midterm: 2.1b-g sketched

Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.md

@@ -148,7 +148,45 @@ monotonically decreases.
 
 ## Point 1
 
-TBD
+### (a) For which kind of minimization problems can the trust region method be used? What are the assumptions on the objective function?
+
+### (b) Write down the quadratic model around a current iterate xk and explain the meaning of each term.
+
+$$m(p) = f + g^T p + \frac12 p^T B p \;\; \text{ s.t. } \|p\| < \Delta$$
+
+$\Delta$ is the trust region radius.
+$p$ is the trust region step.
+$g$ is the gradient at the current iterate $x_k$.
+$B$ is the hessian at the current iterate $x_k$.
+
+### (c) What is the role of the trust region radius?
+
+Limit confidence of model. I.e. it makes the model refrain from taking wide
+quadratic steps when the quadratic model is considerably different from the real
+objective function.
+
+### (d) Explain Cauchy point, sufficient decrease and Dogleg method, and the connection between them.
+
+Cauchy point provides sufficient decrease, but makes method like linear method.
+
+Dogleg method allows for mixing purely linear iteration and purely quadratic one
+along the "dogleg" path picking the furthest point inside or on the edge of the
+region.
+
+Dogleg uses cauchy point if the trust region does not allow for a proper dogleg
+step since it is too slow.
+
+Cauchy provides linear convergence and dogleg superlinear.
+
+### (e) Write down the trust region ratio and explain its meaning.
+
+$$\rho_k = \frac{f(x_k) - f(x_k + p_k)}{m_k(0) - m_k(p_k)}$$
+
+Real decrease over predicted decrease
+
+Test "goodness" of model.
+
+### (f) Does the energy decrease monotonically when Trust Region method is employed? Justify your answer.
 
 ## Point 2