Practical Precautionary Resource Management Using Robust Optimization

Abstract

Uncertainties inherent in fisheries motivate a precautionary approach to management, meaning an approach specifically intended to avoid bad outcomes. Stochastic dynamic optimization models, which have been in the fisheries literature for decades, provide a framework for decision making when uncertain outcomes have known probabilities. However, most such models incorporate population dynamics models for which the parameters are assumed known. In this paper, we apply a robust optimization approach to capture a form of uncertainty nearly universal in fisheries, uncertainty regarding the values of model parameters. Our approach, developed by Nilim and El Ghaoui (Oper Res 53(5):780–798, 2005), establishes bounds on parameter values based on the available data and the degree of precaution that the decision maker chooses. To demonstrate the applicability of the method to fisheries management problems, we use a simple example, the Skeena River sockeye salmon fishery. We show that robust optimization offers a structured and computationally tractable approach to formulating precautionary harvest policies. Moreover, as better information about the resource becomes available, less conservative management is possible without reducing the level of precaution.

Appendix: Detailed Explanation of Solution Algorithm

Nilim and El Ghaoui use the one-to-one mapping between the likelihood function and the approximate probability level to specify their robust optimization problem. Hence, rather than using the likelihood ratio directly, they use the corresponding value of a parameter, β, which is defined such that for a given value of α, $\Pr \left( {p_{i}^{a} } \right) \ge \alpha \Leftrightarrow \ln \left( {L\left( {p_{i}^{a} } \right)} \right) \ge \beta$ with $p_{ij}^{a} \ge 0{\text{ and }}\sum\nolimits_{j} {p_{ij}^{a} = 1}$.

The Lagrangian of the inner problem, (4), can be written as

$${\mathbf{\mathcal{L}}}\left( {V,\zeta ,\mu ,\lambda } \right) = p^{T} V + \zeta^{T} p + \mu \left( {1 - p^{T} {\mathbf{1}}} \right) + \lambda \left( {k^{T} \ln \left( p \right) - \beta } \right),$$

(8)

where we suppress the subscript i and the superscript a so that V, k, and p refer to vectors. There is a separate Lagrangian of this form for each action, a, and starting point in the state space, i = 1,…,n. The Lagrange multipliers are $\zeta$, the shadow price on the inequality constraints; $\mu$, the multiplier on the constraint requiring that the probabilities sum to one; and $\lambda$, the multiplier on the constraint that defines the bounds.

Using the first-order condition of (8) with respect to p _j gives

$$V_{j} + \zeta_{j} - \mu + \lambda \left( {\frac{{k_{j} }}{{p_{j} }}} \right) = 0\,\,{\text{or}}\,\,p_{j} = \left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} - \zeta_{j} }}} \right)$$

(9)

Because of the log function, p _j  > 0 so $\zeta_{j} = 0$ for all j, leaving only two unknown parameters, λ and μ.

Nilim and El Ghaoui solve for λ and μ by first replacing p _j in the Lagrangian (8) using (9),

$${\mathbf{\mathcal{L}}}\left( {V,\zeta ,\mu ,\lambda } \right) = \sum\limits_{j} {\left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)V_{j} } + \mu \left( {1 - \sum\limits_{j} {\left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)} } \right) + \lambda \left( {\sum\limits_{j} {\ln \left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)} k_{j} - \beta } \right).$$

This can be simplified as follows:

$${\mathbf{\mathcal{L}}}\left( {V,\zeta ,\mu ,\lambda } \right) = \mu - \lambda \beta + \sum\limits_{j} {\left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)V_{j} } - \mu \sum\limits_{j} {\left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)} + \lambda \sum\limits_{j} {\ln \left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)} k_{j}$$

$${\mathbf{\mathcal{L}}}\left( {V,\zeta ,\mu ,\lambda } \right) = \mu - \lambda \beta - \sum\limits_{j} {\lambda k_{j} } + \lambda \sum\limits_{j} {\ln \left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)} k_{j} .$$

Letting $K = \sum\nolimits_{j} {k_{j} }$, this becomes

$${\mathbf{\mathcal{L}}}\left( {V,\mu ,\lambda } \right) = \mu + \lambda \left[ {\sum\limits_{j} {\ln \left( {\frac{{\lambda k_{j} }}{{\mu - V_{j} }}} \right)} k_{j} - \beta - K} \right] .$$

(10)

Since the objective of the inner problem is to minimize over p, the solution of the optimization problem is found by maximizing (10) over the shadow prices. Taking the first-order conditions with respect to μ and setting equal to zero,⁸ we obtain

$$\lambda = \left[ {\sum\limits_{j} {\left( {\frac{{k_{j} }}{{\mu - V_{j} }}} \right)} } \right]^{ - 1} .$$

(11)

Since (11) can be substituted into (10), each inner problem can be reduced to finding a single parameter, μ, after which all other parameters can be found algebraically. As long as $V\left( {x_{i} } \right) \ne V\left( {x_{j} } \right)$ for at least one pair, i, j, the inner problem’s solution will occur along the edge of the allowed likelihood range as defined by (3). At the optimal value of μ, (3) holds with an equality or $\ln \left( {L\left( {p_{i}^{a} } \right)} \right) = \beta$. While NEG propose a bisection algorithm to find μ, in our experience, the log likelihood function is monotonic in μ, meaning that a simple root-finding algorithm (Newton’s method) will quickly and accurately find the correct value for μ.⁹