Stochastic Lipschitz Dynamic Programming

and Stochastic augmented Dual Dynamic Programming for Lipschitz problems

This package contains an extension to SDDP.jl-v0 for using SLDP methods to solve Multistage Stochatic MIPs. It is largely inspired in SDDiP, and our Lagrangian submodule is a very reduced version of the original one.

It requires the user to provide an upper bound $\rho_n$ for the Lipschitz constant of each node. This has different meanings according to the cutting method.

More documentation is forthcoming.

Cuts

At present, we use either

• reverse 1-norm cuts
• augmented Lagrangian dual cuts

Reverse 1-norm cuts

Here, the user must provide a valid upper bound for the Lipschitz constant (relative to the 1-norm at the domain). This will be used to construct the cut

$$ \theta \geq Q_n^k(x^k) - \rho_n |x - x^k |. $$

Augmented Lagrangian dual cuts

Here, the user must provide, besides a Lipschitz bound $\rho_n$, a policy for increasing $\rho_n$. This is given by the coefficients $a_n$ and $b_n$ for a linear function, which is then saturated between $0$ and $\rho_n$. The effective augmenting parameter for node $n$ at iteration $k$ will then be $p_n = \text{clip}(a_n k + b_n, 0, \rho_n)$.

Then, we obtain a Benders multiplier $\pi$ by solving the LP relaxation, and solve the augmented Lagrangian problem corresponding to $(\pi, p_n)$. The cut is therefore

$$ \theta \geq Q_n^{k, \text{AL}}(x^k) + \pi^\top(x - x^k) - p_n |x - x^k |. $$

This cut, which we call Strenghtened augmented Benders cut, is valid by construction, so the user-provided $\rho_n$ is only an upper bound to what the algorithm will try. If it is too small, it will not close the duality gap.

Ideally, one would solve for the optimal Lagrange multiplier, given the augmenting term $p_n |x - x^k|$. But another method for producing tight cuts is finding the optimal (lowest) augmenting parameter $\rho$. For the moment, a simple bisection algorithm for $\rho \in (0, \rho_n)$ is provided. Tolerances are still hardcoded.