Scenario tree construction driven by heuristic solutions of the optimization problem

Abstract

We present a new scenario generation process approach driven purely by the out-of-sample performance of a pool of solutions, obtained by some heuristic procedure. We formulate a loss function that measures the discrepancy between out-of-sample and in-sample (in-tree) performance of the solutions. By minimizing such a (usually non-linear, non-convex) loss function for a given number of scenarios, we receive an approximation of the underlying probability distribution with respect to the optimization problem. This approach is especially convenient in cases where the optimization problem is solvable only for a very limited number of scenarios, but an out-of-sample evaluation of the solution is reasonably fast. Another possible usage is the case of binary distributions, where classical scenario generation methods based on fitting the scenario tree and the underlying distribution do not work.

Appendices

Appendix A: Heuristic for loss function minimization

We provide a detailed description of the heuristic used in the example 2.3.1 for the minimization of the loss function. The method is based on sub-gradients of the loss function with respect to the decision variables, denoted $g_{p}$ and $g_{w}$. In order to use them, we derive¹³$\frac{dL}{df}$—the gradient of the loss function with respect to the in-sample evaluation function, further $\frac{df}{de}$—the gradient of the in-sample evaluation function with respect to the exceeded capacity, and so on. By doing so, we get a chain of simple operations (square, multiplication, addition, $\max ()$,¹⁴ etc.), for which we have standard differentiation rules. By simple applications of the chain rule we get the desired sub-gradients $g_{p}$ and $g_{w}$.

Let us note that in other optimization problems it might not be so straightforward to derive sub-gradients as in our case. More complicated functions might come into play. Thus it could require more effort in analytical derivation or the use of numerical sub-gradients, which should always be available, at least in in principle. Or it is possible to choose a different method from non-linear optimization theory, that is not based on gradients [see some textbook on non-linear optimization, for instance Hendrix and Tóth (2010), Boyd and Vandenberghe (2004)].

With the sub-gradient method, we take a small step in the direction of the negative sub-gradient, that is, in the direction of the steepest descent, at every iteration. Such an approach converges to a local minimum, which is also a global minimum in the case of convex minimization. However, not in our case, so we add two features to enhance exploration of the search space in order to avoid termination of our procedure at some low-quality local minimum.

The first feature is to use multiple starts of the procedure from different initial points. The second feature is the recognition and replacement of “useless” scenarios. We recognize these scenarios by evaluating their impact on the loss function. The impact is defined as the change of the loss function values if we remove a particular scenario from the scenario tree. If the change is very small, it means that the particular scenario is not very useful, and we replace the scenario by a new one (chosen randomly). The heuristic is summarized in Algorithm 1.

The parameters p and w are updated (rows 5 and 6 in Algorithm 1) by small steps in the direction of the steepest descent. The step sizes, denoted $\alpha ^p$ and $\alpha ^w$, are decreased with the increasing number of iterations. They are hyper-parameters that enter the procedure and need to be carefully set (too small steps lead to slow convergence, too large to oscillation or divergence). We set these steps based on some trial tests.

The routine of scenario usefulness assessment is computationally expensive. It would require computation of the loss function K times at every iteration. To avoid it, we run a pre-test, where we check the $\infty$-norm of the sub-gradient related to each scenario (row 9), which allows us to break the test once we find one of its element greater than $\epsilon ^w$. Only if the sub-gradient is small, do we proceed to the evaluation of the impact on the loss function.

Initialization and replacement of scenarios is performed by a random draw from the original data ${\hat{w}}$, weights p are randomly set. Algorithm 1 is just a pseudo-code, not the most efficient implementation. Obviously, storing all $L_{jm}$ is not necessary, since at any time, we can store just two best values—a local one for the j cycle and a global one for the m cycle. We can also compute the value of the loss function while evaluating the sub-gradients of the function.

1.1 Note

We do not claim that this is the most effective heuristic for the problem. Most likely it is not. It is based on a simple sub-gradient method. If the main focus was on developing the most efficient algorithm for this task, it would be possible to build it on more sophisticated algorithms for non-linear optimization, such as the adaptive gradient method, possibly with momentum, or methods based on the second sub-derivative.

We believe this can still serve as an inspiration for developing more efficient algorithms, if needed. We pointed out some issues and suggestions how to overcome them. But for some applications, the presented algorithm is “good enough”, as it was in our case. It is important to realize that even if we could guarantee the global optimum of the loss function, the whole framework would still be a heuristic in the sense that there are no guarantees relative to other feasible solutions that are not included in the pool.

Heuristic for (in)feasibility classification

The main issue when considering (in)feasibility is the minimization of the loss function (15). The problem is that we cannot utilize the sub-gradients of the function $u(x, {\mathcal {T}})$. The function is not continuous and returns only two values, that means its derivative is 0 (if it is defined). One needs to either use methods for non-linear optimization that do not utilize derivatives or approximate the function u with some differentiable function. The latter approach is used for the computational test in Sect. 2.5.

In our computational test, the classification of (in)feasibility, i.e., the approximation of $u(x, {\mathcal {T}})$, is performed by a simple neural network model with one hidden layer, sigmoid function as the activation function in both layers and the sum of squares as the measurement of the error. As input we use the vector $e_s$ scaled to (0, 1). The neural network is trained on the training pool of heuristically obtained solutions. This simple model works well in our case and correctly classifies (in)feasibility of solutions.

The main advantage of this approach is that the neural network can back-propagate the sub-gradients of error (miss-classified solutions) via the weights of the neural network to the sub-gradient of the $e_s$ vector and further to scenarios $w_{si}$. Thus, we can, in principal, use Algorithm 1 to find the scenario tree.