cobyqa.subproblems.bound_constrained_cauchy_step

subproblems.bound_constrained_cauchy_step(const, grad, hess_prod, xl, xu, delta, debug)

Maximize approximately the absolute value of a quadratic function subject to bound constraints in a trust region.

This function solves approximately

\[\begin{split}\begin{aligned} \max_{d \in \R^n} & \quad \abs[\bigg]{c + g^{\T}d + \frac{1}{2} d^{\T}Hd}\\ \text{s.t.} & \quad l \le d \le u,\\ & \quad \norm{d} \le \Delta, \end{aligned}\end{split}\]

by maximizing the objective function along the constrained Cauchy direction.

Parameters:

constfloat

Constant \(c\) as shown above.

gradnumpy.ndarray, shape (n,)

Gradient \(g\) as shown above.

hess_prodcallable

Product of the Hessian matrix \(H\) with any vector.

hess_prod(d) -> numpy.ndarray, shape (n,)

returns the product \(Hd\).

xlnumpy.ndarray, shape (n,)

Lower bounds \(l\) as shown above.

xunumpy.ndarray, shape (n,)

Upper bounds \(u\) as shown above.

deltafloat

Trust-region radius \(\Delta\) as shown above.

debugbool

Whether to make debugging tests during the execution.

Returns:

numpy.ndarray, shape (n,)

Approximate solution \(d\).

Notes

This function is described as the first alternative in p. 115 of [1]. It is assumed that the origin is feasible with respect to the bound constraints xl and xu, and that delta is finite and positive.

References

[1]

T. M. Ragonneau. “Model-Based Derivative-Free Optimization Methods and Software.” Ph.D. thesis. Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University, 2022.