The Oracles

We propose two classes for the computation of the first order oracle associated to our DC formulation. Both oracles take as an input an instance of Problem and a few paramters such as the safety probability threshold.

The classes differ in that the second one support numba-compliant problem classes. Given the numba option provided when instantiating an Optimizer, first or second class will be instantiated for solving.

class taco.Oracle(problem, p, pen1, pen2, rho)

Base class that instantiates a first-order oracle for the DC objective of the penalized chance constaint

For two input oracles \(f\) and \(g\) given through their function and gradients, and a sampled dataset from a random variable \(\xi\), this class is an interface to compute the value and the gradient of the function \(x, \eta \mapsto f(x) + \mu \max(\eta,0) + \lambda \left(G(x,\eta) - \text{CVaR}_p(g(x,\xi))\right)\) where \(G(x, \eta) = \eta + \frac{1}{1-p} \mathbb{E}[\max(g(x, \xi) - \eta]\)

Parameters

• problem – Instance of Problem

• p (np.float64) – Real number in [0,1]. Safety probability level

• pen1 (np.float64) – Penalization parameter, must be positive.

• pen2 (np.float64) – Penalization parameter, must be positive.

• rho (np.float64) – Smoothing parameter for the superquantile, must be positive.

f(x)

Value of the DC objective

f1(x)

Value of the convex part of the DC objective \(x, \eta \mapsto f(x) + \mu \max(\eta,0) + \lambda G(x,\eta)\)

f2(x)

Value of the concave part of the DC objective \(x, \eta \mapsto - {CVaR}_p(g(x,{\xi}))\)

g1(x)

Gradient of the convex part of the DC objective \(x, \eta \mapsto f(x) + \mu \max(\eta,0) + \lambda G(x,\eta)\)

g2(x)

Gradient of the concave part of the DC objective \(x, \eta \mapsto - {CVaR}_p(g(x,{\xi}))\)

class taco.FastOracle(*args, **kwargs)

Numba version of the class Oracle.

FastOracle works with numba in full no-python mode. It must take as an input an instance of problem that is a jitclass.

Parameters

• problem – Instance of Problem. Must be a numba jitclass.

• p (np.float64) – Real number in [0,1]. Safety probability level

• pen1 (np.float64) – Penalization parameter, must be positive.

• pen2 (np.float64) – Penalization parameter, must be positive.

• rho (np.float64) – Smoothing parameter for the superquantile, must be positive.