Iterative Risk Allocation Algorithm

• The paper introduces a novel iterative risk allocation algorithm integrating ARSRM to dynamically adjust risk budgets and optimize decision policies via SDDP.
• It leverages state-dependent, randomized spectral risk measures and dual formulations to construct supporting cuts, ensuring convergence under convexity assumptions.
• Applied to asset allocation with transaction costs, the approach improves portfolio return-versus-risk tradeoffs and yields robust solutions under ambiguity.

A multistage risk-aware optimization problem with state- or scenario-dependent preferences introduces significant complexity in both risk modeling and solution methodology. The paper "Multistage Robust Average Randomized Spectral Risk Optimization" (Wu et al., 2 Sep 2024) addresses this by employing the average randomized spectral risk measure (ARSRM), which allows risk preferences to vary stochastically at each stage. The solution is achieved using a stochastic dual dynamic programming (SDDP) approach that iteratively constructs risk-aligned value function approximations and computes optimal policies under distributional uncertainty about the decision maker's risk attitudes. This iterative framework is analyzed both theoretically and computationally, and its performance is validated on asset allocation problems with transaction costs.

1. Multistage Randomized Spectral Risk Models

The model considers a finite-horizon, stage-wise stochastic optimization problem. At each stage $t$ and for each scenario $\xi_t$, a decision $x_t \in \mathcal{X}_t(x_{t-1}, \xi_t)$ is chosen so as to minimize the sum of current cost and the "risk" of future costs. The novelty is in defining this risk recursively with an ARSRM operator: $$V_t(x_{t-1}, \xi_t) = \min_{x_t \in \mathcal{X}_t(x_{t-1}, \xi_t)}\left\{ c_t \cdot x_t + \mathcal{V}_{t+1}(x_t, \xi_t) \right\}$$ where

$$\mathcal{V}_{t+1}(x_t, \xi_t) = \rho_{Q_{t+1}|\mathcal{F}_t}\left(V_{t+1}(x_t, \xi_{t+1})\right)$$

and $\rho_{Q_{t+1}|\mathcal{F}_t}$ is an ARSRM: the expectation (average) of a spectral risk measure (SRM) over a random parameter $s$ with probability law $Q_{t+1}$ conditioned on available information $\mathcal{F}_t$. The SRM is expressed as

$$\rho_{\sigma(s)}(\xi) = \int_0^1 F^{-1}_\xi(z)\, \sigma(z, s) dz$$

with $F^{-1}_\xi$ the quantile function of $\xi$ and $\sigma$ a risk spectrum shape function parameterized by $s$. When $Q_{t+1}$ is random or ambiguous, the risk assessment integrates over possible risk attitudes, capturing state-dependent and inconsistent preferences not representable by a single deterministic SRM.

2. Stochastic Dual Dynamic Programming (SDDP) Approach

To solve the multistage ARSRM (MARSRM) problem, SDDP is employed as the main computational engine. SDDP iteratively constructs lower and upper bounds on the cost-to-go functions at each stage, using:

• Forward pass: Generation of sample paths and candidate primal solutions by simulating scenario trees.
• Backward pass: For each sampled node at each stage, a risk-adjusted subproblem is solved. The dual of this subproblem yields subgradients (via Danskin's theorem) that are used to construct supporting hyperplanes ("cuts") for the stage value function $V_t$:

$$g_{t,i} + G_{t,i} x_{t-1} \preceq V_t(x_{t-1}, \xi_t)$$

Here, $g_{t,i}$ and $G_{t,i}$ are the intercept and subgradient, both computed from LP duals (notably, CVaR can be dualized as a linear program).

• Bounding procedure: At each iteration, upper bounds are produced by aggregating candidate decisions to form approximate feasible policies, while lower bounds are generated from the piecewise linearization of the cost-to-go functions.

Convergence is established under standard SDDP assumptions, and if the recourse and cost functions are convex (as is ensured by CVaR and spectral risk measure dualizations), the SDDP algorithm finds the optimal solution in a finite number of iterations.

3. Risk Allocation and State-Dependent Preferences

Risk allocation in MARSRM is achieved by distributing risk budgets across stages and scenarios according to the realized value of the stochastic risk parameter $s$ drawn from $Q_t$ at each stage. The recursive nature of the problem ensures that optimal policies are dynamically adjusted to current states and historical realizations, faithfully reflecting possible inconsistency or ambiguity in the decision maker's attitude toward risk.

When the distribution $Q$ of $s$ is not fully known (i.e., the risk preference is only partially observable or subject to ambiguity), the model admits a distributionally robust extension (DR-MARSRM). Here, at each stage, the risk assessment is computed as

$\sup_{Q \in \mathcal{Q}_t} \rho_Q(\cdot)$

with $\mathcal{Q}_t$ a moment-based ambiguity set. The dynamic program becomes a min-max formulation, and in the backward pass, the SDDP algorithm constructs multiple cuts per ambiguity scenario ("multi-cut" SDDP) to improve lower bound tightness. This robustification ensures policy safety across a family of possible risk spectrums.

4. Theoretical Properties and Convergence

The paper gives rigorous guarantees:

• Under moderate regularity assumptions, SDDP's sequence of lower–upper bound pairs is proven to converge to the optimal value, with finitely many cuts required for exact solution.
• The ARSRM operator is shown to be convex and piecewise linear in the cost functions when the risk spectrum $\sigma(z, s)$ admits a finite discrete representation or is approximated via CVaR components. This maintains tractability of the subproblems.
• Duality results ensure that all cuts generated are both valid and improve the piecewise linear lower approximations required for SDDP convergence.

5. Computational Schemes

Two computational schemes are detailed:

• For MARSRM (known $Q_t$ at each stage), a single-cut SDDP scheme is sufficient.
• For DR-MARSRM (ambiguous, moment-based $Q_t$), a multi-cut SDDP scheme must be used, introducing a cut for each $Q_t \in \mathcal{Q}_t$ scenario to account for all relevant risk envelope extremes.

Both methods rely on the fact that at each backward pass, the SDDP algorithm can efficiently compute supporting hyperplanes using dual formulations of CVaR (and spectral risk) subproblems.

6. Practical Implementation: Asset Allocation

The framework is applied to a multistage asset allocation problem with transaction costs. The DM's capital is allocated across financial assets whose returns are uncertain, and transaction costs are incurred when rebalancing the portfolio over time. MARSRM and DR-MARSRM are benchmarked against risk-neutral and standard risk-averse (CVaR) multistage stochastic programming models.

Key reported empirical results include:

• The dynamic allocation of risk, as allowed by MARSRM's randomization, improves portfolio return-versus-risk tradeoffs and adapts allocation to scenario-dependent attitudes.
• DR-MARSRM produces more conservative (robust) solutions, especially in the presence of ambiguity on the DM's risk preference distribution.
• All SDDP-based algorithms achieve convergence, empirically validating the finite convergence guarantee.

7. Significance and Future Directions

The MARSRM and DR-MARSRM frameworks generalize classical risk-averse and robust stochastic programming by:

• Encoding time-varying, state-dependent, or inconsistent risk preferences.
• Allowing for ambiguity in the very notion of risk aversion, not only in outcome distributions.
• Maintaining convexity and computational tractability via SDDP.

The conceptual extension to dynamically randomized spectral risk measures enables representation of real-world decision-making where risk attitude may shift unpredictably across time and scenarios. The use of SDDP with duality-based cut generation and multi-cut extensions ensures that practical, large-scale multistage problems with complex risk features are solvable to optimality. Potential areas for future work include tighter ambiguity sets, alternative randomization strategies for risk spectrums, and further integration of learning-based risk identification.