Inverse Optimization: Theory and Applications

1. Introduction

• Observed decisions serve as inputs for inferring forward-model parameters that make them optimal, or, when perfect fit is unnecessary or impossible, maximize a suitable fitness measure.Applications include estimating route-choice preferences, utility functions, and clinical treatment objectives from decision data.Source: §1 · pp. 1–2
• The review responds to recent growth in inverse optimization research and applications, which span transportation, healthcare, and power systems.The paper notes that the prior survey it identifies focused on classical models for particular 0-1 combinatorial forward problems.Source: §1 · pp. 1, 3
• Inverse models vary by forward-problem structure, estimated parameter type, and model-data fit; structure and tractability depend heavily on the first two dimensions.Forward structures include linear, conic, convex, discrete, and sequential models; parameters may lie in the objective, constraints, or both.Source: §1 · p. 2
• Classical inverse optimization assumes perfect model-data fit, whereas data-driven inverse optimization allows imperfect fit, including noisy observations or model misspecification.Data-driven models use loss functions corresponding to different sub-optimality measures.Source: §1 · p. 2
• The paper proposes a framework for describing and categorizing inverse optimization models and reviews methodological techniques, solution methods, model properties, and applications.It covers broad problem structures, developments specific to data-driven models, and modern application areas.Source: §1 · pp. 2–3

2. Fundamentals of Inverse Optimization: A Road Map of This Paper

• Observed decisions are used to estimate parameters of a forward model that makes them approximately or exactly optimal.The forward model represents one or more agents’ decision-generating process.Source: §2 · p. 3

2.1. The forward problem

• The forward model’s parameter can control its objective, feasible set, or both; the objective is typically convex and differentiable, and the feasible set may include integer variables and convex constraints.The paper also lists quadratic and convex-separable objective forms.Source: §2.1 · p. 4

2.2. The inverse problem

• The key distinction is whether inverse-feasibility is enforced strictly: classical formulations minimize an application-specific objective subject to it, while data-driven formulations penalize its violation.Classical formulations are bilevel programs, with techniques available to reformulate them at a single level; κ trades off the application-specific objective and loss in data-driven formulations.Source: §2.2 · p. 5
• Large forward models or data sets motivate alternative learning approaches: Inverse Online Learning processes inputs sequentially, while Inverse Reinforcement Learning addresses large sequential decision-making models.These approaches are discussed as permitting fast algorithms.Source: §2.2 · p. 5

2.3. Choosing an inverse model

• Application choice separates design problems, which use known decision-generating processes to induce agent decisions, from estimation problems, which learn models from observed behavior and can estimate parameters even when not all decisions can be rendered feasible.The paper describes estimation as useful when observed decisions reflect noisy measurements, inconsistent behavior, or unobserved contextual factors.Source: §2.3 · p. 6

3. Classical Inverse Optimization

• Classical inverse optimization develops tractable inverse-feasibility reformulations across linear, conic, discrete, and sequential models, including settings with partial input, because enumerating feasible solutions is generally impractical.The review also adopts a common minimum-distance objective to a prior estimate and assumes a single decision input for exposition.Source: §3 · p. 6

3.1. Linear models

• Linear inverse models generally estimate objective or constraint parameters separately, minimizing distance to prior estimates while assuming the observed decision is feasible for objective estimation.The other model components are treated as known in these separate estimation settings.Source: §3.1 · p. 7

3.1.1. Estimating the objective.

• For linear objective estimation, complementary slackness or strong duality yields equivalent inverse formulations, and p=1 or p=∞ makes the resulting problem a linear program.The complementary-slackness representation uses a dual-feasible vector and a zero slack-weighted product.Source: §3.1.1 · pp. 7–8

3.1.2. Estimating constraint parameters.

• When estimating both the constraint matrix and right-hand side, an objective-implied supporting facet reduces the bilinear inverse problem to a convex formulation.The approach uses the fact that an optimal solution lies on at least one facet of the feasible set.Source: §3.1.2 · pp. 8–9

3.1.3. Jointly estimating objective and constraint parameters.

• Jointly estimating objective and right-hand-side parameters makes complementary slackness bilinear, but auxiliary binary variables and a sufficiently large M yield a mixed integer linear program.The bilinear equation is reformulated into linear constraints using the auxiliary variables.Source: §3.1.3 · p. 9

3.2. Conic and convex models

• For differentiable convex conic forward models, KKT conditions provide a tractable inverse reformulation under a strict-feasibility condition.The stated condition requires a point x in C for which −g(x,θ) lies in the interior of C.Source: §3.2 · p. 10
• For general conic forward models, inverse-feasibility conditions contain bilinear terms, but objective-only parameters yield a conic inverse problem, and affine objective gradients make it convex.When the forward model is convex, the inverse problem is a linear program.Source: §3.2 · p. 10

3.3. Markov decision processes

• For finite-state, finite-action MDPs, inverse optimization uses an observed policy and complementary slackness to characterize rewards; infinite models yield infinite-dimensional LPs, and inverse ADP remains unexplored.The inverse reinforcement-learning literature developed in parallel with limited cross-pollination.Source: §3.3 · pp. 11–12

3.4. Discrete models

• Schaefer (2009) represents integer-program duals with non-decreasing super-additive functions and gives conditions under which a finite-domain representation fully describes inverse-feasibility.The resulting inverse problem is convex but can be intractable at large dimensions because its variables and constraints grow exponentially.Source: Duality via sub-additive functions · pp. 12–13
• For mixed-integer linear models, Wang (2009) generates extreme-point constraints iteratively, while Bodur et al. (2022) show that points near the observed decision can suffice for trust-region cuts.Extreme-point sets are finite but can be exponential, and computing each point may be costly.Source: Cutting plane algorithms · p. 13

3.5. Partially constrained inverse problems

• When only part of a decision is observed, partial inverse optimization seeks parameters making at least one compatible solution optimal; the inverse optimal value problem instead takes the objective value as input.For linear forward models, the partial-decision problem can be formulated as a bilinear program and reformulated as a mixed-integer linear program or solved by decomposition.Source: Inverse optimization with a partial… · p. 14, Inverse optimal value · p. 14
• Under mild forward-model assumptions, including a bounded feasible region, Ahmed and Guan (2005) show that the inverse problem can be reformulated.Source: Inverse optimal value · p. 15
• When no θ ∈ Θ satisfies θ⊤x̂ = ẑ, methods are proposed to minimize this gap.Source: Inverse optimal value · p. 15

3.6. Summary

• Inverse models can be solved by characterizing forward optimality or using complementary slackness, strong duality, KKT, or superadditive duality.Figure 4 summarizes the relationship between forward-problem structure and the optimality conditions used to solve the inverse problem.Source: §3.6 · p. 15

4. Data-Driven Inverse Optimization

• Observed data pair decisions with optimal solution sets, or with their forward models, and the literature mainly estimates objective-function parameters for convex forward models with observed inputs.The inputs capture observable changes between instances.Source: §4 · p. 16

4.1. Distance from the optimal solution set

• The Minimum Distance loss measures the 2-norm distance from an observed decision to the forward model’s optimal solution set.The data-driven problem minimizes empirical loss over observations, with the population version minimizing expected loss.Source: §4.1 · p. 16, §4 · p. 16
• Under the stated regularity and approximation conditions, the empirical estimator converges to the population-risk minimizer, a property called Risk Consistency.The cited result concerns convergence to the best estimate obtainable with full knowledge of the data distribution.Source: §4.1 · p. 17
• The distance loss is discontinuous in θ for most forward models, and the general problem is NP-hard for convex forward models.A relaxation permits slight violations of primal feasibility and strong duality; enumeration applies generally, with efficient decomposition available for strictly convex models.Source: §4.1 · pp. 17–18
• For the shared-instance linear case, the nearest optimal solution changes from x1 to x2 after an infinitesimal cost-vector rotation, while N=1 yields a closed-form solution and reduces enumeration to m cases.The general enumeration count is |N(δ)|, dependent on Θ’s size.Source: Figure 5 · p. 18, §4.1 · p. 19
• Restricting distance minimization to feasible-set vertices addresses instability from discrete parameter estimates, and the resulting estimator is statistically consistent under i.i.d. sampling.The empirical risk converges in probability to expected risk.Source: §4.1 · pp. 16, 19

4.2. Sub-optimality in the objective function value

• Absolute and relative sub-optimality losses measure objective-value gaps and competitive ratios, respectively; these inverse problems often yield tractable formulations, especially for linear forward models.The absolute-suboptimality formulation also yields generalization bounds for strictly convex forward models.Source: §4.2 · p. 19

4.2.1. Absolute sub-optimality loss.

• For linear forward models, the general Inverse Absolute Sub-optimality problem is equivalent to a linear program, extending the classical inverse linear optimization formulation.The formulation allows infeasible observed decisions; its constraints and variables scale with the number of data points.Source: §4.2.1 · p. 20
• When all observations share a forward model, controlling one dual variable instead of N can reduce the absolute-loss problem; constraint-parameter estimation also has an analytic solution under identical models and well-behaved Θ.For large forward instances, the general problem can quickly become difficult to solve.Source: §4.2.1 · pp. 20–21

4.2.2. Relative sub-optimality loss.

• Scaling invariance can remove the fractional component of relative sub-optimality, and with component-wise positive b_i the problem is equivalent to a linear program.For identical forward models, the problem can be drastically reduced in size; the positivity assumption is not necessary for the general problem.Source: §4.2.2 · pp. 21–22
• Jointly estimating objective parameters and constraint matrices under relative sub-optimality yields a non-convex inverse problem, for which enumeration over discretized candidate matrices is proposed.The cited approach discretizes the set of potential Φ matrices.Source: §4.2.2 · p. 22

4.3. Variational inequality loss

• The variational-inequality empirical-risk problem does not immediately guarantee convergence in probability to the expected-risk problem.Its convex-program solution approach applies to a large class of convex forward models and comes with theoretical guarantees on inverse-solution quality.Source: §4.3 · pp. 22, 24
• For conic forward models, a generalization bound relates expected variational-inequality risk to empirical risk, with its constant factor scaling as O(1/√N).As the data set grows, empirical risk increasingly approximates expected risk.Source: §4.3 · pp. 22–23
• For linear forward models, the variational-inequality and absolute-sub-optimality losses are equivalent, so the bound also applies to inverse absolute sub-optimality.The inverse absolute-sub-optimality problem has efficient solution algorithms.Source: §4.3 · p. 23
• A dual-form reformulation makes the variational-inequality inverse problem convex for conic forward models, and convexity also holds when the objective gradient is linear in θ.The theorem’s stated feasibility condition requires a point in the interior of each cone satisfying the equality constraints.Source: §4.3 · pp. 23–24

4.4. Violating the KKT conditions

• The KKT loss measures stationarity and complementary-slackness residuals, and its evaluation for an observed decision requires solving a convex optimization problem.Joint estimation is non-convex because of the complementary-slackness loss, while estimating only objective parameters can be solved with convex programming under the stated conditions.Source: §4.4 · p. 24

4.5. Alternatives to empirical risk minimization

• Alternative risk measures generalize empirical expected-risk models, which are recovered by choosing the expectation as the risk measure.Source: §4.5 · p. 24

4.5.1. Quantile inverse optimization

• The Inverse VaR formulation uses a chosen χ-quantile to reduce sensitivity of parameter estimates to perturbations in the decision data.The reported error has bounded sensitivity; the choice of loss affects solution difficulty, and the original problem is NP-hard, while fixing the target quantile can recast it as a bi-clique problem.Source: §4.5.1 · pp. 25–26
• For linear forward models, the Inverse VaR problem with p = 1 or p = ∞ Minimum Distance loss is a mixed integer linear program.Source: §4.5.1 · p. 25

4.5.2. Distributionally robust inverse optimization

• Distributionally robust inverse optimization provides an out-of-sample guarantee on true risk that applies independently of the risk measure or loss function.The guarantee compares empirical and true risk without additional terms.Source: §4.5.2 · pp. 26–27
• For linear forward models with sub-optimality loss, the CVaR-based distributionally robust problem can be reformulated as a large conic optimization problem.The distributionally robust CVaR problem is NP-hard for many forward-model classes.Source: §4.5.2 · p. 27
• For multi-objective forward models, the distributionally robust formulation can be recast as a semi-infinite optimization problem under certain conditions and solved by constraint generation.Source: §4.5.2 · p. 27

• Representing the objective gradient as a linear combination of kernels at observed decisions permits modeling complex convex objectives without a closed-form objective function.The representation trades interpretability for flexibility and can characterize the gradient of an arbitrary convex function.Source: §4.6 · pp. 27–28
• Kernel representations can accompany loss functions that require forward-objective gradients and can also model constraints when their gradients are needed.Source: §4.6 · p. 28
• Kernel representations are linear in their weights and typically do not increase inverse-problem computational complexity.Source: §4.6 · p. 28
• Depending on the kernel, the represented function spaces can be arbitrarily complex and may overfit training decisions, so regularization is recommended.The cited formulation adds a complexity penalty weighted by κ, interpretable as a penalty on kernel smoothness.Source: §4.6 · p. 28

• Methods contrast in whether they provide useful statistical properties for estimating forward-agent behavior and whether they admit efficient solution algorithms.Source: §4.7 · p. 28
• The review identifies quantile (VaR) and CVaR risk measures as offering additional stability and generalization properties for limited data settings.These properties may be useful when estimates must generalize to new instances or be robust to decision-data sensitivity.Source: §4.7 · p. 29
• Under empirical expected risk, all summarized loss functions can be minimized easily for linear forward models, while Inverse Distance, KKT, and Variational Inequality losses are also efficient for conic models.Source: §4.7 · p. 29
• Inverse Distance has a statistically consistent estimator; Inverse Absolute Sub-optimality and Inverse Variational Inequality have generalization bounds for large and finite data, respectively.Source: §4.7 · p. 29

5.1. Inverse optimization through online learning

• Online inverse optimization updates a parameter estimate from each new decision, avoiding increasingly large batch problems but trading computational efficiency against learning efficiency measured by regret.Update rules use the most recent observation; regret compares cumulative online loss with batch-level loss over the time horizon.Source: §5.1 · pp. 30–31

5.1.1. Updates using a forward optimization oracle.

• Bärmann et al. (2018) use Absolute Sub-optimality loss and a forward optimization oracle, with MWU and OGD updates that yield average-regret convergence guarantees.The update rules are based on multiplicative weights and online gradient descent.Source: §5.1.1 · p. 31
• These guarantees assume noiseless data; Chen and Kılınç-Karzan (2020) extend similar convergence results to certain forward problems with nonlinear objectives.Source: §5.1.1 · p. 32

5.1.2. Updates using an inverse optimization oracle.

• Dong et al. (2018) update estimates by solving a single-point inverse problem under convex bounded feasible regions and strictly convex objectives, with average regret O(1/T) even under qualifying noise distributions.Source: §5.1.2 · p. 32

5.2. Inverse reinforcement learning

• IRL estimates a reward vector that makes an observed policy or trajectories optimal; modern methods draw more on machine learning than inverse optimization.The section notes that IRL techniques may also inspire new data-driven inverse optimization methods.Source: §5.2 · p. 32
• Early Max-Margin IRL estimates rewards so observed-policy actions outperform alternatives in expected reward, using a convex program when rewards depend only on state.The Max-Margin loss compares q-values for observed actions with the next-best action; Ng and Russell (2000) add L1 regularization to encourage smaller-magnitude rewards.Source: §5.2 · pp. 32–33
• For large state-action spaces or trajectory-only data, Max-Margin methods reduce reward variables with fixed basis functions or avoid explicit reward design using value-function margins and candidate policies.These formulations parallel data-driven inverse optimization, while classical IRL can be too large to model for such settings.Source: §5.2 · pp. 33–34
• Maximum Entropy IRL assumes observed trajectories in stochastic MDPs are highest-likelihood under an optimal policy, recasting IRL as maximum likelihood [Ziebart et al. 2008, Boularias et al. 2011].The objective maximizes the expected log-likelihood of observed trajectories.Source: §5.2 · p. 34
• The all-zero reward makes every policy optimal in any MDP, so IRL needs well-designed objectives such as Max-Margin to avoid degenerate estimates.Both inverse optimization and IRL risk uninformative degenerate estimates.Source: §5.2 · p. 34
• Ng et al. (1999) show reward-shaping transformations preserve an MDP’s optimal policy, leaving infinitely many reward functions for the same optimal policy; Fu et al. (2018) propose disentangled rewards.The invariant transformation adds γΦ(s′) − Φ(s) to θ(s,a,s′), for arbitrary Φ.Source: §5.2 · p. 34
• Unlike reward-shaping invariance in IRL, inverse optimization’s scaling invariance is often mitigated by a normalization constraint.The paper characterizes scaling invariance as a similar, though less extensive, property in inverse optimization.Source: §5.2 · p. 34

6. Applications in Design Problems

• The two design settings share the use of prices to induce behavioral change, assuming the original decision-generating processes are known.Source: §6 · p. 35

6.1. Incentive design and pricing in bilevel games

• Bilevel design separates the leader’s target-decision problem from an inverse-optimization subproblem that finds incentives making the target optimal for the follower.The leader’s problem assumes the follower’s decision process is fully observable.Source: §6.1 · p. 35
• Marcotte et al. (2009) study toll design for diverting hazardous-material transport from high-risk and high-population areas; master-subproblem decomposition is more efficient and computes tolls optimizing another criterion besides making a low-risk route optimal.Source: §6.1 · p. 35
• Other bilevel applications use partial inverse optimization for carbon taxes and subsidies, or decompose revenue maximization into inverse problems for given follower decisions.Zhou et al. (2011) consider incentives for renewable generation; Brotcorne et al. (2008, 2011) and Af¸sar et al. (2021) study revenue maximization.Source: §6.1 · p. 36

6.2. Mechanism design in cooperative games

• A centralized operator first maximizes collective network profit and assigns carrier-level target flows; carriers then retain control of their own resources.The model assumes a fully connected network and zero operating costs.Source: §6.2 · p. 37
• In cooperative transportation, inverse optimization computes cost-per-arc prices that incentivize carriers’ target decisions while enforcing budget balance and stability.Stability ensures no carrier subset can profit more by forming a subcoalition.Source: §6.2 · p. 37

7.1.1. Transportation system modeling.

• Estimated traffic costs make observed flows approximately satisfy convex-equilibrium conditions and yield high-quality forecasts across real-world benchmarks (Bertsimas et al. 2015, Zhang et al. 2016a, 2018).The estimated costs also reveal network inefficiency and the value of smart traffic control (Zhang et al. 2018), and can inform infrastructure and policy, including estimating bike-lane effects.Source: §7.1.1 · p. 38

7.1.2. Personalized agent models.

• Chen et al. (2021) calibrate capacitated vehicle-routing models from expert-driver routes using multiplicative weights; a retailer case study reproduces many expert decisions.The model refines network travel-time estimates using routes selected by long-term expert drivers.Source: §7.1.2 · p. 39

7.2.1. Market-clearing models.

• Birge et al. (2017) infer unobservable institutional constraints from public market data under noise-free feasibility and complementary-slackness assumptions; observed dual variables can narrow the search and enable more precise estimation with fewer data points.The inferred constraints can clarify transmission capacities for bidding or infrastructure decisions.Source: §7.2.1 · p. 40

7.2.2. Agent demand models.

• Inverse optimization estimates utility functions for consumer-demand models using known lower and upper consumption bounds; these estimates can generate demand predictions, with marginal utility non-increasing in consumption.Source: §7.2.2 · pp. 40–41
• Calibrated agent-demand models outperform classical and black-box time-series forecasts on synthetic and real data, and their estimated utility functions can be submitted directly as retailer bids.The estimated utility retains the stepwise linear shape required for market-clearing bids.Source: §7.2.2 · p. 42

7.3.1. Healthcare systems.

• Inverse shortest-path optimization measures patient-pathway concordance by finding arc costs that make recommended clinical pathways shortest paths (Chan et al. (2022)).The model minimizes squared duality gaps and normalizes the cost vector to prevent the zero vector from being optimal.Source: §7.3.1 · p. 42
• Using public ACO spending and savings data, inverse optimization estimates ACO-type distributions for subsidy-based contracts that can increase both Medicare savings and ACO payments (Aswani et al. (2019)).The unobserved type represents an ACO’s ability to generate savings.Source: §7.3.1 · p. 43
• Inverse optimization estimates sensitivity and specificity ranges for a hypothetical screening test that would make a given policy optimal (Ayer (2015)).Disease progression is modeled as a partially observable Markov chain, yielding a nonlinear inverse problem with a proposed complete solution method.Source: §7.3.1 · p. 43

7.3.2. Personalized decision making.

• For prostate cancer radiotherapy, inverse-optimized weights in simple linear models recreated treatments designed with complex non-convex models using many objectives (Chan et al. (2014)).The weights were learned from real data on previously treated patients.Source: §7.3.2 · p. 43
• Assuming a control-limit policy over ordered health states, inverse optimization infers patient preferences that make observed liver-transplant acceptance policies optimal (Erkin et al. (2010)).Their formulation uses a weighted 1-norm objective and polyhedral constraints on the cost vector to produce a linear inverse problem.Source: §7.3.2 · p. 44

7.4. Additional applications

• In finance, observed Markowitz investment decisions can be used to infer risk tolerance; Yu et al. (2021) study this with inverse online learning, and Li (2021) develops a broader inference framework.Source: §7.4 · p. 44
• Learned preferences can support individualized robo-advice, while inverse Markowitz variants assess environmental and social considerations in mutual funds across managers.Alsabah et al. (2021) discuss investment support tailored to individual tastes; Utz et al. (2014) assess socially responsible mutual fund products.Source: §7.4 · pp. 44–45
• Inverse models infer objectives in multi-player Nash games using data-driven inverse variational inequalities, inverse KKT conditions, and generalized Nash equilibrium extensions.Bertsimas et al. (2015) use inverse VI; Ratliff et al. (2014) and Risanger et al. (2020) use inverse KKT; Allen et al. (2022) extend inverse KKT to generalized Nash equilibria.Source: §7.4 · p. 45
• Data-driven inverse linear models with sub-optimality loss infer objective functions that explain and predict cellular metabolism behavior.Zhao et al. (2015) and Zhao et al. (2016) apply these models to cellular metabolism.Source: §7.4 · p. 45

8. Conclusion

• Many inverse models do not scale well with data; observations from different feasible regions each add a new set of variables and constraints.Source: §8 · p. 45
• Future work could improve large-scale solvability through decomposition or approximation methods.Source: §8 · p. 45
• The literature has incorporated little to no distributional assumptions about input data, motivating inverse models that better incorporate distributional information.Source: §8 · p. 45

EC.1.1. Toll design for risk mitigation

• A two-stage sequence of linear programs finds a minimum-risk hazardous-material flow and then the tolls that make it optimal for the carrier, minimizing toll collection.The master problem produces a feasible minimum-risk flow; the inverse-optimization subproblem computes tolls that induce it.Source: EC.1.1. Toll design for risk mitigation · p. 56

EC.1.2. Estimating traffic equilibrium models

• Inverse variational inequalities estimate marginal travel-time cost functions from observed flows and demands, addressing the need to derive costs from data rather than assume them.The formulation uses a strongly monotonic cost model and imposes monotonicity over observed flow ranges.Source: EC.1.2. Estimating traffic equilibrium… · pp. 57–58
• The estimated cost functions enable system-optimal routing and one of the first empirical estimates of the price-of-anarchy ratio as a measure of network efficiency.System-optimal solutions are Pareto efficient, unlike the user-optimal traffic-assignment solutions.Source: EC.1.2. Estimating traffic equilibrium… · p. 59

EC.1.3. Inferring constraints of market-clearing models

• For electricity markets, Birge et al. (2017) recover the transmission-constraint matrix and capacities from public data under a noise-free assumption by enforcing optimality conditions.Market operations vary significantly across regions, creating research opportunities; Ruiz et al. (2013) is given as an example.Source: EC.1.3. Inferring constraints of… · p. 60
• A duality-based inverse optimization model infers producers' stepwise bid functions in markets where model constraints, rather than bids, are published.The authors examine the setting where market-clearing constraints are available but bids are not.Source: EC.1.3. Inferring constraints of… · p. 60

EC.1.4. Clinical pathway concordance measurement

• A second inverse problem refines expert-derived arc costs using patient pathways, minimizing sub-optimality for survivors and maximizing it for patients who died among optimal cost vectors.The initial model assumes expert-defined pathways are shortest paths and estimates arc costs to minimize aggregate pathway sub-optimality; the refinement uses real patient pathways.Source: EC.1.4. Clinical pathway concordance… · pp. 60–61
• Clinical-pathway concordance was statistically significantly associated with mortality among colon cancer patients, even after adjusting for patient covariates.The metric compares a patient pathway's cost with a shortest-path cost and normalizes the difference using the longest walk with the same number of steps.Source: EC.1.4. Clinical pathway concordance… · p. 61
• The clinical concordance formulation could be extended to favor lower-cost patient pathways instead of separating patients by survival outcome.The proposed extension would incorporate real-dollar costs for imaging, treatment, and diagnostic tests.Source: EC.1.4. Clinical pathway concordance… · p. 61

EC.1.5. Radiation therapy treatment planning

• Radiation therapy balances tumor-dose escalation against healthy-organ dose through weighted objectives, and inverse optimization can infer objective weights from historical treatments.Both Absolute and Relative Sub-optimality loss functions have been used in inverse formulations of this treatment-planning problem.Source: EC.1.5. Radiation therapy treatment… · pp. 61–62