Cone Model Quasi-Newton Trust Region Algorithm for Unconstrained Optimization Problems

The Cone Model Quasi-Newton Trust Region Algorithm for unconstrained optimization problems is an efficient optimization approach that integrates cone models, quasi-Newton methods, and trust region strategies. Compared to traditional trust region algorithms based on quadratic models, the cone model offers more flexible local approximation capabilities, particularly suitable for objective functions with strong nonlinear characteristics.

Traditional trust region algorithms typically use quadratic models to approximate objective functions, but these models may sometimes fail to accurately capture local function behavior. The cone model enhances adaptability by introducing nonlinear terms, enabling better fitting of complex objective function shapes. The quasi-Newton method reduces computational costs by approximating the Hessian matrix, avoiding the expense of direct second-derivative calculations. In code implementation, this typically involves updating inverse Hessian approximations using formulas like BFGS or DFP.

The core concept of this algorithm involves constructing a cone model as a local approximation of the objective function during each iteration, combined with quasi-Newton method updates for model parameters. The trust region strategy controls step size合理性 to ensure balance between global convergence and local convergence speed. Through dynamic adjustment of the trust region radius, the algorithm can收缩 the region when model accuracy is insufficient and expand the search range when the model proves reliable. A typical implementation would include a trust region subproblem solver and radius update logic based on the actual-to-predicted reduction ratio.

The Cone Model Quasi-Newton Trust Region Algorithm demonstrates excellent performance in solving high-dimensional non-convex optimization problems, particularly when the objective function exhibits sharp curvature changes or when Hessian matrices are difficult to compute accurately. Its advantage lies in providing better fidelity to true function behavior compared to pure quadratic models, while maintaining the computational efficiency of quasi-Newton methods. Such algorithms have potential applications in machine learning, engineering optimization, and financial modeling domains, where they can be implemented using optimization frameworks like MATLAB's fminunc with custom model specifications.