TBD Last update : 2024/03/30

Norm approximationBasic norm approximation problemAssume that the columns of AAA are independent.minimize ∥Ax−b∥\text{minimize }\|Ax - b\| minimize ∥Ax−b∥where A∈Rm×nA\in \R^{m\times n}A∈Rm×n with m≥nm\ge nm≥n, ∥⋅∥\|\cdot\|∥⋅∥ is a norm of Rm\R^mRmGeometric interpretationGeometrically, the solution x∗x^*x∗ is the point such that Ax∗∈R(A)Ax^*\in \mathcal R(A)Ax∗∈R(A) that closest to bbb. ..

LagrangianThe basic idea in Lagrangian duality is to take the constraints into account by augmenting the objective function with a weighted sum of the constraint functions.💡By using the Lagrangian, we can get a hint to solve the original problem which is not convex nor easy to solve.Standard form problem (not necessary convex)minimize f0(x)\text{minimize }f_0(x)minimize f0​(x)subject tofi(x)≤0i=..

Optimization problem in standard formminimize f0(x)\text{minimize } f_0(x)minimize f0​(x)subject tofi(x)≤0,  i=1,…,mhi(x)=0,  i=1,…,pf_i(x) \le 0, \; i = 1, \dots, m \\ h_i(x) = 0, \; i = 1, \dots, pfi​(x)≤0,i=1,…,mhi​(x)=0,i=1,…,px∈Rnx\in \R^nx∈Rn is the optimization variablef0:Rn→Rf_0:\R^n\to \Rf0​:Rn→R is the objective or cost functionfi:Rn→R,  i=1,…,mf_i:\R^n\to \R, \; i = 1, \dots, mfi​:R..

Convex functionf:Rn→Rf:\R^n \to \Rf:Rn→R is convex if dom f\text{dom }f dom f is a convex set andf(θx+(1−θ)y)≤θf(x)+(1−θ)f(y)f(\theta x + (1 - \theta)y ) \le \theta f(x) + (1-\theta)f(y)f(θx+(1−θ)y)≤θf(x)+(1−θ)f(y)for all x,y∈dom f,0≤θ≤1x, y\in \text{dom }f, 0 \le\theta \le 1x,y∈dom f,0≤θ≤1fff is strictly convex if dom f\text{dom }fdom f is convex andf(θx+(1−θ)y)θf(x)+(1−θ)f(y)f(\theta x +..

Affine setLine : all points through x1,x2x_1, x_2x1​,x2​x=θx1+(1−θ)x2x = \theta x_1 + (1 - \theta) x_2x=θx1​+(1−θ)x2​where θ∈R\theta\in \Rθ∈RThis idea can be generalized to more than two points.θ1x1+θ2x2+⋯+θkxk\theta_1x_1 + \theta_2 x_2 + \cdots + \theta_kx_kθ1​x1​+θ2​x2​+⋯+θk​xk​where θ1+⋯+θk=1\theta_1 + \cdots + \theta_k = 1θ1​+⋯+θk​=1We refer to a point can be expressed as the following fo..

Mathematical optimizationmin⁡xfo(x)s.t fi(x)≤bi, i=1,…,m\begin{align}\min_x & f_o(x) \\ \text{s.t }& f_i(x) \le b_i, \ i = 1, \dots, m\end{align}xmin​s.t ​fo​(x)fi​(x)≤bi​, i=1,…,m​​wherex=(x1,…,xn)x = (x_1, \dots, x_n)x=(x1​,…,xn​): optimization variablesfo:Rn→Rf_o : R^n \to Rfo​:Rn→R: objective function (a.k.a the function we want to minimize)💡In deep learning or machine learning perspective..

Creating and Dropping usersTo create a user in MySQLUsername user can be followed by @ and the IP or the hostname from which the user is allowed to log in. Example:To drop a user: DROP USER userAuthorizationcontrol of what users can do in the database what they can see, create, modify, delete etcExercise 1What privileges are needed to be able to execute the following query?Answer:UPDATE (graduat..