Finite

In optimization, instead of directly computing the inverse of the Hessian in each iteration, the Hessian itself can be updated. There are various methods to achieve this, but in this post, I'll explain one of the most commonly used methods the Broyden-Fletcher-Goldfarb-Shanno(BFGS) method. First, estimate the initial design ${x^{(0)}}$ and choose a symmetric positive definite matrix ${{\text{H}}..

Today, I plan to solve an example using the DFP method as explained in the last post(120_Inverse Hessian Updating). Set the initial starting point ${x^{(0)}}$ to (1,2), ${{\text{A}}^{(0)}}$ to the identity matrix, $k=0$ for the first iteration, and the stopping criterion $\varepsilon $ to 0.001. The gradient vector at the starting point can be calculated as follows. Calculate the norm and criter..

The Newton method converges very quickly, but it can be inefficient because it requires calculating $n(n+1)/2$ second-order derivatives to generate the Hessian matrix. Additionally, the Newton method can be challenging to solve if the function's Hessian is singular at any iteration. Similarly, in most engineering problems, calculating second-order derivatives can be very complex or impossible. T..

The Modified Newton Method is a numerical approach for solving optimization problems, designed to overcome the difficulty that the Newton Method may have in guaranteeing convergence under specific conditions. The Newton Method is used to find the extremum of a function and works effectively when the Hessian of the function is continuous and positive definite. In each iteration, it calculates the..

The Conjugate Gradient (CG) algorithm is a method used to solve systems of linear equations or to optimize multivariable functions. It's particularly useful when direct methods are computationally inefficient or for large, sparse systems. The algorithm iteratively improves an initial estimate of the solution until convergence criteria are met. First, set ${x^(0)}$ as the starting point and the i..

Today, solve one of the NAFEMS Benchmark tests for thermal analysis using COMSOL. (ref : A.D. Cameron, J.A. Casey, and G.B. Simpson, NAFEMS Benchmark Tests for Thermal Analyes. NAFEMS Documentation) The problem definition is as follow. The geometry has a height of $1m$ and width of $0.6m$, with the bottom boundary fixed at a temperature of 100 degrees Celsius. The left side is a perfect insulato..

Today, I'll simulate the interference between two slits composed of waves or sound using COMSOL. I'll generate plane waves using two thin wavelengths leading to the slits on the screen and calculate the pattern on the opposite side. For detailed information, please refer to the COMSOL Application Library. Theoretically, the amplitude becomes minimal where the difference in path length is an odd ..

I plan to write and solve the KKT optimality conditions for an LP problem. The LP problem defined in standard form is as follows. The above equations, when expressed in the form of a Lagrangian function, are as follows. Where ${\bf{v}}$ and ${\bf{\lambda }}$ are the Lagrange multipliers for the constraints. The KKT necessary conditions are as follows. The above equations can be solved for ${\bf{..

In optimization problems, Linear Programming (LP) involves finding the values of variables that maximize or minimize a linear function, subject to linear equations or inequalities as constraints. In linear programming problems, the cost function is the objective function that we aim to minimize or maximize. This function is a linear function $f(x)$ with $k$ variables $x$, composed of first-order..

Today, I plan to briefly explain how to apply the dual method through an example, as dexcribed in the previous post(110_More on Duality). The definition of the problem is as follows. Let's find the dual ans solve it. To solve the above primal problem using optimality conditions, First need to fine the Lagrangian. The Lagrangian for the given problem is as follows. Where $v$ is the Lagrange multi..