# cobyqa.subproblems.bound_constrained_xpt_step#

subproblems.bound_constrained_xpt_step(const, grad, hess_prod, xpt, xl, xu, delta, debug)[source]#

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

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 straight lines through the origin and the rows in xpt.

Parameters:
constfloat

Constant $$c$$ as shown above.

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 $$Hd$$.

xptnumpy.ndarray, shape (npt, n)

Points defining the straight lines as shown above.

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 second 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.