. Management Science, 52(3), 311-322. doi:10.1287/mnsc.1050.0451. Paper. Choosing the best-looking option from noisy estimates means the chosen one will, on average, disappoint.
Conejo, Carrión and Morales (2010), listed under Chapter 2. Textbook. Its chapters on risk management apply CVaR to the trading and contracting decisions of electricity producers and retailers, the setting of this chapter.
Birge, J. R. and Louveaux, F. (2011). Introduction to Stochastic Programming, 2nd edition. Springer, New York. doi:10.1007/978-1-4614-0237-4. Textbook. Two-stage models with recourse, value of the stochastic solution and sampling.
Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia. doi:10.1137/1.9780898718751. Textbook. The rigorous treatment, including risk measures inside stochastic programmes and sample-average approximation.
Ben-Tal, A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. Princeton University Press, Princeton, NJ. ISBN 978-0-691-14368-2. Textbook. The alternative to scenarios when you can bound the uncertainty but not describe its distribution.
Bertsimas, D. and Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35-53. doi:10.1287/opre.1030.0065. Paper. A budget of uncertainty that keeps a robust linear programme linear.

Chapter 15: Committing a battery before the prices are known¶

Chapter 15: two-stage stochastic programming, the value of the stochastic solution, Benders (L-shaped) decomposition and sample-average approximation.

Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science, 1(3-4), 197-206. doi:10.1287/mnsc.1.3-4.197. Paper. The first two-stage model: decide now, correct later once the uncertainty is revealed.
Birge, J. R. (1982). The value of the stochastic solution in stochastic linear programs with fixed recourse. Mathematical Programming, 24(1), 314-325. doi:10.1007/BF01585113. Paper. Defines the VSS and proves the ordering EEV ≤ RP ≤ WS used in the chapter.
Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(3), 238-252. doi:10.1007/BF01386316. Paper. Benders decomposition: a master problem plus cuts from the subproblem's duals.
Van Slyke, R. M. and Wets, R. (1969). L-shaped linear programs with applications to optimal control and stochastic programming. SIAM Journal on Applied Mathematics, 17(4), 638-663. doi:10.1137/0117061. Paper. Benders specialised to two-stage stochastic programmes: the L-shaped method.
Mak, W.-K., Morton, D. P. and Wood, R. K. (1999). Monte Carlo bounding techniques for determining solution quality in stochastic programs. Operations Research Letters, 24(1-2), 47-56. doi:10.1016/S0167-6377(98)00054-6. Paper. The optimistic and pessimistic bounds on an SAA solution, with confidence intervals.
Kleywegt, A. J., Shapiro, A. and Homem-de-Mello, T. (2002). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12(2), 479-502. doi:10.1137/S1052623499363220. Paper. How many scenarios SAA needs, and why its optimal value is biased.
Birge and Louveaux (2011), Shapiro, Dentcheva and Ruszczyński (2009) and Conejo, Carrión and Morales (2010), listed under Chapters 14 and 2, are the textbooks for every method in this chapter.

Chapter 16: Robust against the constraint you did not forecast¶

Chapter 16: robust linear programming with budgets of uncertainty, cutting planes, risk-based security and distributionally robust CVaR, on a battery scheme that relieves a corridor after a fault.

Soyster, A. L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21(5), 1154-1157. doi:10.1287/opre.21.5.1154. Paper. The first robust LP: every coefficient at its worst at once, the "box" the chapter starts from.
Ben-Tal, A. and Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25(1), 1-13. doi:10.1016/S0167-6377(99)00016-4. Paper. Shows that a robust LP over a polyhedral or ellipsoidal set is again a tractable convex programme.
Ben-Tal, A. and Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411-424. doi:10.1007/PL00011380. Paper. Small data errors make nominal LP optima badly infeasible on real test problems; the robust counterpart costs little.
Bertsimas and Sim (2004), listed under Chapter 14. Paper. The budget of uncertainty \(\Gamma\), its linear reformulation by duality and its probability bound: the core of this chapter.
Ben-Tal, El Ghaoui and Nemirovski (2009), listed under Chapter 14. Textbook. Uncertainty sets, robust counterparts and adjustable (decision-rule) robustness in one place.
Bertsimas, D., Brown, D. B. and Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464-501. doi:10.1137/080734510. Paper. The survey to read first: sets, tractability, probability guarantees and applications.
Mutapcic, A. and Boyd, S. (2009). Cutting-set methods for robust convex optimization with pessimizing oracles. Optimization Methods and Software, 24(3), 381-406. doi:10.1080/10556780802712889. Paper. Solve, ask an adversary for the worst case, add it, repeat: the second method of the chapter.
Ben-Tal, A., Goryashko, A., Guslitzer, E. and Nemirovski, A. (2004). Adjustable robust solutions of uncertain linear programs. Mathematical Programming, 99(2), 351-376. doi:10.1007/s10107-003-0454-y. Paper. Decisions that wait for some of the uncertainty, as the daily export limit and the arming do here.
Bertsimas, D., Litvinov, E., Sun, X. A., Zhao, J. and Zheng, T. (2013). Adaptive robust optimization for the security constrained unit commitment problem. IEEE Transactions on Power Systems, 28(1), 52-63. doi:10.1109/TPWRS.2012.2205021. Paper. Budgets of uncertainty in power system security, with a system operator's data.
Delage, E. and Ye, Y. (2010). Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 58(3), 595-612. doi:10.1287/opre.1090.0741. Paper. Ambiguity sets built from what the data can support, and the duality that keeps the worst case tractable.
Wiesemann, W., Kuhn, D. and Sim, M. (2014). Distributionally robust convex optimization. Operations Research, 62(6), 1358-1376. doi:10.1287/opre.2014.1314. Paper. A general family of ambiguity sets, including bounds on probabilities of events like "battery \(i\) fails", with tractable reformulations.
Mohajerin Esfahani, P. and Kuhn, D. (2018). Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Mathematical Programming, 171(1-2), 115-166. doi:10.1007/s10107-017-1172-1. Paper. The other main way to build an ambiguity set from samples, with worst-case CVaR as a linear programme.
Sion, M. (1958). On general minimax theorems. Pacific Journal of Mathematics, 8(1), 171-176. doi:10.2140/pjm.1958.8.171. Paper. Why the worst case over distributions and the minimum in the CVaR formula may be swapped.
Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404-413. doi:10.1093/biomet/26.4.404. Paper. The exact confidence bound that turns 4 failures in 120 tests into an upper bound on a battery's failure rate.
Wood, A. J., Wollenberg, B. F. and Sheblé, G. B. (2014). Power Generation, Operation, and Control, 3rd edition. Wiley, Hoboken, NJ. ISBN 978-0-471-79055-6. Textbook. DC power flow, distribution factors and contingency analysis, the network model behind the chapter.
Rockafellar and Uryasev (2000), listed under Chapter 14. Paper. The CVaR formula the distributionally robust model starts from.

Choosing a technique and the simplex method¶

Choosing a technique and T0 · The simplex method by hand: how problem structure decides the method, and how a solver walks the corners of a feasible region.

Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton, NJ. Textbook. The simplex method from its inventor: standard form, bases, pivoting and the tableau.
Charnes, A. (1952). Optimality and degeneracy in linear programming. Econometrica, 20(2), 160-170. doi:10.2307/1907845. Paper. Introduces the penalty on artificial variables now called the big-M method, and treats degeneracy.
Bland, R. G. (1977). New finite pivoting rules for the simplex method. Mathematics of Operations Research, 2(2), 103-107. doi:10.1287/moor.2.2.103. Paper. The smallest-index rule that stops the simplex method from cycling at degenerate corners.
Klee, V. and Minty, G. J. (1972). How good is the simplex algorithm? In Shisha, O. (ed.), Inequalities III, pp. 159-175. Academic Press, New York. Paper. A cube on which the textbook simplex rule visits every corner: exponential in the worst case, though rarely in practice.
Karmarkar, N. (1984). A new polynomial-time algorithm for linear programming. Combinatorica, 4(4), 373-395. doi:10.1007/BF02579150. Paper. The interior-point method: through the inside of the feasible region rather than along its edges.
Land, A. H. and Doig, A. G. (1960). An automatic method of solving discrete programming problems. Econometrica, 28(3), 497-520. doi:10.2307/1910129. Paper. Branch and bound, the method behind every MILP solver in the book.
Bellman, R. (1957). Dynamic Programming. Princeton University Press, Princeton, NJ. Textbook. The principle of optimality behind the DP in Chapter 11.
Bertsimas and Tsitsiklis (1997), listed under Chapter 1, chapters 2-4: the geometry of corners, the simplex method and duality, with proofs.

T1: One model, many solvers¶

T1: the same model in SciPy, Pyomo and OR-Tools, with different solvers.

Postek, K., Zocca, A., Gromicho, J. A. S. and Kantor, J. C. (2025). Hands-On Mathematical Optimization with Python. Cambridge University Press, Cambridge. ISBN 978-1-009-49350-5. Textbook. The model of "one problem, many techniques". The printed book is dated 2025; the notebooks have been developed in the open at mobook.github.io/MO-book.
Huangfu, Q. and Hall, J. A. J. (2018). Parallelizing the dual revised simplex method. Mathematical Programming Computation, 10(1), 119-142. doi:10.1007/s12532-017-0130-5. Paper. The simplex solver behind HiGHS.
Hart, W. E., Watson, J.-P. and Woodruff, D. L. (2011). Pyomo: modeling and solving mathematical programs in Python. Mathematical Programming Computation, 3(3), 219-260. doi:10.1007/s12532-011-0026-8. Paper. The design of the modelling layer we use to keep the model separate from the solver.

T2: Does an optimum exist, and is it unique?¶

T2: infeasibility, unboundedness, optimal faces, degenerate duals and regularisation.

Chinneck, J. W. (2008). Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods. International Series in Operations Research and Management Science, vol. 118. Springer, New York. doi:10.1007/978-0-387-74932-7. Textbook. How to find out why a model is infeasible and what to relax.
Chinneck, J. W. and Dravnieks, E. W. (1991). Locating minimal infeasible constraint sets in linear programs. ORSA Journal on Computing, 3(2), 157-168. doi:10.1287/ijoc.3.2.157. Paper. The irreducible infeasible subsystem idea, which is what a solver reports when it says a model has no solution.
Bertsimas and Tsitsiklis (1997) and Boyd and Vandenberghe (2004), listed under Chapters 1 and 7, cover degeneracy, duality and the optimal face.

T3: Least squares and gradient descent¶

T3: fitting a power curve, conditioning, accelerated and adaptive gradient methods, robust fits.

Golub, G. H. and Van Loan, C. F. (2013). Matrix Computations, 4th edition. Johns Hopkins University Press, Baltimore, MD. ISBN 978-1-4214-0794-4. Textbook. QR, SVD and the normal equations, and why the first two are safer.
Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms, 2nd edition. SIAM, Philadelphia. doi:10.1137/1.9780898718027. Textbook. What conditioning does to a least-squares answer in floating point.
Nocedal, J. and Wright, S. J. (2006). Numerical Optimization, 2nd edition. Springer Series in Operations Research and Financial Engineering. Springer, New York. doi:10.1007/978-0-387-40065-5. Textbook. Step sizes, convergence rates and the line-search and trust-region ideas behind them.
Polyak, B. T. (1964). Some methods of speeding up the convergence of iteration methods. USSR Computational Mathematics and Mathematical Physics, 4(5), 1-17. doi:10.1016/0041-5553(64)90137-5. Paper. The heavy-ball method.
Nesterov, Yu. E. (1983). A method of solving a convex programming problem with convergence rate O(1/k^2). Doklady Akademii Nauk SSSR, 269(3), 543-547. mathnet.ru/eng/dan46009. Paper. The accelerated gradient method.
Kingma, D. P. and Ba, J. (2015). Adam: a method for stochastic optimization. 3rd International Conference on Learning Representations (ICLR). arXiv:1412.6980. Paper. The adaptive step-size method compared in the chapter.
Huber, P. J. (1964). Robust estimation of a location parameter. Annals of Mathematical Statistics, 35(1), 73-101. doi:10.1214/aoms/1177703732. Paper. The Huber loss: quadratic for small errors, linear for large ones.
Koenker, R. and Bassett, G., Jr (1978). Regression quantiles. Econometrica, 46(1), 33-50. doi:10.2307/1913643. Paper. Includes least absolute deviations as the median case, which is a linear programme.

A note on checking¶

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