cobyqa.subsolvers.cauchy_geometry#
- cobyqa.subsolvers.cauchy_geometry(const, grad, curv, xl, xu, delta, debug)[source]#
Maximize approximately the absolute value of a quadratic function subject to bound constraints in a trust region.
This function solves approximately
\[\begin{split}\max_{s \in \R^n} \quad \bigg\lvert c + \transpose{g} s + \frac{1}{2} \transpose{s} H s \bigg\rvert \quad \text{s.t.} \quad \left\{ \begin{array}{l} \xl \le s \le \xu,\\ \lVert s \rVert \le \Delta, \end{array} \right.\end{split}\]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.
- curvcallable
Curvature of \(H\) along any vector.
curv(s) -> float
returns \(\transpose{s} H s\).
- xlnumpy.ndarray, shape (n,)
Lower bounds \(\xl\) as shown above.
- xunumpy.ndarray, shape (n,)
Upper bounds \(\xu\) 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 \(s\).
Notes
This function is described as the first alternative in Section 6.5 of [1]. It is assumed that the origin is feasible with respect to the bound constraints and that delta is finite and positive.
References
[1]T. M. Ragonneau. Model-Based Derivative-Free Optimization Methods and Software. PhD thesis, The Hong Kong Polytechnic University, Hong Kong, China, 2022.