What is the Least square method?The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of nnn point (xi,yi)(x_i, y_i)(xi​,yi​) where xix_ixi​ is an independent variable and yiy_iyi​ is a dependent variable whose value is found by observation. The model function has the form f(x,β)f(x, \beta)f(x,β), where mmm adjustable para..

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..

Natural computationAlgorithms derived from observations of nature phenomenaSimulationsto learn more about these phenomenato learn new ways to solve computational problemsNatural computation and Machine learningWhat is learning?The ability to improve over time, based on experienceWhy? : Solutions to problems are not always programmableExamplesHandwritten character recognitionAdaptive control of p..

Strong vs Weak AIThe Strong AI hypothesis is the philosophical position that a computer program that causes a machine to behave exactly like a human being would also give the machine subjective conscious experience and a mindOn the other hand, Weak AI is the philosophical position that an AI that appears to behave exactly like a human being is only a simulation of human cognitive function, and i..