Finite

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..

Since its first time conducting CFD through COMSOL, I plan to follow the "Flow past a Cylinder" example found in the COMSOL application manual. Fluid flow becomes unstable behind large bodies, making it challenging to compute. The wake formed behind the body generates significant resistance due to all sizes of disordered vorices. However, turbulence in the thin boundary layer beside streamlined ..

Today, I plan to implement the same MNIST classification problem that was previously solved using a DNN in PyTorch(100_DNN Example using PyTorch), but this time using TensorFlow's Keras. After that post, I've made some modifications to the final code(103_Code Modification) to adapt it for Keras. First, Import the necessary libraries. import tensorflow as tf from tensorflow import keras from tens..

In a previous post(109_Duality), we touched on duality, but there were many areas where the explanation was lacking, and establishing a duality relationship requires additional formulations. Therefore, in this post, I intend to explain the necessary additional formulations. ${\bf{x}}({\bf{v}})$ denote the local minimum of the Lagrangian function, as follows. The dual function can be written as f..

When a nonlinear programming problem is given, there exists another similar nonlinear programming problem. The former is referred to as the primal problem, and the latter as the dual problem. Under certain convexity assumptions, since the primal and dual problems have the same optimal objective function values, solving the dual problem can indirectly solve the primal problem. To consider the loc..

In this post, I'll study the second-order necessary and sufficient conditions for optimization probles with constraints. Similar to the unconstrained case, the second part of the Taylor expansion of function at ${x^*}$ is used to determine whether that point is a local minimum. In the presence of constraints, the active constriants at ${x^*}$ must also be considered to determine a feasible chang..

In optimization problems, getting a satisfying answer to "How can the optimal solution be found?" is not straightforward. However, if the optimization problem can be proven to be convex, any local minimum also becomes a global minimum. Furthermore, the KKT necessary conditions are not only necessary but also sufficient for minimum point. In this post, I will study convex functions and their prop..

In this post, I intend to explain the "Karush-Kuhn-Tucker (KKT) Optimality Conditions," widely used in optimization theory. The KKT conditions can be applied to optimization problems with both equality and inequality constraints. They can be considered a more generalized form of the Lagrangian function described in the previous post(105_Lagrange Multiplier Theorem). In this problem, there are bo..

The Lagrange Multiplier Theorem is a technique used to solve the provlem of finding the extrema of multivariable functions under specific constraints. This method is very effective for solving optimization problems with constrains. In this post, I will explain using an example of an optimization problem with equality constraints. Assuming that $x$ is a regular point where it is a local minimum, ..

The concept of Taylor's expansion is essential in optimal design and numerical methods. Taylor's expansion is a mathematical formula that approximates the value of a function near a specific point using the derivatives of the function at that point. In other words, it can be considered a generalization of linear approximation in the form of a polynomial function. It is very useful when attemptin..