IntroductionA principal advantage of Newton’s method is that it converges rapidly when the current estimate of the variables is close to the solution. However, it also has disadvantages, and overcoming these disadvantages has led to many other techniques.In particular, Newton’s method canfail to convergeconverge to a point that is not a stationary pointHowever, this kind of problem can be overco..

Guaranteeing ConvergenceThe auxiliary techniques that are used to guarantee convergence attempt to rein in the optimization method when it is in danger of getting out of control, and they also try to avoid intervening when the optimization method is performing effectively.The term globalization strategy is used to distinguish the method used for the new estimate of the solution from the method f..

Newton’s Methods for MinimizationThe basic idea can be derived using the first-order necessary optimality condition∇f(x)=0\nabla f(x) = 0∇f(x)=0Alternatively, consider second-order Taylor series approximationf(x+p)≈f(x)+∇f(x)p+12pT∇2f(x)pf(x+p) \approx f(x) + \nabla f(x)p + \frac{1}{2}p^T\nabla^2 f(x)pf(x+p)≈f(x)+∇f(x)p+21​pT∇2f(x)pLet q(p)=f(x)+∇f(x)p+12pT∇2f(x)pq(p) = f(x) + \nabla f(x)p + \fr..

Optimality Conditions: PreliminariesLocal minima and maxima have one thing in common∇f(x)=0\nabla f(x) = 0∇f(x)=0First-order Necessary ConditionFrom now on fff is differentiable and that its first and second derivatives are continuous for every x∈Xx\in Xx∈X where XXX is a domain of fffIf x∗x^*x∗ is a local minimum, then ∇f(x∗)=0\nabla f(x^*) = 0∇f(x∗)=0Not a sufficient condition, since it c..

A Generic Optimization Modelmin⁡f(x1,x2,…,xn)\min f(x_1, x_2, \dots, x_n)minf(x1​,x2​,…,xn​)subject to\text{subject to}subject tog1(x1,x2,…,xn)≤b1g2(x1,x2,…,xn)≤b2⋮gm(x1,x2,…,xn)≤bm\begin{align}g_1(x_1, x_2, \dots, x_n) \le b_1 \\ g_2(x_1, x_2, \dots, x_n) \le b_2 \\ \vdots \\ g_m(x_1, x_2, \dots, x_n) \le b_m \end{align}g1​(x1​,x2​,…,xn​)≤b1​g2​(x1​,x2​,…,xn​)≤b2​⋮gm​(x1​,x2​,…,xn​)≤bm​​​In co..