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This function solves approximately .. math:: \min_{s \in \mathbb{R}^n} \quad g^{\mathsf{T}} s + \frac{1}{2} s^{\mathsf{T}} H s \quad \text{s.t.} \quad \left\{ \begin{array}{l} l \le s \le u\\ \lVert s \rVert \le \Delta, \end{array} \right. using an active-set variation of the truncated conjugate gradient method. Parameters ---------- grad : `numpy.ndarray`, shape (n,) Gradient :math:`g` as shown above. hess_prod : callable Product of the Hessian matrix :math:`H` with any vector. ``hess_prod(s) -> `numpy.ndarray`, shape (n,)`` returns the product :math:`H s`. xl : `numpy.ndarray`, shape (n,) Lower bounds :math:`l` as shown above. xu : `numpy.ndarray`, shape (n,) Upper bounds :math:`u` as shown above. delta : float Trust-region radius :math:`\Delta` as shown above. debug : bool Whether to make debugging tests during the execution. Returns ------- `numpy.ndarray`, shape (n,) Approximate solution :math:`s`. Other Parameters ---------------- improve_tcg : bool, optional If True, a solution generated by the truncated conjugate gradient method that is on the boundary of the trust region is improved by moving around the trust-region boundary on the two-dimensional space spanned by the solution and the gradient of the quadratic function at the solution (default is True). Notes ----- This function implements Algorithm 6.2 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, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2022. 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This function solves approximately .. math:: \min_{s \in \mathbb{R}^n} \quad g^{\mathsf{T}} s + \frac{1}{2} s^{\mathsf{T}} H s \quad \text{s.t.} \quad \left\{ \begin{array}{l} l \le s \le u,\\ A_{\scriptscriptstyle I} s \le b_{\scriptscriptstyle I},\\ A_{\scriptscriptstyle E} s = 0,\\ \lVert s \rVert \le \Delta, \end{array} \right. using an active-set variation of the truncated conjugate gradient method. Parameters ---------- grad : `numpy.ndarray`, shape (n,) Gradient :math:`g` as shown above. hess_prod : callable Product of the Hessian matrix :math:`H` with any vector. ``hess_prod(s) -> `numpy.ndarray`, shape (n,)`` returns the product :math:`H s`. xl : `numpy.ndarray`, shape (n,) Lower bounds :math:`l` as shown above. xu : `numpy.ndarray`, shape (n,) Upper bounds :math:`u` as shown above. aub : `numpy.ndarray`, shape (m_linear_ub, n) Coefficient matrix :math:`A_{\scriptscriptstyle I}` as shown above. bub : `numpy.ndarray`, shape (m_linear_ub,) Right-hand side :math:`b_{\scriptscriptstyle I}` as shown above. aeq : `numpy.ndarray`, shape (m_linear_eq, n) Coefficient matrix :math:`A_{\scriptscriptstyle E}` as shown above. delta : float Trust-region radius :math:`\Delta` as shown above. debug : bool Whether to make debugging tests during the execution. Returns ------- `numpy.ndarray`, shape (n,) Approximate solution :math:`s`. Other Parameters ---------------- improve_tcg : bool, optional If True, a solution generated by the truncated conjugate gradient method that is on the boundary of the trust region is improved by moving around the trust-region boundary on the two-dimensional space spanned by the solution and the gradient of the quadratic function at the solution (default is True). Notes ----- This function implements Algorithm 6.3 of [1]_. It is assumed that the origin is feasible with respect to the bound and linear constraints, and that `delta` is finite and positive. References ---------- .. [1] T. M. Ragonneau. *Model-Based Derivative-Free Optimization Methods and Software*. PhD thesis, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2022. 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This function solves approximately .. math:: \min_{s \in \mathbb{R}^n} \quad \frac{1}{2} \big( \lVert \max \{ A_{\scriptscriptstyle I} s - b_{\scriptscriptstyle I}, 0 \} \rVert^2 + \lVert A_{\scriptscriptstyle E} s - b_{\scriptscriptstyle E} \rVert^2 \big) \quad \text{s.t.} \quad \left\{ \begin{array}{l} l \le s \le u,\\ \lVert s \rVert \le \Delta, \end{array} \right. using a variation of the truncated conjugate gradient method. Parameters ---------- aub : `numpy.ndarray`, shape (m_linear_ub, n) Matrix :math:`A_{\scriptscriptstyle I}` as shown above. bub : `numpy.ndarray`, shape (m_linear_ub,) Vector :math:`b_{\scriptscriptstyle I}` as shown above. aeq : `numpy.ndarray`, shape (m_linear_eq, n) Matrix :math:`A_{\scriptscriptstyle E}` as shown above. beq : `numpy.ndarray`, shape (m_linear_eq,) Vector :math:`b_{\scriptscriptstyle E}` as shown above. xl : `numpy.ndarray`, shape (n,) Lower bounds :math:`l` as shown above. xu : `numpy.ndarray`, shape (n,) Upper bounds :math:`u` as shown above. delta : float Trust-region radius :math:`\Delta` as shown above. debug : bool Whether to make debugging tests during the execution. Returns ------- `numpy.ndarray`, shape (n,) Approximate solution :math:`s`. Other Parameters ---------------- improve_tcg : bool, optional If True, a solution generated by the truncated conjugate gradient method that is on the boundary of the trust region is improved by moving around the trust-region boundary on the two-dimensional space spanned by the solution and the gradient of the quadratic function at the solution (default is True). Notes ----- This function implements Algorithm 6.4 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, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2022. 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