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An interior-point method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search method that computes steps by factoring the primal-dual equations and a trust region method that uses a conjugate gradient iteration. Steps computed by direct factorization are always tried first, but if they are deemed ineffective, a trust region iteration that guarantees progress toward stationarity is invoked. To demonstrate its effectiveness, the algorithm is implemented in the Knitro [6,28] software package and is extensively tested on a wide selection of test problems.

We propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for adjusting the trust region radius that avoids undesirable directions and exploits a new strategy to prevent sudden increments of objective function values in nonmonotone trust region techniques. Global convergence of this algorithm is investigated under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the proposed algorithm in solving a collection of unconstrained optimization problems from the CUTEst package.

To understand the trust-region approach to optimization, consider the unconstrained minimization problem, minimize f(x), where the function takes vector arguments and returns scalars. Suppose that the current point is x in n-space and you want to improve by moving to a point with a lower function value. To do so, the algorithm approximates f with a simpler function q, which reasonably reflects the behavior of function f in a neighborhood N around the point x. This neighborhood is the trust region. The solver computes a trial step s by minimizing (or approximately minimizing) over N. The trust-region subproblem is

The key questions in defining a specific trust-region approach to minimizing f(x) are how to choose and compute the approximation q (defined at the current point x), how to choose and modify the trust region N, and how accurately to solve the trust-region subproblem.

In the standard trust-region method ([48]), the quadratic approximation q is defined by the first two terms of the Taylor approximation to F at x. The neighborhood N is usually spherical or ellipsoidal in shape. Mathematically, the trust-region subproblem is typically stated

Such an algorithm provides an accurate solution to Equation 1. However, this requires time proportional to several factorizations of H. Therefore, trust-region problems require a different approach. Several approximation and heuristic strategies, based on Equation 1, have been proposed in the literature ([42] and [50]). Optimization Toolbox solvers follow an approximation approach that restricts the trust-region subproblem to a two-dimensional subspace S ([39] and [42]). After the solver computes the subspace S, the work to solve Equation 1 is trivial because, in the subspace, the problem is only two-dimensional. The dominant work now shifts to the determination of the subspace.

Using trust-region techniques (introduced in Trust-Region Methods for Nonlinear Minimization) handles the case when J(xk) is singular and improves robustness when the starting point is far from the solution. To use a trust-region strategy, you need a merit function to decide if xk + 1 is better or worse than xk. A possible choice is

The trust-region-dogleg algorithm is efficient because it requires only one linear solve per iteration (for the computation of the Gauss-Newton step). Additionally, the algorithm can be more robust than using the Gauss-Newton method with a line search.

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From the user side, there were 13,096,468 downloads of SciPy from the Python Packaging Index (PyPI)95 and 5,776,017 via the default channel of the conda ( -package-data) package manager during the year 2017. These numbers establish a lower bound on the total number of downloads by users given that PyPI and conda are only two of several popular methods for installing SciPy. The SciPy website ( ), which has been the default citation in the absence of a peer-reviewed paper, has been cited over 3,000 times ( =2086009121748039507). Some of the most prominent uses of or demonstrations of credibility for SciPy include the LIGO-Virgo scientific collaboration that lead to the observation of gravitational waves96, the fact that SciPy is shipped directly with macOS and in the Intel distribution for Python97, and that SciPy is used by 47% of all machine learning projects on GitHub ( -01-24-the-state-of-the-octoverse-machine-learning/). 1e1e36bf2d