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MotivationSo far, all methods considered require at least first-order derivativeBut what ifThe function is not differentiable everywhere?It is very time-consuming to perform derivative calculation?That is why Derivative-Free methods came.→ search methods without the use of any derivative informationCompass SearchIntuition : Try a few points around the current solutionBetter point found : move, a..
7. Derivative-Free MethodsMotivationSo far, all methods considered require at least first-order derivativeBut what ifThe function is not differentiable everywhere?It is very time-consuming to perform derivative calculation?That is why Derivative-Free methods came.→ search methods without the use of any derivative informationCompass SearchIntuition : Try a few points around the current solutionBetter point found : move, a..
2023.12.13 -
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..
6. Least-Squares Data FittingWhat 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..
2023.11.27 -
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..
5. Methods for Unconstrained OptimizationIntroductionA 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..
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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..
4. Guaranteeing ConvergenceGuaranteeing 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..
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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+21pT∇2f(x)pLet q(p)=f(x)+∇f(x)p+12pT∇2f(x)pq(p) = f(x) + \nabla f(x)p + \fr..
3. Guaranteeing DescentNewton’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+21pT∇2f(x)pLet q(p)=f(x)+∇f(x)p+12pT∇2f(x)pq(p) = f(x) + \nabla f(x)p + \fr..
2023.11.27 -
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..
2. Optimality Conditions, Convexity, Newton’s Method for EquationsOptimality 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..
2023.11.27 -
A Generic Optimization Modelminf(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)≤b1g2(x1,x2,…,xn)≤b2⋮gm(x1,x2,…,xn)≤bmIn co..
1. BasicsA Generic Optimization Modelminf(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)≤b1g2(x1,x2,…,xn)≤b2⋮gm(x1,x2,…,xn)≤bmIn co..
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Last Update : 2023/11/4
Principle of Mathematical Analysis SolutionLast Update : 2023/11/4
2023.11.05