optimization_formulas.tex

% ================================================================================
% GENERAL OPTIMIZATION FORMULAS
% Mathematical formulation for gradient-based shape optimization
% ================================================================================

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}

\begin{document}

% ================================================================================
\section{Optimization Problem Formulation}
% ================================================================================

\subsection{General Objective Function}
\begin{equation}
\min_{\mathbf{x}} \quad f(\mathbf{x})
\end{equation}

where $\mathbf{x} \in \mathbb{R}^n$ is the design variable vector.

\subsection{Bound Constraints}
\begin{equation}
\mathbf{x}^L \leq \mathbf{x} \leq \mathbf{x}^U
\end{equation}

where $\mathbf{x}^L$ and $\mathbf{x}^U$ are the lower and upper bounds, respectively.

\subsection{General Constrained Optimization}
\begin{align}
\min_{\mathbf{x}} \quad &f(\mathbf{x}) \\
\text{subject to} \quad &g_i(\mathbf{x}) \leq 0, \quad i = 1, \ldots, m \\
&h_j(\mathbf{x}) = 0, \quad j = 1, \ldots, p \\
&\mathbf{x}^L \leq \mathbf{x} \leq \mathbf{x}^U
\end{align}

% ================================================================================
\section{FFD Parameterization}
% ================================================================================

\subsection{Control Point Grid}
\begin{equation}
\mathbf{P}_{i,j,k} = \mathbf{P}_{i,j,k}^0 + \delta \mathbf{P}_{i,j,k}
\end{equation}

where $\mathbf{P}_{i,j,k}^0$ are the initial control point positions and $\delta \mathbf{P}_{i,j,k}$ are the design-dependent displacements.

\subsection{Design Variable Mapping}
\begin{equation}
\delta \mathbf{P}_{i,j,k} = \mathbf{D}_{i,j,k} \mathbf{x}
\end{equation}

where $\mathbf{D}_{i,j,k}$ is the design variable mapping matrix.

\subsection{Trilinear Interpolation}
\begin{equation}
\delta \mathbf{r}(\boldsymbol{\xi}) = \sum_{i=0}^{n_i-1} \sum_{j=0}^{n_j-1} \sum_{k=0}^{n_k-1} B_i^{p_i}(\xi) B_j^{p_j}(\eta) B_k^{p_k}(\zeta) \delta \mathbf{P}_{i,j,k}
\end{equation}

where $\boldsymbol{\xi} = (\xi, \eta, \zeta)$ are the parametric coordinates and $B_i^p$ are Bernstein polynomials of degree $p$:
\begin{equation}
B_i^p(t) = \binom{p}{i} t^i (1-t)^{p-i}
\end{equation}

\subsection{Mesh Point Deformation}
\begin{equation}
\mathbf{r}^{new} = \mathbf{r}^{old} + \delta \mathbf{r}(\boldsymbol{\xi}(\mathbf{r}^{old}))
\end{equation}

% ================================================================================
\section{Gradient Computation Methods}
% ================================================================================

\subsection{Forward Finite Differences}
\begin{equation}
\frac{\partial f}{\partial x_i} \approx \frac{f(\mathbf{x} + h \mathbf{e}_i) - f(\mathbf{x})}{h}
\end{equation}

\subsection{Backward Finite Differences}
\begin{equation}
\frac{\partial f}{\partial x_i} \approx \frac{f(\mathbf{x}) - f(\mathbf{x} - h \mathbf{e}_i)}{h}
\end{equation}

\subsection{Central Finite Differences}
\begin{equation}
\frac{\partial f}{\partial x_i} \approx \frac{f(\mathbf{x} + h \mathbf{e}_i) - f(\mathbf{x} - h \mathbf{e}_i)}{2h}
\end{equation}

\subsection{Gradient Vector}
\begin{equation}
\nabla f(\mathbf{x}) = \begin{bmatrix}
\frac{\partial f}{\partial x_1} \\
\frac{\partial f}{\partial x_2} \\
\vdots \\
\frac{\partial f}{\partial x_n}
\end{bmatrix}
\end{equation}

\subsection{Computational Cost}
For forward finite differences:
\begin{equation}
N_{eval} = 1 + n
\end{equation}

For central finite differences:
\begin{equation}
N_{eval} = 2n
\end{equation}

where $n$ is the number of design variables.

% ================================================================================
\section{Sequential Quadratic Programming (SQP)}
% ================================================================================

\subsection{Quadratic Subproblem}
At each iteration $k$, solve:
\begin{align}
\min_{\mathbf{p}} \quad &\nabla f(\mathbf{x}^k)^T \mathbf{p} + \frac{1}{2} \mathbf{p}^T \mathbf{B}^k \mathbf{p} \\
\text{subject to} \quad &\nabla g_i(\mathbf{x}^k)^T \mathbf{p} + g_i(\mathbf{x}^k) \leq 0, \quad i = 1, \ldots, m \\
&\nabla h_j(\mathbf{x}^k)^T \mathbf{p} + h_j(\mathbf{x}^k) = 0, \quad j = 1, \ldots, p
\end{align}

where $\mathbf{p}$ is the search direction and $\mathbf{B}^k$ is the Hessian approximation.

\subsection{Lagrangian Function}
\begin{equation}
\mathcal{L}(\mathbf{x}, \boldsymbol{\lambda}, \boldsymbol{\mu}) = f(\mathbf{x}) + \sum_{i=1}^{m} \lambda_i g_i(\mathbf{x}) + \sum_{j=1}^{p} \mu_j h_j(\mathbf{x})
\end{equation}

\subsection{KKT Conditions}
\begin{align}
\nabla_{\mathbf{x}} \mathcal{L} &= \nabla f(\mathbf{x}) + \sum_{i=1}^{m} \lambda_i \nabla g_i(\mathbf{x}) + \sum_{j=1}^{p} \mu_j \nabla h_j(\mathbf{x}) = \mathbf{0} \\
g_i(\mathbf{x}) &\leq 0, \quad i = 1, \ldots, m \\
h_j(\mathbf{x}) &= 0, \quad j = 1, \ldots, p \\
\lambda_i &\geq 0, \quad i = 1, \ldots, m \\
\lambda_i g_i(\mathbf{x}) &= 0, \quad i = 1, \ldots, m
\end{align}

% ================================================================================
\section{Hessian Approximation Methods}
% ================================================================================

\subsection{BFGS Update}
\begin{equation}
\mathbf{B}^{k+1} = \mathbf{B}^k + \frac{\mathbf{y}^k (\mathbf{y}^k)^T}{(\mathbf{y}^k)^T \mathbf{s}^k} - \frac{\mathbf{B}^k \mathbf{s}^k (\mathbf{s}^k)^T \mathbf{B}^k}{(\mathbf{s}^k)^T \mathbf{B}^k \mathbf{s}^k}
\end{equation}

where:
\begin{align}
\mathbf{s}^k &= \mathbf{x}^{k+1} - \mathbf{x}^k \\
\mathbf{y}^k &= \nabla f(\mathbf{x}^{k+1}) - \nabla f(\mathbf{x}^k)
\end{align}

\subsection{DFP Update}
\begin{equation}
\mathbf{B}^{k+1} = \mathbf{B}^k - \frac{\mathbf{B}^k \mathbf{s}^k (\mathbf{s}^k)^T \mathbf{B}^k}{(\mathbf{s}^k)^T \mathbf{B}^k \mathbf{s}^k} + \frac{\mathbf{y}^k (\mathbf{y}^k)^T}{(\mathbf{y}^k)^T \mathbf{s}^k}
\end{equation}

\subsection{SR1 Update}
\begin{equation}
\mathbf{B}^{k+1} = \mathbf{B}^k + \frac{(\mathbf{y}^k - \mathbf{B}^k \mathbf{s}^k)(\mathbf{y}^k - \mathbf{B}^k \mathbf{s}^k)^T}{(\mathbf{y}^k - \mathbf{B}^k \mathbf{s}^k)^T \mathbf{s}^k}
\end{equation}

% ================================================================================
\section{Line Search Methods}
% ================================================================================

\subsection{Armijo Condition}
\begin{equation}
f(\mathbf{x}^k + \alpha \mathbf{p}^k) \leq f(\mathbf{x}^k) + c_1 \alpha (\nabla f(\mathbf{x}^k))^T \mathbf{p}^k
\end{equation}

where $0 < c_1 < 1$ is the Armijo parameter.

\subsection{Wolfe Conditions}
Strong Wolfe conditions:
\begin{align}
f(\mathbf{x}^k + \alpha \mathbf{p}^k) &\leq f(\mathbf{x}^k) + c_1 \alpha (\nabla f(\mathbf{x}^k))^T \mathbf{p}^k \\
|(\nabla f(\mathbf{x}^k + \alpha \mathbf{p}^k))^T \mathbf{p}^k| &\leq c_2 |(\nabla f(\mathbf{x}^k))^T \mathbf{p}^k|
\end{align}

where $0 < c_1 < c_2 < 1$.

\subsection{Backtracking Line Search}
\begin{equation}
\alpha^{k+1} = \rho \alpha^k
\end{equation}

where $0 < \rho < 1$ is the reduction factor.

\subsection{Design Variable Update}
\begin{equation}
\mathbf{x}^{k+1} = \mathbf{x}^k + \alpha^k \mathbf{p}^k
\end{equation}

% ================================================================================
\section{Convergence Criteria}
% ================================================================================

\subsection{Gradient Norm Tolerance}
\begin{equation}
\|\nabla f(\mathbf{x}^k)\| < \epsilon_g
\end{equation}

\subsection{Step Size Tolerance}
\begin{equation}
\|\mathbf{x}^{k+1} - \mathbf{x}^k\| < \epsilon_x
\end{equation}

\subsection{Objective Function Tolerance}
\begin{equation}
\frac{|f(\mathbf{x}^{k+1}) - f(\mathbf{x}^k)|}{|f(\mathbf{x}^k)| + \epsilon_{abs}} < \epsilon_f
\end{equation}

where $\epsilon_{abs}$ prevents division by zero.

\subsection{Maximum Iterations}
\begin{equation}
k \leq k_{max}
\end{equation}

\subsection{Constraint Violation Tolerance}
\begin{align}
\max_{i=1,\ldots,m} \{g_i(\mathbf{x}^k)\} &< \epsilon_g \\
\max_{j=1,\ldots,p} \{|h_j(\mathbf{x}^k)|\} &< \epsilon_h
\end{align}

% ================================================================================
\section{Shape Sensitivity Analysis}
% ================================================================================

\subsection{General Chain Rule}
\begin{equation}
\frac{\partial f}{\partial x_i} = \frac{\partial f}{\partial \mathbf{r}} \cdot \frac{\partial \mathbf{r}}{\partial x_i}
\end{equation}

where $\mathbf{r}$ represents the mesh coordinates.

\subsection{FFD Shape Sensitivity}
\begin{equation}
\frac{\partial \mathbf{r}}{\partial x_i} = \frac{\partial}{\partial x_i}\left[\mathbf{r}^0 + \delta \mathbf{r}(\boldsymbol{\xi}(\mathbf{r}^0))\right]
\end{equation}

\subsection{Parametric Sensitivity}
\begin{equation}
\frac{\partial \delta \mathbf{r}}{\partial x_i} = \sum_{j=0}^{n_j-1} \sum_{k=0}^{n_k-1} \sum_{l=0}^{n_l-1} B_j(\xi) B_k(\eta) B_l(\zeta) \frac{\partial \delta \mathbf{P}_{j,k,l}}{\partial x_i}
\end{equation}

% ================================================================================
\section{Trust Region Methods}
% ================================================================================

\subsection{Trust Region Subproblem}
\begin{align}
\min_{\mathbf{p}} \quad &\nabla f(\mathbf{x}^k)^T \mathbf{p} + \frac{1}{2} \mathbf{p}^T \mathbf{B}^k \mathbf{p} \\
\text{subject to} \quad &\|\mathbf{p}\| \leq \Delta^k
\end{align}

where $\Delta^k$ is the trust region radius.

\subsection{Trust Region Update}
\begin{equation}
\rho^k = \frac{f(\mathbf{x}^k) - f(\mathbf{x}^k + \mathbf{p}^k)}{m^k(\mathbf{0}) - m^k(\mathbf{p}^k)}
\end{equation}

where $m^k(\mathbf{p})$ is the quadratic model.

Trust region radius update:
\begin{equation}
\Delta^{k+1} = \begin{cases}
\gamma_1 \Delta^k & \text{if } \rho^k < \eta_1 \\
\Delta^k & \text{if } \eta_1 \leq \rho^k < \eta_2 \\
\min(\gamma_2 \Delta^k, \Delta_{max}) & \text{if } \rho^k \geq \eta_2
\end{cases}
\end{equation}

% ================================================================================
\section{Multi-Objective Optimization}
% ================================================================================

\subsection{Weighted Sum Method}
\begin{equation}
\min_{\mathbf{x}} \quad \sum_{i=1}^{n_obj} w_i f_i(\mathbf{x})
\end{equation}

where $w_i \geq 0$ and $\sum_{i=1}^{n_obj} w_i = 1$.

\subsection{Pareto Optimality}
A point $\mathbf{x}^*$ is Pareto optimal if there exists no $\mathbf{x}$ such that:
\begin{align}
f_i(\mathbf{x}) &\leq f_i(\mathbf{x}^*) \quad \forall i = 1, \ldots, n_obj \\
f_j(\mathbf{x}) &< f_j(\mathbf{x}^*) \quad \text{for at least one } j
\end{align}

\subsection{$\epsilon$-Constraint Method}
\begin{align}
\min_{\mathbf{x}} \quad &f_1(\mathbf{x}) \\
\text{subject to} \quad &f_i(\mathbf{x}) \leq \epsilon_i, \quad i = 2, \ldots, n_obj
\end{align}

% ================================================================================
\section{Uncertainty Quantification}
% ================================================================================

\subsection{Robust Optimization}
\begin{equation}
\min_{\mathbf{x}} \quad \mathbb{E}[f(\mathbf{x}, \boldsymbol{\xi})] + \beta \sqrt{\text{Var}[f(\mathbf{x}, \boldsymbol{\xi})]}
\end{equation}

where $\boldsymbol{\xi}$ represents uncertain parameters and $\beta$ is the risk aversion parameter.

\subsection{Chance Constraints}
\begin{equation}
\mathbb{P}[g_i(\mathbf{x}, \boldsymbol{\xi}) \leq 0] \geq 1 - \alpha_i
\end{equation}

where $\alpha_i$ is the allowable violation probability.

\end{document}